All the methods that ara available at jqGrid, jqEdit and jqRender classes can be used within TreeGrid class
boolean deleteTreeNode( array $data )
Delete the data into the database according the table element and tree grid model. Use this method instead of delete method in edit class
A primaryKey should be set. If the key is not set It can be obtained from jqGridDB::getPrimaryKey Return true on success, false when the operation is not succefull
Parameters:
- array $data: associative array which key values correspond to the names in the delete command
API Tags:
Access: public
editTree( [ $summary = null], [ $params = null], [string $oper = false], [ $echo = true] )
Perform the all CRUD operations depending on the oper param send from the grid, the table element and the treegrid model If the primaryKey is not set we try to obtain it using jqGridDB::getPrimaryKey If the primary key is not set or can not be obtained the operation is aborted.
Also the method call the queryTree to perform the tree ouput
Parameters:
- array $summary: - set which columns should be sumarized in order to be displayed to the grid By default this parameter uses SQL SUM function: array("colmodelname"=>"sqlname"); It can be set to use the other one this way array("colmodelname"=>array("sqlname"=>"AVG")); By default the first field correspond to the name of colModel the second to the database name
- array $params: additional parameters that can be passed to the query
- string $oper: if set the requested oper operation is performed without to check the parameter sended from the grid.
- boolean $echo: should the result be echoed or returned
API Tags:
Access: public
array getTableConfig( )
Return the curent configuration for the model used
API Tags:
Access: public
string getTreeModel( )
Returns the currently used tree model
API Tags:
Access: public
boolean insertTreeNode( array $data )
Insert the data array into the database according to the table element and tree model.
A primaryKey should be set. If the key is not set It can be obtained from jqGridDB::getPrimaryKey. The table config for the tree should be set.
Parameters:
- array $data: associative array which key values correspond to the names in the table.
API Tags:
See: setTableConfig Return true on succes, false otherwiese.
Access: public
mixed queryTree( [ $summary = null], [ $params = null], [boolen $echo = true] )
Return the result of the query to jqTreeGrid. Currently does not support searching.
Parameters:
- array $summary: - set which columns should be sumarized in order to be displayed to the grid By default this parameter uses SQL SUM function: array("colmodelname"=>"sqlname"); It can be set to use the other one this way array("colmodelname"=>array("sqlname"=>"AVG")); By default the first field correspond to the name of colModel the second to the database name
- array $params: - parameter values passed to the sql
- boolen $echo: if set to false return the records as object, otherwiese json encoded or xml string depending on the dataType variable
API Tags:
Access: public
mixed. renderTree( [string $tblelement = ''], [string $pager = ''], [boolean $script = true], [ $summary = null], [ $params = null], [boolean $createtbl = false], [boolean $createpg = false], [boolean $echo = true] )
Main method which do allmost everthing for the tree grid.
Construct the tree grid, perform CRUD operations, perform Query set a jqGrid method, and javascript code.
Parameters:
- string $tblelement: the id of the table element to costrict the grid
- string $pager: the id for the pager element
- boolean $script: if set to true add a script tag before constructin the grid.
- array $summary: - set which columns should be sumarized in order to be displayed to the grid By default this parameter uses SQL SUM function: array("colmodelname"=>"sqlname"); It can be set to use other one this way : array("colmodelname"=>array("sqlname"=>"AVG")); By default the first field correspond to the name of colModel the second to the database name
- array $params: parameters passed to the query
- boolean $createtbl: if set to true the table element is created automatically from this method. Default is false
- boolean $createpg: if set to true the pager element is created automatically from this script. Default false.
- boolean $echo: if set to false the function return the string representing the grid
API Tags:
Access: public
setTableConfig( array $aconfig )
Set the configuration fields for the model used. For the adjacency model a 'id' and 'parent' should be set For the nested model 'id', 'left', 'right' and 'level' should be set Optional fields for all models : expanded, icon, leaf, loaded
Parameters:
API Tags:
Access: public
setTreeModel( [string $model = 'nested'] )
Set the tree mode used. Can be adjacency or nested. Deafault is nested
Parameters:
API Tags:
Access: public
boolean updateTreeNode( array $data )
Update the data into the database according the table element A primaryKey should be set. If the key is not set It can be obtained from jqGridDB::getPrimaryKey.Return true on success, false when the operation is not succefull
Parameters:
- array $data: associative array which key values correspond to the names in the table
API Tags:
Access: public


- Account - Class in Files
-
This class links a name with a set of wins, losses and ties
- Account(String) - Constructor for class Files.Account
-
Creates an account with base stats
- AccountManagementPanel - Class in GUI
-
Contains the panel which allows you to create and delete accounts
- AccountManagementPanel(SendChangeSignal) - Constructor for class GUI.AccountManagementPanel
-
Creates new form AccountManagement
- AccountSetPanel - Class in GUI
-
Contains panel which allows you to customize your match in terms of who vs who
- AccountSetPanel(Account[]) - Constructor for class GUI.AccountSetPanel
-
Constructor for the AccountSetPanel
- addPropertyChangeListener(PropertyChangeListener) - Method in interface PropertyChangers.SendChangeSignal
-
- addPropertyChangeListener(PropertyChangeListener) - Method in class PropertyChangers.SendChangeString
-
adds a ProperyChangeListener to listeners


OpenTableClone provides a web app for guests to directly make a reservation at your restaurant. Manage your reservations and information online.
