Inserting log requests into the application code requires a fair amount of planning and effort. Observation shows that approximately 4 percent of code is dedicated to logging. Consequently, even moderately sized applications will have thousands of logging statements embedded within their code. Given their number, it becomes imperative to manage these log statements without the need to modify them manually.
The log4net environment is fully configurable programmatically. However, it is far more flexible to configure log4net using configuration files. Currently, configuration files are written in XML.
Let us give a taste of how this is done with the help of an imaginary application MyApp that uses log4net.
using Com.Foo; // Import log4net classes. using log4net; using log4net.Config; public class MyApp { // Define a static logger variable so that it references the // Logger instance named "MyApp". private static readonly ILog log = LogManager.GetLogger(typeof(MyApp)); static void Main(string[] args) { // Set up a simple configuration that logs on the console. BasicConfigurator.Configure(); log.Info("Entering application."); Bar bar = new Bar(); bar.DoIt(); log.Info("Exiting application."); } }
MyApp begins by importing log4net related classes. It then defines a static logger variable with the name MyApp which happens to be the fully qualified name of the class.
MyApp uses the following Bar class:
// Import log4net classes. using log4net; namespace Com.Foo { public class Bar { private static readonly ILog log = LogManager.GetLogger(typeof(Bar)); public void DoIt() { log.Debug("Did it again!"); } } }
The invocation of the BasicConfigurator.Configure() method creates a rather simple log4net setup. This method is hardwired to add to the root logger a ConsoleAppender. The output will be formatted using a PatternLayout set to the pattern "%-4timestamp [%thread] %-5level %logger %ndc - %message%newline".
Note that by default, the root logger is assigned to Level.DEBUG.
The output of MyApp is:
0 [main] INFO MyApp - Entering application. 36 [main] DEBUG Com.Foo.Bar - Did it again! 51 [main] INFO MyApp - Exiting application.
As a side note, let me mention that in log4net child loggers link only to their existing ancestors. In particular, the logger named Com.Foo.Bar is linked directly to the root logger, thereby circumventing the unused Com or Com.Foo loggers. This significantly increases performance and reduces log4net's memory footprint.
The MyApp class configures log4net by invoking BasicConfigurator.Configure() method. Other classes only need to import the log4net namespace, retrieve the loggers they wish to use, and log away.
The previous example always outputs the same log information. Fortunately, it is easy to modify MyApp so that the log output can be controlled at run-time. Here is a slightly modified version.
using Com.Foo; // Import log4net classes. using log4net; using log4net.Config; public class MyApp { private static readonly ILog log = LogManager.GetLogger(typeof(MyApp)); static void Main(string[] args) { // BasicConfigurator replaced with XmlConfigurator. XmlConfigurator.Configure(new System.IO.FileInfo(args[0])); log.Info("Entering application."); Bar bar = new Bar(); bar.DoIt(); log.Info("Exiting application."); } }
This version of MyApp instructs the XmlConfigurator to parse a configuration file and set up logging accordingly. The path to the configuration file is specified on the command line.
Here is a sample configuration file that results in exactly same output as the previous BasicConfigurator based example.
<log4net>  <appender name="A1" type="log4net.Appender.ConsoleAppender">  <layout type="log4net.Layout.PatternLayout"> <conversionPattern value="%-4timestamp [%thread] %-5level %logger %ndc - %message%newline" /> </layout> </appender>  <root> <level value="DEBUG" /> <appender-ref ref="A1" /> </root> </log4net>
Suppose we are no longer interested in seeing the output of any component belonging to the Com.Foo package. The following configuration file shows one possible way of achieving this.
<log4net>  <appender name="A1" type="log4net.Appender.ConsoleAppender">  <layout type="log4net.Layout.PatternLayout">  <conversionPattern value="%date [%thread] %-5level %logger %ndc - %message%newline" /> </layout> </appender>  <root> <level value="DEBUG" /> <appender-ref ref="A1" /> </root>  <logger name="Com.Foo"> <level value="WARN" /> </logger> </log4net>
The output of MyApp configured with this file is shown below.
2000-09-07 14:07:41,508 [main] INFO MyApp - Entering application. 2000-09-07 14:07:41,529 [main] INFO MyApp - Exiting application.
As the logger Com.Foo.Bar does not have an assigned level, it inherits its level from Com.Foo, which was set to WARN in the configuration file. The log statement from the Bar.DoIt method has the level DEBUG, lower than the logger level WARN. Consequently, DoIt() method's log request is suppressed.
Here is another configuration file that uses multiple appenders.
<log4net> <appender name="Console" type="log4net.Appender.ConsoleAppender"> <layout type="log4net.Layout.PatternLayout">  <conversionPattern value="%5level [%thread] (%file:%line) - %message%newline" /> </layout> </appender> <appender name="RollingFile" type="log4net.Appender.RollingFileAppender"> <file value="example.log" /> <appendToFile value="true" /> <maximumFileSize value="100KB" /> <maxSizeRollBackups value="2" /> <layout type="log4net.Layout.PatternLayout"> <conversionPattern value="%level %thread %logger - %message%newline" /> </layout> </appender> <root> <level value="DEBUG" /> <appender-ref ref="Console" /> <appender-ref ref="RollingFile" /> </root> </log4net>
Calling the enhanced MyApp with the this configuration file will output the following on the console.
INFO [main] (MyApp.cs:16) - Entering application. DEBUG [main] (Bar.cs:12) - Doing it again! INFO [main] (MyApp.cs:19) - Exiting application.
In addition, as the root logger has been allocated a second appender, output will also be directed to the example.log file. This file will be rolled over when it reaches 100KB. When roll-over occurs, the old version of example.log is automatically moved to example.log.1.
Note that to obtain these different logging behaviors we did not need to recompile code. We could just as easily have logged to an email address, redirected all Com.Foo output to an NT Event logger, or forwarded logging events to a remote log4net server, which would log according to local server policy.
For more examples of configuring appenders using the XmlConfigurator see the Example Appender Configuration document.
The log4net configuration can be configured using assembly-level attributes rather than specified programmatically.
The log4net.Config.XmlConfiguratorAttribute Allows the XmlConfigurator to be configured using the following properties:
If specified, this is the filename of the configuration file to use with the XmlConfigurator. This file path is relative to the application base directory (AppDomain.CurrentDomain.BaseDirectory).
This property cannot be used in conjunction with the ConfigFileExtension property.
If specified, this is the extension for the configuration file. The assembly file name is used as the base name with the this extension appended. For example if the assembly is loaded from the a file TestApp.exe and the ConfigFileExtension property is set to log4net then the configuration file name is TestApp.exe.log4net. This is equivalent to setting the ConfigFile property to TestApp.exe.log4net.
The path to the configuration file is build by using the application base directory (AppDomain.CurrentDomain.BaseDirectory), the assembly file name and the configuration file extension.
This property cannot be used in conjunction with the ConfigFile property.
If this flag is specified and set to true then the framework will watch the configuration file and will reload the config each time the file is modified.
If neither of the ConfigFile or ConfigFileExtension properties are specified, the application configuration file (e.g. TestApp.exe.config) will be used as the log4net configuration file.
Example usage:
// Configure log4net using the .config file [assembly: log4net.Config.XmlConfigurator(Watch=true)] // This will cause log4net to look for a configuration file // called TestApp.exe.config in the application base // directory (i.e. the directory containing TestApp.exe) // The config file will be watched for changes.
// Configure log4net using the .log4net file [assembly: log4net.Config.XmlConfigurator(ConfigFileExtension="log4net",Watch=true)] // This will cause log4net to look for a configuration file // called TestApp.exe.log4net in the application base // directory (i.e. the directory containing TestApp.exe) // The config file will be watched for changes.
This attribute may only be used once per assembly.
Using attributes can be a clearer method for defining where the application's configuration will be loaded from. However it is worth noting that attributes are purely passive. They are information only. Therefore if you use configuration attributes you must invoke log4net to allow it to read the attributes. A simple call to LogManager.GetLogger will cause the attributes on the calling assembly to be read and processed. Therefore it is imperative to make a logging call as early as possible during the application start-up, and certainly before any external assemblies have been loaded and invoked.
Typically the log4net configuration is specified using a file. This file can be read in one of two ways:
The System.Configuration API is only available if the configuration data is in the application's config file; the file named MyApp.exe.config or Web.config. Because the System.Configuration API does not support reloading of the config file the configuration settings cannot be watched using the log4net.Config.XmlConfigurator.ConfigureAndWatch methods. The main advantage of using the System.Configuration APIs to read the configuration data is that it requires less permissions than accessing the configuration file directly.
The only way to configure an application using the System.Configuration APIs is to call the log4net.Config.XmlConfigurator.Configure() method or the log4net.Config.XmlConfigurator.Configure(ILoggerRepository) method.
In order to embed the configuration data in the .config file the section name must be identified to the .NET config file parser using a configSections element. The section must specify the log4net.Config.Log4NetConfigurationSectionHandler that will be used to parse the config section. This type must be fully assembly qualified because it is being loaded by the .NET config file parser not by log4net. The correct assembly name for the log4net assembly must be specified. The following is a simple example configuration file that specifies the correct section handler to use for the log4net section.
<?xml version="1.0" encoding="utf-8" ?> <configuration> <configSections>
 </configSections> <log4net> <appender name="ConsoleAppender" type="log4net.Appender.ConsoleAppender" > <layout type="log4net.Layout.PatternLayout"> <conversionPattern value="%date [%thread] %-5level %logger [%ndc] - %message%newline" /> </layout> </appender> <root> <level value="INFO" /> <appender-ref ref="ConsoleAppender" /> </root> </log4net> </configuration>
In the above example the log4net assembly is specified. This assembly must be located where the .NET runtime can find it. For example it could be located in the same directory as the application. If the log4net assembly is stored in the GAC then the fully qualified assembly name must be specified including the culture, version and public key.
When using the .config file to specify the configuration the section name and XML element name must be log4net.
The XmlConfigurator can directly read any XML file and use it to configure log4net. This includes the application's .config file; the file named MyApp.exe.config or Web.config. The only reason not to read the configuration file directly is if the application does not have sufficient permissions to read the file, then the configuration must be loaded using the .NET configuration APIs (see above).
The file to read the configuration from can be specified using any of the log4net.Config.XmlConfigurator methods that accept a System.IO.FileInfo object. Because the file system can be monitored for file change notifications the ConfigureAndWatch methods can be used to monitor the configuration file for modifications and automatically reconfigure log4net.
Additionally the log4net.Config.XmlConfiguratorAttribute can be used to specify the file to read the configuration from.
The configuration is read from the log4net element in the file. Only one log4net element can be specified in the file but it may be located anywhere in the XML hierarchy. For example it may be the root element:
<?xml version="1.0" encoding="utf-8" ?> <log4net> <appender name="ConsoleAppender" type="log4net.Appender.ConsoleAppender" > <layout type="log4net.Layout.PatternLayout"> <conversionPattern value="%date [%thread] %-5level %logger [%ndc] - %message%newline" /> </layout> </appender> <root> <level value="INFO" /> <appender-ref ref="ConsoleAppender" /> </root> </log4net>
Or it may be nested within other elements:
<?xml version="1.0" encoding="utf-8" ?> <configuration> <configSections>
 </configSections> <log4net> <appender name="ConsoleAppender" type="log4net.Appender.ConsoleAppender" > <layout type="log4net.Layout.PatternLayout"> <conversionPattern value="%date [%thread] %-5level %logger [%ndc] - %message%newline" /> </layout> </appender> <root> <level value="INFO" /> <appender-ref ref="ConsoleAppender" /> </root> </log4net> </configuration>
The above example shows how the configuration data can be embedded inside a .config file even though the file is being read directly by log4net. An important note is that the .NET config file parser will throw an exception if it finds an element that has not been registered using the configSections element. Therefore in the above example the log4net section name is registered, but the type specified to handle the section is System.Configuration.IgnoreSectionHandler. This is a built-in class that indicates that another method for reading the config section will be employed.
log4net includes a configuration reader that parses an XML DOM, the log4net.Config.XmlConfigurator. This section defines the syntax accepted by the configurator.
This is an example of a valid XML configuration. The root element must be <log4net>. Note that this does not mean that this element cannot be embedded in another XML document. See the section above on Configuration Files for more information on how to embed the XmlConfigurator XML in a configuration file.
<log4net> <appender name="ConsoleAppender" type="log4net.Appender.ConsoleAppender" > <layout type="log4net.Layout.PatternLayout"> <conversionPattern value="%date [%thread] %-5level %logger [%ndc] - %message%newline" /> </layout> </appender> <root> <level value="INFO" /> <appender-ref ref="ConsoleAppender" /> </root> </log4net>
The <log4net> element supports the following attributes:
| Attribute | Description |
|---|---|
| debug | Optional attribute. Value must be either true or false. The default value is false. Set this attribute to true to enable internal log4net debugging for this configuration. |
| update | Optional attribute. Value must be either Merge or Overwrite. The default value is Merge. Set this attribute to Overwrite to reset the configuration of the repository being configured before applying this configuration. |
| threshold | Optional attribute. Value must be the name of a level registered on the repository. The default value is ALL. Set this attribute to limit the messages that are logged across the whole repository, regardless of the logger that the message is logged to. |
The <log4net> element supports the following child elements:
| Element | Description |
|---|---|
| appender | Zero or more elements allowed. Defines an appender. |
| logger | Zero or more elements allowed. Defines the configuration of a logger. |
| renderer | Zero or more elements allowed. Defines an object renderer. |
| root | Optional element, maximum of one allowed. Defines the configuration of the root logger. |
| param | Zero or more elements allowed. Repository specific parameters |
Appenders may only be defined as child elements of the <log4net> element. Each appender must be uniquely named. The implementing type for the appender must be specified.
This example shows an appender of type log4net.Appender.ConsoleAppender being defined. The appender will be known as ConsoleAppender.
<appender name="ConsoleAppender" type="log4net.Appender.ConsoleAppender" > <layout type="log4net.Layout.PatternLayout"> <conversionPattern value="%date [%thread] %-5level %logger [%ndc] - %message%newline" /> </layout> </appender>
The <appender> element supports the following attributes:
| Attribute | Description |
|---|---|
| name | Required attribute. Value must be a string name for this appender. The name must be unique among all the appenders defined in this configuration file. This name is used by the <appender-ref> element of a Logger to reference an appender. |
| type | Required attribute. Value must be the type name for this appender. If the appender is not defined in the log4net assembly this type name must be fully assembly qualified. |
The <appender> element supports the following child elements:
| Element | Description |
|---|---|
| appender-ref | Zero or more elements allowed. Allows the appender to reference other appenders. Not supported by all appenders. |
| filter | Zero or more elements allowed. Defines the filters used by this appender. |
| layout | Optional element, maximum of one allowed. Defines the layout used by this appender. |
| param | Zero or more elements allowed. Appender specific parameters. |
For examples of configuring appenders see the Example Appender Configuration document.
Filters elements may only be defined as children of <appender> elements.
The <filter> element supports the following attributes:
| Attribute | Description |
|---|---|
| type | Required attribute. Value must be the type name for this filter. If the filter is not defined in the log4net assembly this type name must be fully assembly qualified. |
The <filter> element supports the following child elements:
| Element | Description |
|---|---|
| param | Zero or more elements allowed. Filter specific parameters. |
Filters form a chain that the event has to pass through. Any filter along the way can accept the event and stop processing, deny the event and stop processing, or allow the event on to the next filter. If the event gets to the end of the filter chain without being denied it is implicitly accepted and will be logged.
<filter type="log4net.Filter.LevelRangeFilter"> <levelMin value="INFO" /> <levelMax value="FATAL" /> </filter>
This filter will deny events that have a level that is lower than INFO or higher than FATAL. All events between INFO and FATAL will be logged.
If we want to only allow messages through that have a specific substring (e.g. 'database') then we need to specify the following filters:
<filter type="log4net.Filter.StringMatchFilter"> <stringToMatch value="database" /> </filter> <filter type="log4net.Filter.DenyAllFilter" />
The first filter will look for the substring 'database' in the message text of the event. If the text is found the filter will accept the message and filter processing will stop, the message will be logged. If the substring is not found the event will be passed to the next filter to process. If there is no next filter the event would be implicitly accepted and would be logged. But because we don't want the non matching events to be logged we need to use a log4net.Filter.DenyAllFilter that will just deny all events that reach it. This filter is only useful at the end of the filter chain.
If we want to allow events that have either 'database' or 'ldap' in the message text we can use the following filters:
<filter type="log4net.Filter.StringMatchFilter"> <stringToMatch value="database"/> </filter> <filter type="log4net.Filter.StringMatchFilter"> <stringToMatch value="ldap"/> </filter> <filter type="log4net.Filter.DenyAllFilter" />
Layout elements may only be defined as children of <appender> elements.
The <layout> element supports the following attributes:
| Attribute | Description |
|---|---|
| type | Required attribute. Value must be the type name for this layout. If the layout is not defined in the log4net assembly this type name must be fully assembly qualified. |
The <layout> element supports the following child elements:
| Element | Description |
|---|---|
| param | Zero or more elements allowed. Layout specific parameters. |
This example shows how to configure a layout that uses the log4net.Layout.PatternLayout.
<layout type="log4net.Layout.PatternLayout"> <conversionPattern value="%date [%thread] %-5level %logger [%ndc] - %message%newline" /> </layout>
Only one root logger element may only be defined and it must be a child of <log4net> element. The root logger is the root of the logger hierarchy. All loggers ultimately inherit from this logger.
An example root logger:
<root> <level value="INFO" /> <appender-ref ref="ConsoleAppender" /> </root>
The <root> element supports no attributes.
The <root> element supports the following child elements:
| Element | Description |
|---|---|
| appender-ref | Zero or more elements allowed. Allows the logger to reference appenders by name. |
| level | Optional element, maximum of one allowed. Defines the logging level for this logger. This logger will only accept event that are at this level or above. |
| param | Zero or more elements allowed. Logger specific parameters. |
Logger elements may only be defined as children of the <log4net> element.
An example logger:
<logger name="LoggerName"> <level value="DEBUG" /> <appender-ref ref="ConsoleAppender" /> </logger>
The <logger> element supports the following attributes.
| Attribute | Description |
|---|---|
| name | Required attribute. Value must be the name of the logger. |
| additivity | Optional attribute. Value may be either true or false. The default value is true. Set this attribute to false to prevent this logger from inheriting the appenders defined on parent loggers. |
The <logger> element supports the following child elements:
| Element | Description |
|---|---|
| appender-ref | Zero or more elements allowed. Allows the logger to reference appenders by name. |
| level | Optional element, maximum of one allowed. Defines the logging level for this logger. This logger will only accept event that are at this level or above. |
| param | Zero or more elements allowed. Logger specific parameters. |
Renderer elements may only be defined as children of the <log4net> element.
An example renderer:
<renderer renderingClass="MyClass.MyRenderer" renderedClass="MyClass.MyFunkyObject" />
The <renderer> element supports the following attributes.
| Attribute | Description |
|---|---|
| renderingClass | Required attribute. Value must be the type name for this renderer. If the type is not defined in the log4net assembly this type name must be fully assembly qualified. This is the type of the object that will take responsibility for rendering the renderedClass. |
| renderedClass | Required attribute. Value must be the type name for the target type for this renderer. If the type is not defined in the log4net assembly this type name must be fully assembly qualified. This is the name of the type that this renderer will render. |
The <renderer> element supports no child elements.
Parameter elements may be children of many elements. See the specific elements above for details.
An example param:
<param name="ConversionPattern" value="%date [%thread] %-5level %logger [%ndc] - %message%newline" />
The <param> element supports the following attributes.
| Attribute | Description |
|---|---|
| name | Required attribute. Value must be the name of the parameter to set on the parent object. |
| value | Optional attribute. One of value or type attributes must be specified. The value of this attribute is a string that can be converted to the value of the parameter. |
| type | Optional attribute. One of value or type attributes must be specified. The value of this attribute is a type name to create and set as the value of the parameter. If the type is not defined in the log4net assembly this type name must be fully assembly qualified. |
The <param> element supports the following child elements:
| Element | Description |
|---|---|
| param | Zero or more elements allowed. Parameter specific parameters. |
An example param that uses nested param elements:
<param name="evaluator" type="log4net.spi.LevelEvaluator"> <param name="Threshold" value="WARN"/> <param>
Configuration parameters map directly to writable properties on an object. The properties available depend on the actual type of the object being configured. The log4net SDK documentation contains the API reference for all the components included in the log4net assembly.
For 3rd party components please see their relevant API reference for details of the properties available.
All parameters may alternately be specified using the parameter name as the element name rather than using the param element and name attribute.
For example a param:
<param name="evaluator" type="log4net.spi.LevelEvaluator"> <param name="Threshold" value="WARN"/> <param>
may be written as:
<evaluator type="log4net.spi.LevelEvaluator"> <threshold value="WARN"/> <evaluator>


Superhero W
W is a superhero like no other, who struggles between the hunger for Justice and the need to use violence to achieve that very ideal of justice.
Through the episodes, you'll follow W, a shadow running in the darkness of lonely New York City corners, chasing criminals, helping lost souls. Some nights are memorable for W and they are filled with successes in fighting crime; some nights W holds on to the daytime identity of Guinn McGovern, trying to deal with a normal social existence and the struggles of modern dating. Some nights W has to wear both hats: the job of the superhero doesn’t allow any day off, even when some truths about Guinn, and about how some of the main characters of the series became who they are, could be waiting around a corner.
Reviewers say it is “a classic action story but somehow very different from anything else”, with a “writing style packed of action and visual clues as if it was taken right out of a Marvel comic strip”.
The first ten episodes of THE W SERIES have been collected in W IS FOR WONDER.
Discover HERE how W was born.
Sign up to be the first to receive breaking news about upcoming episodes, special promotions, exclusive content, and much more.


`
Here you will learn all there is to learn about skiing. Below there is a link to the basics of learning to ski.
How to SkiThere can be a lot of skiers on one run. You have to the flow of traffic and how to act saftley while skiing
You have to make sure your skis fit properly so you can control them
Different runs have different symbols which indicate their difficulty level.
Check to see if your jackets and ski pants are waterproof and breathable. Make sure you have layers.
Its chilly in the mountains so its important to stay warm. You will need a base layer, an insulating layer, and a shell.
You have to know what type of skiing your skis are best suited for.


17:19
CALLS
CHATS
22
GROUPS
CONTACT
Hooking up and staying hooked
The thunder down under
The round pegs in squre hole
The round pegs in squre hole


The Best Bob Haircuts of the 2010s, From Taylor Swift to Dua Lipa
by Sarah SiegelOver the last decade, hair trends have come and gone, but one has remained popular throughout the 2010s: the bob. While the styles have varied from sleek to curly, blunt to layered, and with or without bangs, this is the one look that is always in fashion.
Some of the most iconic bobs are still as memorable as the day we first saw them. From Taylor Swift's edgy textured bob at the iHeartRadio Music Awards in 2016 to Nicki Minaj's sleek black bob in 2011, many of these daring haircuts still inspire us today.
Ahead, check out the best bob haircuts of the last decade. But be warned, these photos might tempt you to get a haircut.
Hayley Williams's Orange Hair and Baby Bangs in 2010
We're totally obsessed with Hayley Williams's orange hair, baby bangs, and bob haircut from the Grammys in 2010.
Carrie Underwood's Layered Bob in 2010
We're tempted to use Carrie Underwood's layered bob haircut at the Grammys as inspiration for our next cut.
Nicki Minaj's 1920s-Style Bob in 2011
At the MTV Movie Awards in 2011, Nicki Minaj stunned with a chic black bob and thick bangs in a 1920s style.
Rihanna's Bob Haircut in 2011
It's hard to forget the vivid red hue of Rihanna's bob in 2011.
Katy Perry's Inverted Bob in 2012
Katy Perry's blue inverted bob from 2012 might inspire your next haircut.