Manage your bookingsAll future reservations:
Ready to start your next project with us? Give us a call or send us an email and we will get back to you as soon as possible!


Using the Skupper CLI
Using the skupper
command-line interface (CLI) allows you to create and manage Skupper sites from the context of the current namespace.
A typical workflow is to create a site, link sites together, and expose services to the service network.
Installing the skupper
CLI
Installing the skupper
command-line interface (CLI) provides a simple method to get started with Skupper.
-
Install the
skupper
command-line interface.For Linux:
$ curl -fL https://github.com/skupperproject/skupper/releases/download/0.5.2/skupper-cli-0.5.2-linux-amd64.tgz | tar -xzf -
For MacOS:
$ curl -fL https://github.com/skupperproject/skupper/releases/download/0.5.2/skupper-cli-0.5.2-mac-amd64.tgz | tar -xzf -
-
Copy the
skupper
executable to a directory in your $PATH:$ mkdir -p $HOME/bin $ export PATH=$PATH:$HOME/bin $ mv skupper $HOME/bin
-
Verify the installation.
$ skupper version client version 0.5.2
Creating a site using the CLI
A service network consists of Skupper sites. This section describes how to create a site using the default settings.
-
The
skupper
CLI is installed. -
You are logged into the cluster.
-
The services you want to expose on the service network are in the active namespace.
-
Create a default site:
$ skupper init
-
Check the site:
$ skupper status Skupper is enabled for namespace "west" in interior mode. It is not connected to any other sites.
The default settings include:
-
console - The Skupper console is provisioned with a single user. The password for the
admin
user is stored in theskupper-console-users
secret. For more information on the console, see Monitoring Skupper sites using the console. -
site name - The site name defaults to the namespace name, for example,
west
.
Custom sites
The default skupper init
creates sites that satisfy typical requirements.
If you require a custom configuration, note the following options:
-
Creating a site without a console:
$ skupper init --enable-console false
-
Configuring console authentication. There are a number of
skupper
options regarding authentication for the console:--console-auth <authentication-mode>
-
Set the authentication mode for the console:
-
openshift
- Use OpenShift authentication, so that users who have permission to log into OpenShift and view the Project (namespace) can view the console. -
internal
- Use Skupper authentication, see theconsole-user
andconsole-password
options. -
unsecured
- No authentication, anyone with the URL can view the console.
-
--console-user <username>
-
Username for the console user when authentication mode is set to
internal
. Defaults toadmin
. --console-password <password>
-
Password for the console user when authentication mode is set to
internal
. If not specified, a random passwords is generated.
Linking sites
A service network consists of Skupper sites. This section describes how to link sites to form a service network.
Linking two sites requires a single initial directional connection. However:
-
Communication between the two sites is bidirectional, only the initial linking is directional.
-
The choice of direction for linking is typically determined by accessibility. For example, if you are linking an OpenShift Dedicated cluster with a CodeReady Containers cluster, you must link from the CodeReady Containers cluster to the OpenShift Dedicated cluster because that route is accessible.
-
Determine the direction of the link. If both clusters are publicly addressable, then the direction is not significant. If one of the clusters is addressable from the other cluster, perform step 2 below on the addressable cluster.
-
Generate a token on the cluster that you want to link to:
$ skupper token create <filename>
where
<filename>
is the name of a YAML file that is saved on your local filesystem.This file contains a key and the location of the site that created it.
Access to this file provides access to the service network. Protect it appropriately. -
Use a token on the cluster that you want to connect from:
-
Create a link to the service network:
$ skupper link create <filename> [-name <link-name>]
where
<filename>
is the name of a YAML file generated from theskupper token create
command and<link-name>
is the name of the link. -
Check the link:
$ skupper link status Connection for conn1 not active
In this example no <link-name> was specified, the name defaulted to
conn1
.
-
-
If you want to delete a link:
$ skupper link delete <link-name>
where
<link-name>
is the name of the link specified during creation.
Exposing services on the service network
After creating a service network, exposed services can communicate across that network.
The skupper
CLI has two options for exposing services:
-
expose
supports simple use cases, for example, a deployment with a single service. See Exposing simple services on the service network for instructions. -
service create
andservice bind
is a more flexible method of exposing services, for example, if you have multiple services for a deployment. See Exposing complex services on the service network for instructions.
Exposing simple services on the service network
This section describes how services can be enabled for a service network for simple use cases.
-
Create a deployment, some pods, or a service in one of your sites, for example to create the backend service for the tutorial:
$ kubectl create deployment hello-world-backend --image quay.io/skupper/hello-world-backend
This step is not Skupper-specific, that is, this process is unchanged from standard processes for your cluster.
-
Create a service that can communicate on the service network:
$ skupper expose [deployment <name>|pods <selector>|statefulset <statefulsetname>|service <name>]
where
-
<name>
is the name of your deployment -
<selector>
is a pod selector -
<statefulsetname>
is the name of a statefulset
For the example deployment in step 1, you create a service using the following command:
$ skupper expose deployment/hello-world-backend --port 8080
Options for this command include:
--port <port-number>
-
Specify the port number that this service is available on the service network.
--target-port <port-number>
-
Specify the port number of pods that you want to expose.
-
--protocol <protocol>
allows you specify the protocol you want to use,tcp
,http
orhttp2
-
-
If you do not specify ports, skupper uses the containerPort value of the deployment.
|
Exposing complex services on the service network
This section describes how services can be enabled for a service network for more complex use cases.
-
Create a deployment, some pods, or a service in one of your sites, for example to create the backend service for the tutorial:
$ kubectl create deployment hello-world-backend --image quay.io/skupper/hello-world-backend
This step is not Skupper-specific, that is, this process is unchanged from standard processes for your cluster.
-
Create a service that can communicate on the service network:
$ skupper service create <name> <port>
where
-
<name>
is the name of the service you want to create -
<port>
is the port the service uses
For the example deployment in step 1, you create a service using the following command:
$ skupper service create hello-world-backend 8080
-
-
Bind the service to a cluster service:
$ skupper service bind <service-name> <target-type> <target-name>
where
-
<service-name>
is the name of the service on the service network -
<target-type>
is the object you want to expose,deployment
,statefulset
,pods
, orservice
. -
<target-name>
is the name of the cluster service -
--protocol <protocol>
allows you specify the protocol you want to use,tcp
,http
orhttp2
For the example deployment in step 1, you bind the service using the following command:
$ skupper service bind hello-world-backend deployment hello-world-backend
-
CLI options for working with different clusters
By default, all skupper
commands apply to the cluster you are logged into and the current namespace.
The following skupper
options allow you to override that behavior and apply to all commands:
--namespace <namespace-name>
-
Apply command to
<namespace-name>
. For example, if you are currently working onfrontend
namespace and want to initialize a site in thebackend
namespace:$ skupper init --namespace backend
--kubeconfig <kubeconfig-path>
-
Path to the kubeconfig file - This allows you run multiple sessions to a cluster from the same client. An alternative is to set the
KUBECONFIG
environment variable. See the tutorial for an example of using kubeconfig files. --context <context-name>
-
The kubeconfig file can contain defined contexts, and this option allows you to use those contexts.


Interested in Contributing?
- Check out available resources.
- Create an account and start submitting your own systems.