Rose Byrne's Sleek Bob in 2012
Rose Byrne's sleek and shiny bob with bangs at the Oscars has been on our Pinterest inspiration board since 2012.
January Jones's Edgy Bob in 2013
January Jones's textured bob with a deep side part was an edgy change from her Mad Men character, Betty Draper.
Beyoncé's Bob Haircut in 2014
Beyoncé arrived to the Grammys in 2014 with an angular bob haircut with blunt ends.
Sarah Hyland's Bob Haircut in 2015
Sarah Hyland's caramel-colored and chin-length bob looked stunning at the Emmys in 2015.
Taylor Swift's Textured Bob and Bangs in 2016
At the iHeartRadio Music Awards in 2016, Taylor Swift convinced us that chopping off our hair was a great idea with her sexy textured bob and bangs.
Kylie Jenner's Asymmetrical Bob in 2016
Kylie Jenner's asymmetrical bob in 2016 made a big statement.
Zendaya's Bob Haircut in 2017
Zendaya looked glamorous at the MTV Movie & TV Awards in 2017 with her curled bob.
Kiernan Shipka's Bob Haircut in 2017
Kiernan Shipka showed off a newly cut blunt bob haircut at the MTV Movie & TV Awards in 2017.
Yara Shahidi's Bob Haircut in 2018
Yara Shahidi's blunt, curly bob looked so cool at the MTV Movie & TV Awards in 2018.
Amandla Stenberg's Bob Haircut in 2018
A post shared by Amandla Stenberg (@amandlasponsored) on Oct 20, 2018 at 6:08am PDT
We couldn't look away from Amandla Stenberg's multicolor braided bob in 2018.
Rowan Blanchard's Bob Haircut in 2018
We loved Rowan Blanchard's wavy brown bob haircut at the premiere of A Wrinkle in Time in 2018.
Jhené Aiko's Bob Haircut in 2018
Singer Jhené Aiko showed off her wavy bob while performing in 2018.
Kim Kardashian's Bob Haircut in 2019
Kim Kardashian stunned her fans when she went shorter than ever before in 2019.
Maisie Williams's Bob Haircut in 2019
Maisie Williams debuted a short brown bob and baby bangs at the Emmys in 2019, and we could barely contain our excitement.
Barbie Ferreira's Bob Haircut in 2019
A post shared by barbie ferreira (@barbieferreira) on Jul 31, 2019 at 10:50am PDT
We audibly gasped when we saw how short Barbie Ferreira had her hair cut in 2019. She added short bangs to the mix, too.


Alberta private members bill to reopen debate on physician conscience rights
by Dean Bennett, The Canadian PressEDMONTON - Alberta politicians are to debate the role of conscience rights and the responsibilities of physicians asked to assist or advise on abortions, contraception or medically assisted deaths.
United Conservative backbencher Dan Williams is to introduce Thursday a private member's bill to reassert the Charter-protected freedom of conscience and religion for health providers.
Williams says the bill is in response to an Appeal Court of Ontario ruling this spring.
Ontario's high court affirmed a lower court ruling that found physicians who object on moral grounds to contentious issues like abortion must offer patients an "effective referral" to another health provider.
The court called it a fair compromise, but groups such as Canadian Physicians for Life say it is still an intolerable violation of their rights.
Currently, the College of Physicians of Surgeons of Alberta directs doctors who do not wish to provide these services to offer patients timely access to either a member or a service that can.


Well, my real name is Babak (BOB-ack). You can call me that. Or Bob. And my last name is Hagh, pronounced exactly like the capital of South Holland, The Hague. I’m a full-stack developer based out of Raleigh, NC. I’ve always been fascinated with technology from a young age, mostly curious about how and why things work. I’m a huge game show fan, and when I couldn’t find online games of my favorite shows to play, I started to learn programming to make them exist and allow me to play them at any time (my first game was Card Sharks in Visual Basic.) I'm fascinated with all things media, technology, social, and the internet. My professional career started in wireless communications, and from then on, all my roles involved working with technology to make products better and efficient. My goal is to build things that can make a difference in people’s lives. With my passion to learn something new every day, I want to push myself in doing better than yesterday.


Black Beauty
by Anna Sewell
Part 1, Chapter 19: Only Ignorance
Additional Information
- Year Published: 1870
- Language: English
- Country of Origin: England
- Source: Sewell, A. (1870) Black Beauty New York: F.M. Lupton Publishing Co.
-
Readability:
- Flesch–Kincaid Level: 4.2
- Word Count: 793
- Genre: Memoir
- Keywords: first person narrative, horses
- ✎ Cite This
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I do not know how long I was ill. Mr. Bond, the horse-doctor, came every day. One day he bled me; John held a pail for the blood. I felt very faint after it and thought I should die, and I believe they all thought so too.
Ginger and Merrylegs had been moved into the other stable, so that I might be quiet, for the fever made me very quick of hearing; any little noise seemed quite loud, and I could tell every one's footstep going to and from the house. I knew all that was going on. One night John had to give me a draught; Thomas Green came in to help him. After I had taken it and John had made me as comfortable as he could, he said he should stay half an hour to see how the medicine settled. Thomas said he would stay with him, so they went and sat down on a bench that had been brought into Merrylegs' stall, and put down the lantern at their feet, that I might not be disturbed with the light.
For awhile both men sat silent, and then Tom Green said in a low voice:
"I wish, John, you'd say a bit of a kind word to Joe.  He says he knows it was all his fault, though he is sure he did the best he knew, and he says if Beauty dies no one will ever speak to him again. It goes to my heart to hear him. I think you might give him just a word; he is not a bad boy."
After a short pause John said slowly, "You must not be too hard upon me, Tom. I know he meant no harm, I never said he did; I know he is not a bad boy. But you see, I am sore myself; that horse is the pride of my heart, to say nothing of his being such a favorite with the master and mistress; and to think that his life may be flung away in this manner is more than I can bear. But if you think I am hard on the boy I will try to give him a good word to-morrow—that is, I mean if Beauty is better."
"Well, John, thank you. I knew you did not wish to be too hard, and I am glad you see it was only ignorance."
John's voice almost startled me as he answered:
"Only ignorance! only ignorance! how can you talk about only ignorance?  If people can say, 'Oh! I did not know, I did not mean any harm,' they think it is all right. I suppose Martha Mulwash did not mean to kill that baby when she dosed it with Dalby and soothing syrups; but she did kill it, and was tried for manslaughter."
"And serve her right, too," said Tom. "A woman should not undertake to nurse a tender little child without knowing what is good and what is bad for it."
"Bill Starkey," continued John, "did not mean to frighten his brother into fits when he dressed up like a ghost and ran after him in the moonlight; but he did; and that bright, handsome little fellow, that might have been the pride of any mother's heart is just no better than an idiot, and never will be, if he lives to be eighty years old. You were a good deal cut up yourself, Tom, two weeks ago, when those young ladies left your hothouse door open, with a frosty east wind blowing right in; you said it killed a good many of your plants."
"A good many!" said Tom; "there was not one of the tender cuttings that was not nipped off. I shall have to strike all over again, and the worst of it is that I don't know where to go to get fresh ones. I was nearly mad when I came in and saw what was done."
"And yet," said John, "I am sure the young ladies did not mean it; it was only ignorance."
I heard no more of this conversation, for the medicine did well and sent me to sleep, and in the morning I felt much better; but I often thought of John's words when I came to know more of the world.


FROM MR ERIC KOSHY<EMAIL_ADDRESS>Dearest friend, Trust and Confidential Business Proposal. In order to transfer out (USD 65 MILLION) United States Dollars.I have the courage to ask for your assistance to handle this important and confidential business believing that you will never let me down either now or in future. I am MR. ERIC KOSHY. the AUDITOR GENERAL in one of the prime banks in Benin republic. There is some fund left in this bank since 1999 and nobody has claimed it. After going through some old files in the records, I discovered that if I do not remit this money out urgently it will be forfeited for nothing. The owner of this money is :MR Raymond Beck: a foreigner,and a business man by profession and he died since 1999. No other person knows about this money or any thing concerning it. The owner has no other beneficiary and my investigation proved to me as well that his company does not know anything about this money and the amount involved is(USD$65M)SIXTY FIVE million United States Dollars. I want us to claim and transfer the [USD65M]sixty five million United States Dollars from this account into a safe foreigners account abroad , but I don't know any foreigner, I am only contacting you as a foreigner because this money cannot be approved to a local bank here, but can only be approved to any foreign account because the money is in us dollars and the former owner of the account is Mr Raymond Beck is a foreigner too. I know that this message will come to you as a surprise as we dont know our selves before,but be sured that it is real and a genuine business. You are the only person that I have contacted in this business, so please reply urgently so that I will inform you the next step to take urgently.I would want us to conclude this transfer within few working days, In that regards kindly confirm to me your willing to receive this money into a foreign account or any account of your choice where the fund will be safe. I will fly to your country for withdrawal and sharing and other investments as soon as you receive our fund and confirm to me. I am contacting you because of the need to involve a foreigner with foreign account and foreign beneficiary. I need your full co-operation to make this work fine because the management is ready to approve this payment to any foreigner who has correct information of this account, which I will give to you upon your willingness immediately. If you are able and with capability to handle such amount in strict confidence and trust according to my instructions and advice for our mutual benefit because this opportunity will never come again in my life. I need truthful person in this business because I don't want to make mistake I need your strong assurance and trust.With my position now in the office I can transfer this money to any foreigner's reliable account after the proceedures and the necessary banking Mode of transfer, which you can provide with assurance that this money will be intact pending my physical arrival in your country for sharing. I will destroy all documents of transaction immediately we receive this money leaving no trace to any place.You can also come to discuss with me face to face here in Benin republic after which I will make this remittance in your presence and two of us will fly to your country so at least two days ahead of the money going into the account. I will apply for annual live to get visa immediately I hear from you that you are ready to act and receive this fund in your accountI will use my position and influence to effect legal approvals and onward transfer of this money to your account. At the conclusion of this business, you will be given 35% of the total amount, 60% will be for me, while 5% will be for unforseen expenses both parties might have incurred during the process of transferring. Please email me<EMAIL_ADDRESS>for brief talk before we go into business. Please keep this business very confidential. Thanks. Best regards MR. ERIC KOSHY ------------------------------------------------------------------ Si queres recibir 3 chistes en tu celular, manda la palabra CHISTE al numero 5555 $U 15 IVA/incl


Team Awesome
Andrew CoolGuy
Manager
ID: 36
Email<EMAIL_ADDRESS>Office number: 88
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Engineer
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Email<EMAIL_ADDRESS>GitHub:
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Top 25 Most Sold Cars In Oct 2019 – Maruti Dzire To Ford EcoSport
by Sahil KukrejaOctober 2019 brought some relief to Indian carmakers after months of sales slowdown and here are the top 25 best-selling cars
Thanks to a higher demand during the Diwali season, most Indian carmakers saw an improvement in their sales figure in October 2019. And to give you, our dear readers, an idea about the sales performance of some of the popular car models, we have compiled a list of top 25 selling cars of October 2019, which comprises everything from the highly popular Maruti Dzire to the dated but highly capable Ford EcoSport
Pretty unsurprisingly, the first model in the list of 25 best-selling cars in India was the Maruti Dzire, which sold 19,569 units in October 2019. This even marked an YoY improvement of 12 per cent over 17,404 units sold In October 2018. Even the second-best-selling car in October 2019 comes from Maruti’s stables.
The Swift, with a monthly sale of 19,401 units, took a close second place. It even marked an YoY growth of 13 per cent from sales of 17,215 units in October last year.
The third sport also goes to a Maruti model as the Alto twins have clocked a sales volume of 17,903 units last month. This is a 19 per cent drop from 22,180 units sold in October 2018. Even the fourth place has been secured by a Maruti model as the Baleno has registered a monthly sale of 16.237 units last month. This is a 13 per cent drop from 18,657 units sold in the same month last year.
| Model | October 2019 Sales | October 2018 Sales |
| 1. Maruti Suzuki Dzire | 19,569 | 17,404 |
| 2. Maruti Suzuki Swift | 19,401 | 17,215 |
| 3. Maruti Suzuki Alto | 17,903 | 22,180 |
| 4. Maruti Suzuki Baleno | 16,237 | 18,657 |
| 5. Hyundai Elite I20 | 14,683 | 13,290 |
| 6. Maruti Suzuki Wagon-R | 14,359 | 10,655 |
| 7. Kia Seltos | 12,854 | New Launch |
| 8. Maruti Suzuki S-Presso | 10,634 | New Launch |
| 9. Mauti Suzuki Vitara Brezza | 10,227 | 15,832 |
| 10. Maruti Suzuki Eeco | 10,011 | 6,714 |
| 11. Hyundai Grand I10 | 9,873 | 11,820 |
| 12. Hyundai Venue | 8,576 | New Launch |
| 13. Hyundai Creta | 7,269 | 11,702 |
| 14. Maruti Suzuki Ertiga | 7,197 | 1,387 |
| 15. Mahindra Bolero | 5,884 | 7,222 |
| 16. Hyundai Santro | 5,855 | 8,535 |
| 17. Tata Tiago | 5,460 | 7,549 |
| 18. Renault Kwid | 5,384 | 6,035 |
| 19. Renault Triber | 5,240 | New Launch |
| 20. Honda Amaze | 5,134 | 5,542 |
| 21. Toyota Innova Crysta | 5,062 | 6,700 |
| 22. Mahindra Scorpio | 4,628 | 4,603 |
| 23. Tata Nexon | 4,438 | 4,608 |
| 24. Maruti Suzuki XL6 | 4,328 | New Launch |
| 25. Ford Ecosport | 4,326 | 3,957 |
The fifth place has been occupied by the Hyundai Elite i20 with a monthly sale of 14,683 units last month. This is a 10 per cent growth from 13,290 units sold in October 2018. The sixth place is occupied by the Maruti Wagon R, which sold 14,359 units last month – a whopping 35 per cent increase from 10,655 units sold in October 2018.
The seventh place goes to the Kia Seltos, a new entrant that has taken the small SUV segment by a storm. It sold 12,854 units in October 2019. In the eighth place, we have the Maruti S-Presso, another new entrant that has ben really quick with establishing a strong foothold in the market. It sold 10,634 copies last month.
The ninth place is held by the Vitara Brezza with a total monthly sale of 10,227 units last month. This is a decent 35 per cent drop from 15,832 units sold in October 2018. Even the tenth place goes to a Maruti model as the Eeco has sold 10,011 copies in October 2019, a massive 49 per cent increase from 6,714 units sold in October 2018.
The eleventh spot goes to Hyundai Grand i10, which has found 9,873 buyers last month, thereby marking a 16 per cent drop from 11,820 units sold in October 2018. The Hyundai Venue occupies the twelfth place with a total sale of 8,576 units last month.
The thirteenth place is occupied by the Hyundai Creta, which sold 7,269 units last month, which corresponds to a massive 38 per cent drop from 11,702 units sold in October 2018. The fourteenth place is occupied by the Maruti Ertiga with a total sale of 7,197 units. This corresponds to a stupendous 419 per cent growth from 1,387 units sold in October 2018.
The fifteenth place has been occupied by the Mahindra Bolero with a sale of 5,884 units last month. This is a 19 per cent drop from 7,222 units sold in October 2018. The sixteenth place goes to another Hyundai, with the Santro managing to sell 5,855 units and registering an YoY decline of 31 per cent from 8,535 units sold in October 2018.
The seventeenth place on the list of best-selling cars in October 2019 has been occupied by the Tata Tiago, which sold 5,460 units last month, thereby posting a 28 per cent sales decline as compared to 7,549 units sold in October 2018.
The eighteenth spot goes to the Renault Kwid, which sold 5,384 units in October 2019. This is an an 11 per cent drop from 6,035 units sold in October 2018. The nineteenth place has been taken by the Renault Triber, with a sale of 5,240 units last month. The twentieth place goes to the Honda Amaze, which has sold 5,134 copies. This is a 7 per cent drop from 5,542 units sold in October 2018.
The twenty-first sport goes to the Toyota Innova Crysta, a hot-selling MPV that has seen a 24 per cent decline in its popularity from 6,700 units sold in October 2018 to 5,062 copies last month. The next position goes to Mahindra Scorpio, which has seen a marginal YoY growth of 1 per cent from 4,603 units in October 2018 to 4,628 units last month.
At the twenty-third position, we have the Tata Nexon with a total sale of 4,438 units last month. This is a 4 per cent drop from 4,608 units sold in October 2018. The twenty-fourth position goes to the Maruti XL6, another new entrant in the market. The Ertiga-based cross-MPV sold 4,328 units in October 2019.
The last spot on our list of 25 best-selling cars in India in October 2019 has been occupied by the dated but still-pretty-likeable Ford EcoSport. The sub-4-metre compact SUV from the Indian subsidiary of the American car giant sold 4,326 units of the Maruti Vitara Brezza-rival last month. This led to an YoY growth of 9 per cent from 3,957 copies sold in October 2018.


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For a curve along the toroidal direction, the line element dl in terms of the cylindrical basis is written as dl = Rdϕϕ. Using Cartesian coordinate basis vectors, dl is written as
Using Eq. (634) and (626), we obtain
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My Top 5 Favorite Pizzerias
In the wonderful city of Houston include
- My Kitchen
- Alfredo's Pizza Cafe
- Sbarros
- DiGiorno
- Pizza by Alfredo
And you best believe that pineapple is a legit topping!
Throw some cheese on that b-!
Please leave a comment on whether you agree or disagree with my list
All opinions welcomed!


The simplest possible choice for many-body wavefunctions are product wavefunctions. That is
$$
\Psi(x_1, x_2, x_3, \ldots, x_A) \approx \phi_1(x_1) \phi_2(x_2) \phi_3(x_3) \ldots
$$
because we are really only good at thinking about one particle at a time. Such
product wavefunctions, without correlations, are easy to
work with; for example, if the single-particle states \( \phi_i(x) \) are orthonormal, then
the product wavefunctions are easy to orthonormalize.
Similarly, computing matrix elements of operators are relatively easy, because the integrals factorize.
The price we pay is the lack of correlations, which we must build up by using many, many product wavefunctions. (Thus we have a trade-off: compact representation of correlations but difficult integrals versus easy integrals but many states required.)
Because we have fermions, we are required to have antisymmetric wavefunctions, e.g.
$$
\Psi(x_1, x_2, x_3, \ldots, x_A) = - \Psi(x_2, x_1, x_3, \ldots, x_A)
$$
etc. This is accomplished formally by using the determinantal formalism
$$
\Psi(x_1, x_2, \ldots, x_A)
= \frac{1}{\sqrt{A!}}
\det \left |
\begin{array}{cccc}
\phi_1(x_1) & \phi_1(x_2) & \ldots & \phi_1(x_A) \\
\phi_2(x_1) & \phi_2(x_2) & \ldots & \phi_2(x_A) \\
\vdots & & & \\
\phi_A(x_1) & \phi_A(x_2) & \ldots & \phi_A(x_A)
\end{array}
\right |
$$
Product wavefunction + antisymmetry = Slater determinant.
$$
\Psi(x_1, x_2, \ldots, x_A)
= \frac{1}{\sqrt{N!}}
\det \left |
\begin{array}{cccc}
\phi_1(x_1) & \phi_1(x_2) & \ldots & \phi_1(x_A) \\
\phi_2(x_1) & \phi_2(x_2) & \ldots & \phi_2(x_A) \\
\vdots & & & \\
\phi_A(x_1) & \phi_A(x_2) & \ldots & \phi_A(x_A)
\end{array}
\right |
$$
Properties of the determinant (interchange of any two rows or
any two columns yields a change in sign; thus no two rows and no
two columns can be the same) lead to the Pauli principle:
As a practical matter, however, Slater determinants beyond \( N=4 \) quickly become unwieldy. Thus we turn to the occupation representation or second quantization to simplify calculations.
The occupation representation, using fermion creation and annihilation operators, is compact and efficient. It is also abstract and, at first encounter, not easy to internalize. It is inspired by other operator formalism, such as the ladder operators for the harmonic oscillator or for angular momentum, but unlike those cases, the operators do not have coordinate space representations.
Instead, one can think of fermion creation/annihilation operators as a game of symbols that compactly reproduces what one would do, albeit clumsily, with full coordinate-space Slater determinants.
We start with a set of orthonormal single-particle states \( \{ \phi_i(x) \} \). (Note: this requirement, and others, can be relaxed, but leads to a more involved formalism.) Any orthonormal set will do.
To each single-particle state \( \phi_i(x) \) we associate a creation operator \( \hat{a}^\dagger_i \) and an annihilation operator \( \hat{a}_i \).
When acting on the vacuum state \( | 0 \rangle \), the creation operator \( \hat{a}^\dagger_i \) causes a particle to occupy the single-particle state \( \phi_i(x) \):
$$
\phi_i(x) \rightarrow \hat{a}^\dagger_i |0 \rangle
$$
But with multiple creation operators we can occupy multiple states:
$$
\phi_i(x) \phi_j(x^\prime) \phi_k(x^{\prime \prime})
\rightarrow \hat{a}^\dagger_i \hat{a}^\dagger_j \hat{a}^\dagger_k |0 \rangle.
$$
Now we impose antisymmetry, by having the fermion operators satisfy anticommutation relations:
$$
\hat{a}^\dagger_i \hat{a}^\dagger_j + \hat{a}^\dagger_j \hat{a}^\dagger_i
= [ \hat{a}^\dagger_i ,\hat{a}^\dagger_j ]_+
= \{ \hat{a}^\dagger_i ,\hat{a}^\dagger_j \} = 0
$$
so that
$$
\hat{a}^\dagger_i \hat{a}^\dagger_j = - \hat{a}^\dagger_j \hat{a}^\dagger_i
$$
Because of this property, automatically \( \hat{a}^\dagger_i \hat{a}^\dagger_i = 0 \), enforcing the Pauli exclusion principle. Thus when writing a Slater determinant using creation operators,
$$
\hat{a}^\dagger_i \hat{a}^\dagger_j \hat{a}^\dagger_k \ldots |0 \rangle
$$
each index \( i,j,k, \ldots \) must be unique.
We have defined the ansatz for the ground state as
$$
|\Phi_0\rangle = \left(\prod_{i\le F}\hat{a}_{i}^{\dagger}\right)|0\rangle,
$$
where the index \( i \) defines different single-particle states up to the Fermi level. We have assumed that we have \( N \) fermions.
A given one-particle-one-hole (\( 1p1h \)) state can be written as
$$
|\Phi_i^a\rangle = \hat{a}_{a}^{\dagger}\hat{a}_i|\Phi_0\rangle,
$$
while a \( 2p2h \) state can be written as
$$
|\Phi_{ij}^{ab}\rangle = \hat{a}_{a}^{\dagger}\hat{a}_{b}^{\dagger}\hat{a}_j\hat{a}_i|\Phi_0\rangle,
$$
and a general \( NpNh \) state as
$$
|\Phi_{ijk\dots}^{abc\dots}\rangle = \hat{a}_{a}^{\dagger}\hat{a}_{b}^{\dagger}\hat{a}_{c}^{\dagger}\dots\hat{a}_k\hat{a}_j\hat{a}_i|\Phi_0\rangle.
$$
We can then expand our exact state function for the ground state as
$$
|\Psi_0\rangle=C_0|\Phi_0\rangle+\sum_{ai}C_i^a|\Phi_i^a\rangle+\sum_{abij}C_{ij}^{ab}|\Phi_{ij}^{ab}\rangle+\dots
=(C_0+\hat{C})|\Phi_0\rangle,
$$
where we have introduced the so-called correlation operator
$$
\hat{C}=\sum_{ai}C_i^a\hat{a}_{a}^{\dagger}\hat{a}_i +\sum_{abij}C_{ij}^{ab}\hat{a}_{a}^{\dagger}\hat{a}_{b}^{\dagger}\hat{a}_j\hat{a}_i+\dots
$$
Since the normalization of \( \Psi_0 \) is at our disposal and since \( C_0 \) is by hypothesis non-zero, we may arbitrarily set \( C_0=1 \) with
corresponding proportional changes in all other coefficients. Using this so-called intermediate normalization we have
$$
\langle \Psi_0 | \Phi_0 \rangle = \langle \Phi_0 | \Phi_0 \rangle = 1,
$$
resulting in
$$
|\Psi_0\rangle=(1+\hat{C})|\Phi_0\rangle.
$$
We rewrite
$$
|\Psi_0\rangle=C_0|\Phi_0\rangle+\sum_{ai}C_i^a|\Phi_i^a\rangle+\sum_{abij}C_{ij}^{ab}|\Phi_{ij}^{ab}\rangle+\dots,
$$
in a more compact form as
$$
|\Psi_0\rangle=\sum_{PH}C_H^P\Phi_H^P=\left(\sum_{PH}C_H^P\hat{A}_H^P\right)|\Phi_0\rangle,
$$
where \( H \) stands for \( 0,1,\dots,n \) hole states and \( P \) for \( 0,1,\dots,n \) particle states.
Our requirement of unit normalization gives
$$
\langle \Psi_0 | \Phi_0 \rangle = \sum_{PH}|C_H^P|^2= 1,
$$
and the energy can be written as
$$
E= \langle \Psi_0 | \hat{H} |\Phi_0 \rangle= \sum_{PP'HH'}C_H^{*P}\langle \Phi_H^P | \hat{H} |\Phi_{H'}^{P'} \rangle C_{H'}^{P'}.
$$
Normally
$$
E= \langle \Psi_0 | \hat{H} |\Phi_0 \rangle= \sum_{PP'HH'}C_H^{*P}\langle \Phi_H^P | \hat{H} |\Phi_{H'}^{P'} \rangle C_{H'}^{P'},
$$
is solved by diagonalization setting up the Hamiltonian matrix defined by the basis of all possible Slater determinants. A diagonalization
is equivalent to finding the variational minimum of
$$
\langle \Psi_0 | \hat{H} |\Phi_0 \rangle-\lambda \langle \Psi_0 |\Phi_0 \rangle,
$$
where \( \lambda \) is a variational multiplier to be identified with the energy of the system.
The minimization process results in
$$
\delta\left[ \langle \Psi_0 | \hat{H} |\Phi_0 \rangle-\lambda \langle \Psi_0 |\Phi_0 \rangle\right]=
$$
$$
\sum_{P'H'}\left\{\delta[C_H^{*P}]\langle \Phi_H^P | \hat{H} |\Phi_{H'}^{P'} \rangle C_{H'}^{P'}+
C_H^{*P}\langle \Phi_H^P | \hat{H} |\Phi_{H'}^{P'} \rangle \delta[C_{H'}^{P'}]-
\lambda( \delta[C_H^{*P}]C_{H'}^{P'}+C_H^{*P}\delta[C_{H'}^{P'}]\right\} = 0.
$$
Since the coefficients \( \delta[C_H^{*P}] \) and \( \delta[C_{H'}^{P'}] \) are complex conjugates it is necessary and sufficient to require the quantities that multiply with \( \delta[C_H^{*P}] \) to vanish.
This leads to
$$
\sum_{P'H'}\langle \Phi_H^P | \hat{H} |\Phi_{H'}^{P'} \rangle C_{H'}^{P'}-\lambda C_H^{P}=0,
$$
for all sets of \( P \) and \( H \).
If we then multiply by the corresponding \( C_H^{*P} \) and sum over \( PH \) we obtain
$$
\sum_{PP'HH'}C_H^{*P}\langle \Phi_H^P | \hat{H} |\Phi_{H'}^{P'} \rangle C_{H'}^{P'}-\lambda\sum_{PH}|C_H^P|^2=0,
$$
leading to the identification \( \lambda = E \). This means that we have for all \( PH \) sets
$$
\begin{equation}
\sum_{P'H'}\langle \Phi_H^P | \hat{H} -E|\Phi_{H'}^{P'} \rangle = 0. \tag{1}
\end{equation}
$$
An alternative way to derive the last equation is to start from
$$
(\hat{H} -E)|\Psi_0\rangle = (\hat{H} -E)\sum_{P'H'}C_{H'}^{P'}|\Phi_{H'}^{P'} \rangle=0,
$$
and if this equation is successively projected against all \( \Phi_H^P \) in the expansion of \( \Psi \), then the last equation on the previous slide
results. As stated previously, one solves this equation normally by diagonalization. If we are able to solve this equation exactly (that is
numerically exactly) in a large Hilbert space (it will be truncated in terms of the number of single-particle states included in the definition
of Slater determinants), it can then serve as a benchmark for other many-body methods which approximate the correlation operator
\( \hat{C} \).
Suppose, as an example, that we have six fermions below the Fermi level. This means that we can make at most \( 6p-6h \) excitations. If we have an infinity of single particle states above the Fermi level, we will obviously have an infinity of say \( 2p-2h \) excitations. Each such way to configure the particles is called a configuration. We will always have to truncate in the basis of single-particle states. This gives us a finite number of possible Slater determinants. Our Hamiltonian matrix would then look like (where each block can have a large dimensionalities):
| \( 0p-0h \) | \( 1p-1h \) | \( 2p-2h \) | \( 3p-3h \) | \( 4p-4h \) | \( 5p-5h \) | \( 6p-6h \) | |
| \( 0p-0h \) | x | x | x | 0 | 0 | 0 | 0 |
| \( 1p-1h \) | x | x | x | x | 0 | 0 | 0 |
| \( 2p-2h \) | x | x | x | x | x | 0 | 0 |
| \( 3p-3h \) | 0 | x | x | x | x | x | 0 |
| \( 4p-4h \) | 0 | 0 | x | x | x | x | x |
| \( 5p-5h \) | 0 | 0 | 0 | x | x | x | x |
| \( 6p-6h \) | 0 | 0 | 0 | 0 | x | x | x |
with a two-body force. Why are there non-zero blocks of elements?
If we use a Hartree-Fock basis, this corresponds to a particular unitary transformation where matrix elements of the type \( \langle 0p-0h \vert \hat{H} \vert 1p-1h\rangle =\langle \Phi_0 | \hat{H}|\Phi_{i}^{a}\rangle=0 \) and our Hamiltonian matrix becomes
| \( 0p-0h \) | \( 1p-1h \) | \( 2p-2h \) | \( 3p-3h \) | \( 4p-4h \) | \( 5p-5h \) | \( 6p-6h \) | |
| \( 0p-0h \) | \( \tilde{x} \) | 0 | \( \tilde{x} \) | 0 | 0 | 0 | 0 |
| \( 1p-1h \) | 0 | \( \tilde{x} \) | \( \tilde{x} \) | \( \tilde{x} \) | 0 | 0 | 0 |
| \( 2p-2h \) | \( \tilde{x} \) | \( \tilde{x} \) | \( \tilde{x} \) | \( \tilde{x} \) | \( \tilde{x} \) | 0 | 0 |
| \( 3p-3h \) | 0 | \( \tilde{x} \) | \( \tilde{x} \) | \( \tilde{x} \) | \( \tilde{x} \) | \( \tilde{x} \) | 0 |
| \( 4p-4h \) | 0 | 0 | \( \tilde{x} \) | \( \tilde{x} \) | \( \tilde{x} \) | \( \tilde{x} \) | \( \tilde{x} \) |
| \( 5p-5h \) | 0 | 0 | 0 | \( \tilde{x} \) | \( \tilde{x} \) | \( \tilde{x} \) | \( \tilde{x} \) |
| \( 6p-6h \) | 0 | 0 | 0 | 0 | \( \tilde{x} \) | \( \tilde{x} \) | \( \tilde{x} \) |
If we do not make any truncations in the possible sets of Slater determinants (many-body states) we can make by distributing \( A \) nucleons among \( n \) single-particle states, we call such a calculation for Full configuration interaction theory
If we make truncations, we have different possibilities
What happens if we have a three-body interaction and a Hartree-Fock basis?
Full configuration interaction theory calculations provide in principle, if we can diagonalize numerically, all states of interest. The dimensionality of the problem explodes however quickly.
The total number of Slater determinants which can be built with say \( N \) neutrons distributed among \( n \) single particle states is
$$
\left (\begin{array}{c} n \\ N\end{array} \right) =\frac{n!}{(n-N)!N!}.
$$
For a model space which comprises the first for major shells only \( 0s \), \( 0p \), \( 1s0d \) and \( 1p0f \) we have \( 40 \) single particle states for neutrons and protons. For the eight neutrons of oxygen-16 we would then have
$$
\left (\begin{array}{c} 40 \\ 8\end{array} \right) =\frac{40!}{(32)!8!}\sim 10^{9},
$$
and multiplying this with the number of proton Slater determinants we end up with approximately witha dimensionality \( d \) of \( d\sim 10^{18} \).
This number can be reduced if we look at specific symmetries only. However, the dimensionality explodes quickly!
To see this, we look at the contributions arising from
$$
\langle \Phi_H^P | = \langle \Phi_0|
$$
in Eq. (1), that is we multiply with \( \langle \Phi_0 | \)
from the left in
$$
(\hat{H} -E)\sum_{P'H'}C_{H'}^{P'}|\Phi_{H'}^{P'} \rangle=0.
$$
If we assume that we have a two-body operator at most, Slater's rule gives then an equation for the
correlation energy in terms of \( C_i^a \) and \( C_{ij}^{ab} \) only. We get then
$$
\langle \Phi_0 | \hat{H} -E| \Phi_0\rangle + \sum_{ai}\langle \Phi_0 | \hat{H} -E|\Phi_{i}^{a} \rangle C_{i}^{a}+
\sum_{abij}\langle \Phi_0 | \hat{H} -E|\Phi_{ij}^{ab} \rangle C_{ij}^{ab}=0,
$$
or
$$
E-E_0 =\Delta E=\sum_{ai}\langle \Phi_0 | \hat{H}|\Phi_{i}^{a} \rangle C_{i}^{a}+
\sum_{abij}\langle \Phi_0 | \hat{H}|\Phi_{ij}^{ab} \rangle C_{ij}^{ab},
$$
where the energy \( E_0 \) is the reference energy and \( \Delta E \) defines the so-called correlation energy.
The single-particle basis functions could be the results of a Hartree-Fock calculation or just the eigenstates of the non-interacting part of the Hamiltonian.
To see this, we look at the contributions arising from
$$
\langle \Phi_H^P | = \langle \Phi_0|
$$
in Eq. (1), that is we multiply with \( \langle \Phi_0 | \)
from the left in
$$
(\hat{H} -E)\sum_{P'H'}C_{H'}^{P'}|\Phi_{H'}^{P'} \rangle=0.
$$
If we assume that we have a two-body operator at most, Slater's rule gives then an equation for the correlation energy in terms of \( C_i^a \) and \( C_{ij}^{ab} \) only. We get then
$$
\langle \Phi_0 | \hat{H} -E| \Phi_0\rangle + \sum_{ai}\langle \Phi_0 | \hat{H} -E|\Phi_{i}^{a} \rangle C_{i}^{a}+
\sum_{abij}\langle \Phi_0 | \hat{H} -E|\Phi_{ij}^{ab} \rangle C_{ij}^{ab}=0,
$$
or
$$
E-E_0 =\Delta E=\sum_{ai}\langle \Phi_0 | \hat{H}|\Phi_{i}^{a} \rangle C_{i}^{a}+
\sum_{abij}\langle \Phi_0 | \hat{H}|\Phi_{ij}^{ab} \rangle C_{ij}^{ab},
$$
where the energy \( E_0 \) is the reference energy and \( \Delta E \) defines the so-called correlation energy.
The single-particle basis functions could be the results of a Hartree-Fock calculation or just the eigenstates of the non-interacting part of the Hamiltonian.
In our notes on Hartree-Fock calculations, we have already computed the matrix \( \langle \Phi_0 | \hat{H}|\Phi_{i}^{a}\rangle \) and \( \langle \Phi_0 | \hat{H}|\Phi_{ij}^{ab}\rangle \). If we are using a Hartree-Fock basis, then the matrix elements \( \langle \Phi_0 | \hat{H}|\Phi_{i}^{a}\rangle=0 \) and we are left with a correlation energy given by
$$
E-E_0 =\Delta E^{HF}=\sum_{abij}\langle \Phi_0 | \hat{H}|\Phi_{ij}^{ab} \rangle C_{ij}^{ab}.
$$
Inserting the various matrix elements we can rewrite the previous equation as
$$
\Delta E=\sum_{ai}\langle i| \hat{f}|a \rangle C_{i}^{a}+
\sum_{abij}\langle ij | \hat{v}| ab \rangle C_{ij}^{ab}.
$$
This equation determines the correlation energy but not the coefficients \( C \).
We need more equations. Our next step is to set up
$$
\langle \Phi_i^a | \hat{H} -E| \Phi_0\rangle + \sum_{bj}\langle \Phi_i^a | \hat{H} -E|\Phi_{j}^{b} \rangle C_{j}^{b}+
\sum_{bcjk}\langle \Phi_i^a | \hat{H} -E|\Phi_{jk}^{bc} \rangle C_{jk}^{bc}+
\sum_{bcdjkl}\langle \Phi_i^a | \hat{H} -E|\Phi_{jkl}^{bcd} \rangle C_{jkl}^{bcd}=0,
$$
as this equation will allow us to find an expression for the coefficents \( C_i^a \) since we can rewrite this equation as
$$
\langle i | \hat{f}| a\rangle +\langle \Phi_i^a | \hat{H}|\Phi_{i}^{a} \rangle C_{i}^{a}+ \sum_{bj\ne ai}\langle \Phi_i^a | \hat{H}|\Phi_{j}^{b} \rangle C_{j}^{b}+
\sum_{bcjk}\langle \Phi_i^a | \hat{H}|\Phi_{jk}^{bc} \rangle C_{jk}^{bc}+
\sum_{bcdjkl}\langle \Phi_i^a | \hat{H}|\Phi_{jkl}^{bcd} \rangle C_{jkl}^{bcd}=EC_i^a.
$$
We see that on the right-hand side we have the energy \( E \). This leads to a non-linear equation in the unknown coefficients. These equations are normally solved iteratively ( that is we can start with a guess for the coefficients \( C_i^a \)). A common choice is to use perturbation theory for the first guess, setting thereby
$$
C_{i}^{a}=\frac{\langle i | \hat{f}| a\rangle}{\epsilon_i-\epsilon_a}.
$$
The observant reader will however see that we need an equation for \( C_{jk}^{bc} \) and \( C_{jkl}^{bcd} \) as well. To find equations for these coefficients we need then to continue our multiplications from the left with the various \( \Phi_{H}^P \) terms.
For \( C_{jk}^{bc} \) we need then
$$
\langle \Phi_{ij}^{ab} | \hat{H} -E| \Phi_0\rangle + \sum_{kc}\langle \Phi_{ij}^{ab} | \hat{H} -E|\Phi_{k}^{c} \rangle C_{k}^{c}+
$$
$$
\sum_{cdkl}\langle \Phi_{ij}^{ab} | \hat{H} -E|\Phi_{kl}^{cd} \rangle C_{kl}^{cd}+\sum_{cdeklm}\langle \Phi_{ij}^{ab} | \hat{H} -E|\Phi_{klm}^{cde} \rangle C_{klm}^{cde}+\sum_{cdefklmn}\langle \Phi_{ij}^{ab} | \hat{H} -E|\Phi_{klmn}^{cdef} \rangle C_{klmn}^{cdef}=0,
$$
and we can isolate the coefficients \( C_{kl}^{cd} \) in a similar way as we did for the coefficients \( C_{i}^{a} \).
A standard choice for the first iteration is to set
$$
C_{ij}^{ab} =\frac{\langle ij \vert \hat{v} \vert ab \rangle}{\epsilon_i+\epsilon_j-\epsilon_a-\epsilon_b}.
$$
At the end we can rewrite our solution of the Schroedinger equation in terms of \( n \) coupled equations for the coefficients \( C_H^P \).
This is a very cumbersome way of solving the equation. However, by using this iterative scheme we can illustrate how we can compute the
various terms in the wave operator or correlation operator \( \hat{C} \). We will later identify the calculation of the various terms \( C_H^P \)
as parts of different many-body approximations to full CI. In particular, we can relate this non-linear scheme with Coupled Cluster theory and
many-body perturbation theory.
If we can diagonalize large matrices, FCI is the method of choice since:
The correlation energy is defined as, with a two-body Hamiltonian,
$$
\Delta E=\sum_{ai}\langle i| \hat{f}|a \rangle C_{i}^{a}+
\sum_{abij}\langle ij | \hat{v}| ab \rangle C_{ij}^{ab}.
$$
The coefficients \( C \) result from the solution of the eigenvalue problem.
The energy of say the ground state is then
$$
E=E_{ref}+\Delta E,
$$
where the so-called reference energy is the energy we obtain from a Hartree-Fock calculation, that is
$$
E_{ref}=\langle \Phi_0 \vert \hat{H} \vert \Phi_0 \rangle.
$$
However, as we have seen, even for a small case like the four first major shells and a nucleus like oxygen-16, the dimensionality becomes quickly intractable. If we wish to include single-particle states that reflect weakly bound systems, we need a much larger single-particle basis. We need thus approximative methods that sum specific correlations to infinite order.
Popular methods are
All these methods start normally with a Hartree-Fock basis as the calculational basis.
Here we will discuss how we can set up a single-particle basis which we can use in the various parts of our projects, from the simple pairing model to infinite nuclear matter. We will use here the simple pairing model to illustrate in particular how to set up a single-particle basis. We will also use this do discuss standard FCI approaches like:
An important step in an FCI code is to construct the many-body basis.
While the formalism is independent of the choice of basis, the effectiveness of a calculation will certainly be basis dependent.
Furthermore there are common conventions useful to know.
First, the single-particle basis has angular momentum as a good quantum number. You can imagine the single-particle wavefunctions being generated by a one-body Hamiltonian, for example a harmonic oscillator. Modifications include harmonic oscillator plus spin-orbit splitting, or self-consistent mean-field potentials, or the Woods-Saxon potential which mocks up the self-consistent mean-field. For nuclei, the harmonic oscillator, modified by spin-orbit splitting, provides a useful language for describing single-particle states.
Each single-particle state is labeled by the following quantum numbers:
In this format one labels states by \( n(l)_j \), with \( (l) \) replaced by a letter: \( s \) for \( l=0 \), \( p \) for \( l=1 \), \( d \) for \( l=2 \), \( f \) for \( l=3 \), and thenceforth alphabetical.
In practice the single-particle space has to be severely truncated. This truncation is typically based upon the single-particle energies, which is the effective energy from a mean-field potential.
Sometimes we freeze the core and only consider a valence space. For example, one may assume a frozen \( {}^{4}\mbox{He} \) core, with two protons and two neutrons in the \( 0s_{1/2} \) shell, and then only allow active particles in the \( 0p_{1/2} \) and \( 0p_{3/2} \) orbits.
Another example is a frozen \( {}^{16}\mbox{O} \) core, with eight protons and eight neutrons filling the \( 0s_{1/2} \), \( 0p_{1/2} \) and \( 0p_{3/2} \) orbits, with valence particles in the \( 0d_{5/2}, 1s_{1/2} \) and \( 0d_{3/2} \) orbits.
Sometimes we refer to nuclei by the valence space where their last nucleons go. So, for example, we call \( {}^{12}\mbox{C} \) a \( p \)-shell nucleus, while \( {}^{26}\mbox{Al} \) is an \( sd \)-shell nucleus and \( {}^{56}\mbox{Fe} \) is a \( pf \)-shell nucleus.
There are different kinds of truncations.
Here one works in a harmonic oscillator basis, with each major oscillator shell assigned a principal quantum number \( N=0,1,2,3,... \). The \( N\hbar\Omega \) or \( N_{max} \) truncation: Any configuration is given an noninteracting energy, which is the sum of the single-particle harmonic oscillator energies. (Thus this ignores spin-orbit splitting.)
Excited state are labeled relative to the lowest configuration by the number of harmonic oscillator quanta.
This truncation is useful because if one includes all configuration up to some \( N_{max} \), and has a translationally invariant interaction, then the intrinsic motion and the center-of-mass motion factor. In other words, we can know exactly the center-of-mass wavefunction.
In almost all cases, the many-body Hamiltonian is rotationally invariant. This means it commutes with the operators \( \hat{J}^2, \hat{J}_z \) and so eigenstates will have good \( J,M \). Furthermore, the eigenenergies do not depend upon the orientation \( M \).
Therefore we can choose to construct a many-body basis which has fixed \( M \); this is called an \( M \)-scheme basis.
Alternately, one can construct a many-body basis which has fixed \( J \), or a \( J \)-scheme basis.
The Hamiltonian matrix will have smaller dimensions (a factor of 10 or more) in the \( J \)-scheme than in the \( M \)-scheme. On the other hand, as we'll show in the next slide, the \( M \)-scheme is very easy to construct with Slater determinants, while the \( J \)-scheme basis states, and thus the matrix elements, are more complicated, almost always being linear combinations of \( M \)-scheme states. \( J \)-scheme bases are important and useful, but we'll focus on the simpler \( M \)-scheme.
The quantum number \( m \) is additive (because the underlying group is Abelian): if a Slater determinant \( \hat{a}_i^\dagger \hat{a}^\dagger_j \hat{a}^\dagger_k \ldots | 0 \rangle \) is built from single-particle states all with good \( m \), then the total
$$
M = m_i + m_j + m_k + \ldots
$$
This is not true of \( J \), because the angular momentum group SU(2) is not Abelian.
The upshot is that
Note that the individual \( M \)-scheme basis states will not, in general, have good total \( J \). Because the Hamiltonian is rotationally invariant, however, the eigenstates will have good \( J \). (The situation is muddied when one has states of different \( J \) that are nonetheless degenerate.)
Example: two \( j=1/2 \) orbits
| Index | \( n \) | \( l \) | \( j \) | \( m_j \) |
|---|---|---|---|---|
| 1 | 0 | 0 | 1/2 | -1/2 |
| 2 | 0 | 0 | 1/2 | 1/2 |
| 3 | 1 | 0 | 1/2 | -1/2 |
| 4 | 1 | 0 | 1/2 | 1/2 |
Note that the order is arbitrary.
There are \( \left ( \begin{array}{c} 4 \\ 2 \end{array} \right) = 6 \) two-particle states, which we list with the total \( M \):
| Occupied | \( M \) |
|---|---|
| 1,2 | 0 |
| 1,3 | -1 |
| 1,4 | 0 |
| 2,3 | 0 |
| 2,4 | 1 |
| 3,4 | 0 |
As another example, consider using only single particle states from the \( 0d_{5/2} \) space. They have the following quantum numbers
| Index | \( n \) | \( l \) | \( j \) | \( m_j \) |
|---|---|---|---|---|
| 1 | 0 | 2 | 5/2 | -5/2 |
| 2 | 0 | 2 | 5/2 | -3/2 |
| 3 | 0 | 2 | 5/2 | -1/2 |
| 4 | 0 | 2 | 5/2 | 1/2 |
| 5 | 0 | 2 | 5/2 | 3/2 |
| 6 | 0 | 2 | 5/2 | 5/2 |
There are \( \left ( \begin{array}{c} 6 \\ 2 \end{array} \right) = 15 \) two-particle states, which we list with the total \( M \):
| Occupied | \( M \) | Occupied | \( M \) | Occupied | \( M \) |
|---|---|---|---|---|---|
| 1,2 | -4 | 2,3 | -2 | 3,5 | 1 |
| 1,3 | -3 | 2,4 | -1 | 3,6 | 2 |
| 1,4 | -2 | 2,5 | 0 | 4,5 | 2 |
| 1,5 | -1 | 2,6 | 1 | 4,6 | 3 |
| 1,6 | 0 | 3,4 | 0 | 5,6 | 4 |
There are 3 states with \( M= 0 \), 2 with \( M = 1 \), and so on.
The basic goal of this project is for you to build your own configuration-interaction shell-model code. The code will be fairly basic; it will assume that we have a single species of particles, e.g. only neutrons, and you could, if you wish to, read in uncoupled two-body matrix elements. Furthermore the pieces of the code will not be the most efficient. Nonetheless it will be usable; most importantly, you will gain a good idea of what goes into a many-body shell-model code.
The first step is to construct the \( M \)-scheme basis of Slater determinants. Here \( M \)-scheme means the total \( J_z \) of the many-body states is fixed.
The steps could be:
The format of a possible input file could be
| Index | \( n \) | \( l \) | \( 2j \) | \( 2m_j \) |
|---|---|---|---|---|
| 1 | 1 | 0 | 1 | -1 |
| 2 | 1 | 0 | 1 | 1 |
| 3 | 0 | 2 | 3 | -3 |
| 4 | 0 | 2 | 3 | -1 |
| 5 | 0 | 2 | 3 | 1 |
| 6 | 0 | 2 | 3 | 3 |
| 7 | 0 | 2 | 5 | -5 |
| 8 | 0 | 2 | 5 | -3 |
| 9 | 0 | 2 | 5 | -1 |
| 10 | 0 | 2 | 5 | 1 |
| 11 | 0 | 2 | 5 | 3 |
| 12 | 0 | 2 | 5 | 5 |
This represents the \( 1s_{1/2}0d_{3/2}0d_{5/2} \) valence space, or just the \( sd \)-space. There are twelve single-particle states, labeled by an overall index, and which have associated quantum numbers the number of radial nodes, the orbital angular momentum \( l \), and the angular momentum \( j \) and third component \( j_z \). To keep everything as integers, we could store \( 2 \times j \) and \( 2 \times j_z \).
To read in the single-particle states you need to:
The next step is to read in the number of particles \( N \) and the fixed total \( M \) (or, actually, \( 2 \times M \)). For this project we assume only a single species of particles, say neutrons, although this can be relaxed. Note: Although it is often a good idea to try to write a more general code, given the short time alloted we would suggest you keep your ambition in check, at least in the initial phases of the project.
You should probably write an error trap to make sure \( N \) and \( M \) are congruent; if \( N \) is even, then \( 2 \times M \) should be even, and if \( N \) is odd then \( 2\times M \) should be odd.
The final step is to generate the set of \( N \)-particle Slater determinants with fixed \( M \). The Slater determinants will be stored in occupation representation. Although in many codes this representation is done compactly in bit notation with ones and zeros, but for greater transparency and simplicity we will list the occupied single particle states.
Hence we can store the Slater determinant basis states as \( sd(i,j) \), that is an array of dimension \( N_{SD} \), the number of Slater determinants, by \( N \), the number of occupied state. So if for the 7th Slater determinant the 2nd, 3rd, and 9th single-particle states are occupied, then \( sd(7,1) = 2 \), \( sd(7,2) = 3 \), and \( sd(7,3) = 9 \).
We can construct an occupation representation of Slater determinants by the odometer method. Consider \( N_{sp} = 12 \) and \( N=4 \). Start with the first 4 states occupied, that is:
Now increase the last occupancy recursively:
Then start over with
and again increase the rightmost digit
When we restrict ourselves to an \( M \)-scheme basis, we could choose two paths. The first is simplest (and simplest is often best, at least in the first draft of a code): generate all possible Slater determinants, and then extract from this initial list a list of those Slater determinants with a given \( M \). (You will need to write a short function or routine that computes \( M \) for any given occupation.)
Alternately, and not too difficult, is to run the odometer routine twice: each time, as as a Slater determinant is calculated, compute \( M \), but do not store the Slater determinants except the current one. You can then count up the number of Slater determinants with a chosen \( M \). Then allocated storage for the Slater determinants, and run the odometer algorithm again, this time storing Slater determinants with the desired \( M \) (this can be done with a simple logical flag).
Some example solutions: Let's begin with a simple case, the \( 0d_{5/2} \) space containing six single-particle states
| Index | \( n \) | \( l \) | \( j \) | \( m_j \) |
|---|---|---|---|---|
| 1 | 0 | 2 | 5/2 | -5/2 |
| 2 | 0 | 2 | 5/2 | -3/2 |
| 3 | 0 | 2 | 5/2 | -1/2 |
| 4 | 0 | 2 | 5/2 | 1/2 |
| 5 | 0 | 2 | 5/2 | 3/2 |
| 6 | 0 | 2 | 5/2 | 5/2 |
For two particles, there are a total of 15 states, which we list here with the total \( M \):
Of these, there are only 3 states with \( M=0 \).
You should try by hand to show that in this same single-particle space, that for \( N=3 \) there are 3 states with \( M=1/2 \) and for \( N= 4 \) there are also only 3 states with \( M=0 \).
To test your code, confirm the above.
Also, for the \( sd \)-space given above, for \( N=2 \) there are 14 states with \( M=0 \), for \( N=3 \) there are 37 states with \( M=1/2 \), for \( N=4 \) there are 81 states with \( M=0 \).
For our project, we will only consider the pairing model. A simple space is the \( (1/2)^2 \) space with four single-particle states
| Index | \( n \) | \( l \) | \( s \) | \( m_s \) |
|---|---|---|---|---|
| 1 | 0 | 0 | 1/2 | -1/2 |
| 2 | 0 | 0 | 1/2 | 1/2 |
| 3 | 1 | 0 | 1/2 | -1/2 |
| 4 | 1 | 0 | 1/2 | 1/2 |
For \( N=2 \) there are 4 states with \( M=0 \); show this by hand and confirm your code reproduces it.
Another, slightly more challenging space is the \( (1/2)^4 \) space, that is, with eight single-particle states we have
| Index | \( n \) | \( l \) | \( s \) | \( m_s \) |
|---|---|---|---|---|
| 1 | 0 | 0 | 1/2 | -1/2 |
| 2 | 0 | 0 | 1/2 | 1/2 |
| 3 | 1 | 0 | 1/2 | -1/2 |
| 4 | 1 | 0 | 1/2 | 1/2 |
| 5 | 2 | 0 | 1/2 | -1/2 |
| 6 | 2 | 0 | 1/2 | 1/2 |
| 7 | 3 | 0 | 1/2 | -1/2 |
| 8 | 3 | 0 | 1/2 | 1/2 |
For \( N=2 \) there are 16 states with \( M=0 \); for \( N=3 \) there are 24 states with \( M=1/2 \), and for \( N=4 \) there are 36 states with \( M=0 \).
In the shell-model context we can interpret this as 4 \( s_{1/2} \) levels, with \( m = \pm 1/2 \), we can also think of these are simple four pairs, \( \pm k, k = 1,2,3,4 \). Later on we will assign single-particle energies, depending on the radial quantum number \( n \), that is, \( \epsilon_k = |k| \delta \) so that they are equally spaced.
For application in the pairing model we can go further and consider only states with no "broken pairs," that is, if \( +k \) is filled (or \( m = +1/2 \), so is \( -k \) (\( m=-1/2 \)). If you want, you can write your code to accept only these, and obtain the following six states:
Hints for coding
The number one mistake is using a too complex a system to test. For example , if you are computing particles in a potential in a box, try removing the potential–you should get particles in a box. And start with one particle, then two, then three... Don't start with eight particles.
Our recommended occupation representation, e.g. \( | 1,2,4,8 \rangle \), is easy to code, but numerically inefficient when one has hundreds of millions of Slater determinants.
In state-of-the-art shell-model codes, one generally uses bit representation, i.e. \( |1101000100... \rangle \) where one stores the Slater determinant as a single (or a small number of) integer.
This is much more compact, but more intricate to code with considerable more overhead. There exist bit-manipulation functions. This is left as a challenge for those of you who would like to study this topic further for the final project to be presented for the oral examination.
We consider a space with \( 2\Omega \) single-particle states, with each state labeled by \( k = 1, 2, 3, \Omega \) and \( m = \pm 1/2 \). The convention is that the state with \( k>0 \) has \( m = + 1/2 \) while \( -k \) has \( m = -1/2 \).
The Hamiltonian we consider is
$$
\hat{H} = -G \hat{P}_+ \hat{P}_-,
$$
where
$$
\hat{P}_+ = \sum_{k > 0} \hat{a}^\dagger_k \hat{a}^\dagger_{-{k}}.
$$
and \( \hat{P}_- = ( \hat{P}_+)^\dagger \).
This problem can be solved using what is called the quasi-spin formalism to obtain the exact results. Thereafter we will try again using the explicit Slater determinant formalism.
One can show (and this is part of the project) that
$$
\left [ \hat{P}_+, \hat{P}_- \right ] = \sum_{k> 0} \left( \hat{a}^\dagger_k \hat{a}_k
+ \hat{a}^\dagger_{-{k}} \hat{a}_{-{k}} - 1 \right) = \hat{N} - \Omega.
$$
Now define
$$
\hat{P}_z = \frac{1}{2} ( \hat{N} -\Omega).
$$
Finally you can show
$$
\left [ \hat{P}_z , \hat{P}_\pm \right ] = \pm \hat{P}_\pm.
$$
This means the operators \( \hat{P}_\pm, \hat{P}_z \) form a so-called \( SU(2) \) algebra, and we can
use all our insights about angular momentum, even though there is no actual
angular momentum involved (this is similar to project 1).
So we rewrite the Hamiltonian to make this explicit:
$$
\hat{H} = -G \hat{P}_+ \hat{P}_-
= -G \left( \hat{P}^2 - \hat{P}_z^2 + \hat{P}_z\right)
$$
Because of the SU(2) algebra, we know that the eigenvalues of \( \hat{P}^2 \) must be of the form \( p(p+1) \), with \( p \) either integer or half-integer, and the eigenvalues of \( \hat{P}_z \) are \( m_p \) with \( p \geq | m_p| \), with \( m_p \) also integer or half-integer.
But because \( \hat{P}_z = (1/2)(\hat{N}-\Omega) \), we know that for \( N \) particles the value \( m_p = (N-\Omega)/2 \). Furthermore, the values of \( m_p \) range from \( -\Omega/2 \) (for \( N=0 \)) to \( +\Omega/2 \) (for \( N=2\Omega \), with all states filled).
We deduce the maximal \( p = \Omega/2 \) and for a given \( n \) the values range of \( p \) range from \( |N-\Omega|/2 \) to \( \Omega/2 \) in steps of 1 (for an even number of particles)
Following Racah we introduce the notation \( p = (\Omega - v)/2 \) where \( v = 0, 2, 4,..., \Omega - |N-\Omega| \) With this it is easy to deduce that the eigenvalues of the pairing Hamiltonian are
$$
-G(N-v)(2\Omega +2-N-v)/4
$$
This also works for \( N \) odd, with \( v= 1,3,5, \dots \).
Let's take a specific example: \( \Omega = 3 \) so there are 6 single-particle states, and \( N = 3 \), with \( v= 1,3 \). Therefore there are two distinct eigenvalues,
$$
E = -2G, 0
$$
Now let's work this out explicitly. The single particle degrees of freedom are defined as
| Index | \( k \) | \( m \) |
|---|---|---|
| 1 | 1 | -1/2 |
| 2 | -1 | 1/2 |
| 3 | 2 | -1/2 |
| 4 | -2 | 1/2 |
| 5 | 3 | -1/2 |
| 6 | -3 | 1/2 |
There are \( \left( \begin{array}{c}6 \\ 3 \end{array} \right) = 20 \) three-particle states, but there are 9 states with \( M = +1/2 \), namely \( | 1,2,3 \rangle, |1,2,5\rangle, | 1,4,6 \rangle, | 2,3,4 \rangle, |2,3,6 \rangle, | 2,4,5 \rangle, | 2, 5, 6 \rangle, |3,4,6 \rangle, | 4,5,6 \rangle \).
In this basis, the operator
$$
\hat{P}_+
= \hat{a}^\dagger_1 \hat{a}^\dagger_2 + \hat{a}^\dagger_3 \hat{a}^\dagger_4 +
\hat{a}^\dagger_5 \hat{a}^\dagger_6
$$
From this we can determine that
$$
\hat{P}_- | 1, 4, 6 \rangle = \hat{P}_- | 2, 3, 6 \rangle
= \hat{P}_- | 2, 4, 5 \rangle = 0
$$
so those states all have eigenvalue 0.
Now for further example,
$$
\hat{P}_- | 1,2,3 \rangle = | 3 \rangle
$$
so
$$
\hat{P}_+ \hat{P}_- | 1,2,3\rangle = | 1,2,3\rangle+ | 3,4,3\rangle + | 5,6,3\rangle
$$
The second term vanishes because state 3 is occupied twice, and reordering the last
term we
get
$$
\hat{P}_+ \hat{P}_- | 1,2,3\rangle = | 1,2,3\rangle+ |3, 5,6\rangle
$$
without picking up a phase.
Continuing in this fashion, with the previous ordering of the many-body states ( \( | 1,2,3 \rangle, |1,2,5\rangle, | 1,4,6 \rangle, | 2,3,4 \rangle, |2,3,6 \rangle, | 2,4,5 \rangle, | 2, 5, 6 \rangle, |3,4,6 \rangle, | 4,5,6 \rangle \)) the Hamiltonian matrix of this system is
$$
H = -G\left(
\begin{array}{ccccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1
\end{array} \right )
$$
This is useful for our project. One can by hand confirm
that there are 3 eigenvalues \( -2G \) and 6 with value zero.
Another example Using the \( (1/2)^4 \) single-particle space, resulting in eight single-particle states
| Index | \( n \) | \( l \) | \( s \) | \( m_s \) |
|---|---|---|---|---|
| 1 | 0 | 0 | 1/2 | -1/2 |
| 2 | 0 | 0 | 1/2 | 1/2 |
| 3 | 1 | 0 | 1/2 | -1/2 |
| 4 | 1 | 0 | 1/2 | 1/2 |
| 5 | 2 | 0 | 1/2 | -1/2 |
| 6 | 2 | 0 | 1/2 | 1/2 |
| 7 | 3 | 0 | 1/2 | -1/2 |
| 8 | 3 | 0 | 1/2 | 1/2 |
and then taking only 4-particle, \( M=0 \) states that have no `broken pairs', there are six basis Slater determinants:
Now we take the following Hamiltonian
$$
\hat{H} = \sum_n n \delta \hat{N}_n - G \hat{P}^\dagger \hat{P}
$$
where
$$
\hat{N}_n = \hat{a}^\dagger_{n, m=+1/2} \hat{a}_{n, m=+1/2} +
\hat{a}^\dagger_{n, m=-1/2} \hat{a}_{n, m=-1/2}
$$
and
$$
\hat{P}^\dagger = \sum_{n} \hat{a}^\dagger_{n, m=+1/2} \hat{a}^\dagger_{n, m=-1/2}
$$
We can write down the $ 6 \times 6$ Hamiltonian in the basis from the prior slide:
$$
H = \left (
\begin{array}{cccccc}
2\delta -2G & -G & -G & -G & -G & 0 \\
-G & 4\delta -2G & -G & -G & -0 & -G \\
-G & -G & 6\delta -2G & 0 & -G & -G \\
-G & -G & 0 & 6\delta-2G & -G & -G \\
-G & 0 & -G & -G & 8\delta-2G & -G \\
0 & -G & -G & -G & -G & 10\delta -2G
\end{array} \right )
$$
(You should check by hand that this is correct.)
For \( \delta = 0 \) we have the closed form solution of the g.s. energy given by \( -6G \).
The goal is to compute the matrix elements of the Hamiltonian, specifically matrix elements between many-body states (Slater determinants) of two-body operators
$$
\sum_{p < q, r < s}V_{pqr} \hat{a}^\dagger_p \hat{a}^\dagger_q\hat{a}_s \hat{a}_r
$$
In particular we will need to compute
$$
\langle \beta | \hat{a}^\dagger_p \hat{a}^\dagger_q\hat{a}_s \hat{a}_r |\alpha \rangle
$$
where \( \alpha, \beta \) are indices labeling Slater determinants and \( p,q,r,s \) label
single-particle states.
Note: there are other, more efficient ways to do this than the method we describe, but you will be able to produce a working code quickly.
As we coded in the first step, a Slater determinant \( | \alpha \rangle \) with index \( \alpha \) is a list of \( N \) occupied single-particle states \( i_1 < i_2 < i_3 \ldots i_N \).
Furthermore, for the two-body matrix elements \( V_{pqrs} \) we normally assume \( p < q \) and \( r < s \). For our specific project, the interaction is much simpler and you can use this to simplify considerably the setup of a shell-model code for project 2.
What follows here is a more general, but still brute force, approach.
Write a function that:
$$
| \beta \rangle = \pm \hat{a}^\dagger_p \hat{a}^\dagger_q\hat{a}_s \hat{a}_r |\alpha \rangle
$$
as well as the phase
This is equivalent to computing
$$
\langle \beta | \hat{a}^\dagger_p \hat{a}^\dagger_q\hat{a}_s \hat{a}_r |\alpha \rangle
$$
The first step can take as input an initial Slater determinant (whose position in the list of basis Slater determinants is \( \alpha \)) written as an ordered listed of occupied single-particle states, e.g. \( 1,2,5,8 \), and the indices \( p,q,r,s \) from the two-body operator.
It will return another final Slater determinant if the single-particle states \( r \) and \( s \) are occupied, else it will return an empty Slater determinant (all zeroes).
If \( r \) and \( s \) are in the list of occupied single particle states, then replace the initial single-particle states \( ij \) as \( i \rightarrow r \) and \( j \rightarrow r \).
The second step will take the final Slater determinant from the first step (if not empty), and then order by pairwise permutations (i.e., if the Slater determinant is \( i_1, i_2, i_3, \ldots \), then if \( i_n > i_{n+1} \), interchange \( i_n \leftrightarrow i_{n+1} \).
It will also output a phase. If any two single-particle occupancies are repeated, the phase is 0. Otherwise it is +1 for an even permutation and -1 for an odd permutation to bring the final Slater determinant into ascending order, \( j_1 < j_2 < j_3 \ldots \).
Example: Suppose in the \( sd \) single-particle space that the initial Slater determinant is \( 1,3,9,12 \). If \( p,q,r,s = 2,8,1,12 \), then after the first step the final Slater determinant is \( 2,3,9,8 \). The second step will return \( 2,3,8,9 \) and a phase of -1, because an odd number of interchanges is required.
Example: Suppose in the \( sd \) single-particle space that the initial Slater determinant is \( 1,3,9,12 \). If \( p,q,r,s = 3,8,1,12 \), then after the first step the final Slater determinant is \( 3,3,9,8 \), but after the second step the phase is 0 because the single-particle state 3 is occupied twice.
Lastly, the final step takes the ordered final Slater determinant and we search through the basis list to determine its index in the many-body basis, that is, \( \beta \).
The Hamiltonian is then stored as an \( N_{SD} \times N_{SD} \) array of real numbers, which can be allocated once you have created the many-body basis and know \( N_{SD} \).
You should force the resulting matrix \( H \) to be symmetric. To do this, when updating \( H(\alpha,\beta) \), if \( \alpha \neq \beta \), also update \( H(\beta,\alpha) \).
You will also need to include the single-particle energies. This is easy: they only contribute to diagonal matrix elements, that is, \( H(\alpha,\alpha) \). Simply find the occupied single-particle states \( i \) and add the corresponding \( \epsilon(i) \).


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Babu Jayendran has over 30 years experience in Information Systems Auditing and Controls. After completing his B.Sc (Hons) in Mathematics, from the Indian Institute of Technology, Kharagpur, he qualified as a Chartered Accountant and then became a Certified Information Systems Auditor (CISA) in 1985. Babu has also been a speaker at several meetings and seminars on Information Systems Audit in India & abroad.
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The design of the modern and custom-made icon system for communication is based on the Scania symbol, the Scania Sans font family and the product DNA. It has been customised with the goal of unifying the Scania brand experience throughout all applications. The icons enable clear and instantaneous communication for information graphics and sales support. The icons should always be supported by text (except for some UI icons used in web applications). Use the icons for information, never for decoration or as logotypes.
Restrictions: The icons may only be used for their intended purposes. Any restrictions must be followed. The icons may not be modified. Any questions may be submitted to Scania identity helpdesk via<EMAIL_ADDRESS>| No. | Name | Icon | Category | Usage | Restrictions | Downloads | Keywords | Notes | Contact |
|---|---|---|---|---|---|---|---|---|---|
| 85 | Smart transport | Adobe Illustrator SVG Vector PNG Bitmap | Mikael Person | ||||||
| 86 | Date | Date icon. | Adobe Illustrator SVG Vector PNG Bitmap | calendar | NO AI FILE | NOE | |||
| 100 | Rough roads | To illustrate bad/rough road conditions. | Adobe Illustrator SVG Vector PNG Bitmap | tyre | Used in animation for "Fleet maintenance with flexible plans", original colour is orange. | Mikael Jakobsson | |||
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| 103 | Error | Use no.79 Emergency instead. | Use no.79 Emergency instead. | Adobe Illustrator SVG Vector PNG Bitmap | exclamation mark | Used in animation for "Fleet maintenance with flexible plans", original colour is orange. | Mikael Jakobsson | ||
| 104 | Heart | To illustrate that that something is good/in order. | Adobe Illustrator SVG Vector PNG Bitmap | OK | Used in animation for "Fleet maintenance with flexible plans", original colour is orange. | Mikael Jakobsson | |||
| 105 | Caret left | UI | Adobe Illustrator SVG Vector PNG Bitmap | DISH | Miku Kimber | ||||
| 106 | Caret right | UI | Adobe Illustrator SVG Vector PNG Bitmap | DISH | Miku Kimber | ||||
| 107 | Open/upload document | UI | Adobe Illustrator SVG Vector PNG Bitmap | DISH | Miku Kimber | ||||
| 108 | Quotation | Adobe Illustrator SVG Vector PNG Bitmap | DISH | Miku Kimber | |||||
| 109 | Summary | Adobe Illustrator SVG Vector PNG Bitmap | DISH | Miku Kimber | |||||
| 110 | Zoom in | UI | Adobe Illustrator SVG Vector PNG Bitmap | DISH | Miku Kimber | ||||
| 111 | Zoom out | UI | Adobe Illustrator SVG Vector PNG Bitmap | DISH | Miku Kimber | ||||
Under development |
|||||||||
| 112 | News/Newsletter | UI | To illustrate the possibility to recieve a newsletter / Link to newspage ??? | Adobe Illustrator SVG Vector PNG Bitmap | Paper | NO AI FILE | NOE | ||
| Propulsion | Product/service | To illustrate the application Propulsion (marine applications, all-speed engines). | Adobe Illustrator SVG Vector PNG Bitmap | Propeller | NO AI FILE | Johan Sporre | |||
| Auxiliary | Product/service | To illustrate the application Auxiliary (marine applications, single-speed engines). | Adobe Illustrator SVG Vector PNG Bitmap | Flash | NO AI FILE | Johan Sporre | |||
| "Autonomos driving" | Product/service | Adobe Illustrator SVG Vector PNG Bitmap | NO AI FILE | Susanne Ekdahl | |||||
| "Electrified vehicles" | Product/service | Adobe Illustrator SVG Vector PNG Bitmap | NO AI FILE | Susanne Ekdahl |


Float Property of CSS Style is used to put an image on left/right side of the text in same horizon!!
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TextNode 1 - The Scala supported IDE is one of the few pain points of developers who want to start using Scala in their Java project. On existing long term project developed by a team its hard to step in and introduce a new language that is not supported by the existing IDE. On way to go about it is to hid the fact that you use Scala from the Java world by using one way dependency injection. Still, if you wish to truly absorb Scala into your existing java environment then you'll soon introduced cross language dependencies.
Python-Goose: useful #python library for extracting
body text + metadata from a news article or article-type page: https://t.co/OGKBvxxunu
— Derek Greene (@derekgreene) October
30, 2014
Python-Goose: useful #python library for extracting body text + metadata
from a news article or article-type page: https://t.co/OGKBvxxunu
—
Derek Greene (@derekgreene) October 30,
2014


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