Scored Systems
| System | Submitter | System Notes | Constraint | Run Notes | BLEU | BLEU (11b) | BLEU-cased | BLEU-cased (11b) | TER | z BLEU-cased-norm | zz BEER 2.0 | zzz CharactTER |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| uedin-wmt11-en-es (Details) | Edinburgh University of Edinburgh |
no | Run 26 (normalised) |
31.9 |
31.9 |
30.8 |
30.8 |
0.578 |
||||
| uedin-wmt11-en-es (Details) | Edinburgh University of Edinburgh |
no | Run 26 |
31.9 |
31.9 |
30.8 |
30.8 |
0.578 |
||||
| UCH-UPV-2011 (Details) | pakozm Universidad CEU-Cardenal Herrera and Universitat Politécnica de València |
Trained on Europarl+NewsCommentary+News+UnitedNations. Combination of independent phrase-tables over each corpus, plus Neural Network Language Models. NO LDC DATA | no | Linear combination of phrase-tables and rescoring with 6gram Nueral Network Language Model. PRIMARY SUBMISSION |
31.5 |
31.5 |
29.4 |
29.4 |
0.579 |
|||
| UU-baseline (Details) | jorgtied University of Helsinki |
Phrase-based SMT (constrained) | no | standard PBSMT system, EP+News+UN (no LDC). |
31.0 |
31.0 |
29.8 |
29.8 |
0.583 |
|||
| UCH-UPV-2011 (Details) | pakozm Universidad CEU-Cardenal Herrera and Universitat Politécnica de València |
Trained on Europarl+NewsCommentary+News+UnitedNations. Combination of independent phrase-tables over each corpus, plus Neural Network Language Models. NO LDC DATA | no | Smoothing phrase tables by counting CONTRASTIVE SUBMISSION |
30.9 |
30.9 |
28.9 |
28.9 |
0.585 |
|||
| UCH-UPV-2011 (Details) | pakozm Universidad CEU-Cardenal Herrera and Universitat Politécnica de València |
Trained on Europarl+NewsCommentary+News+UnitedNations. Combination of independent phrase-tables over each corpus, plus Neural Network Language Models. NO LDC DATA | no | Linear phrase-tables combination. CONTRASTIVE SUBMISSION |
30.9 |
30.9 |
28.8 |
28.8 |
0.588 |
|||
| UCH-UPV-2011 (Details) | pakozm Universidad CEU-Cardenal Herrera and Universitat Politécnica de València |
Trained on Europarl+NewsCommentary+News+UnitedNations. Combination of independent phrase-tables over each corpus, plus Neural Network Language Models. NO LDC DATA | no | Phrase-based baseline system. Concatenation of all available corpora. CONTRASTIVE SUBMISSION |
30.5 |
30.5 |
28.5 |
28.5 |
0.589 |
|||
| UoW (Details) | wilkeraziz University of Wolverhampton |
Tree-to-string model trained using the entire Europarl-v6. The English side was shallow parsed and annotated with semantic role labels. The semantic labels were used to specialize the vocabulary of non terminals of the shallow parsing. | no | constrained submission / predicate arguments were not used to improve shallow parsing in this run |
30.3 |
30.3 |
29.3 |
29.3 |
0.584 |
|||
| UoW-c (Details) | wilkeraziz University of Wolverhampton |
Tree-to-string model trained using the entire Europarl-v6. The English side was shallow parsed and the chunks were relaxed following ideas of syntax-augmented machine translation (Zollmann and Venugopal). | no | constrained submission |
30.3 |
30.3 |
29.3 |
29.3 |
0.584 |
|||
| PROMT DeepHybrid (Details) | Alexander Molchanov PROMT LLC |
PROMT DeepHybrid is the first version of PROMT hybrid system. The system is based on PROMT RBMT-engine combined with statistical approach. Trained on Europarliament + NewsCommentary. | no | second run (Europarliament + NewsComm), different training parameters |
29.6 |
29.6 |
28.6 |
28.6 |
0.592 |
|||
| UoW (Details) | wilkeraziz University of Wolverhampton |
Tree-to-string model trained using the entire Europarl-v6. The English side was shallow parsed and annotated with semantic role labels. The semantic labels were used to specialize the vocabulary of non terminals of the shallow parsing. | no | [primary], constrained submission |
29.0 |
29.0 |
28.1 |
28.1 |
0.601 |
|||
| TransductiveMoses (Details) | ergunbicici Koc University |
Moses system with training set selected using a transductive learning approach. | no | Secondary |
28.4 |
28.4 |
25.5 |
25.5 |
0.608 |
|||
| TransductiveMoses (Details) | ergunbicici Koc University |
Moses system with training set selected using a transductive learning approach. | no | Primary |
28.3 |
28.3 |
25.5 |
25.5 |
0.606 |
|||
| uk-dan (Details) | zeman Charles University in Prague, ÚFAL |
no | en-es contrastive stc |
27.6 |
27.6 |
26.5 |
26.5 |
0.621 |
||||
| uk-dan (Details) | zeman Charles University in Prague, ÚFAL |
no | en-es primary |
27.4 |
27.4 |
22.1 |
22.1 |
0.621 |
||||
| GTH-UPM Moses with corpus selection (Details) | lapiz Speech Technology Group at UPM |
PRIMARY Moses with train corpus selection - Moses based system - Reordering: msd-bidirectional-fe - Max phrase size 7 - Train corpus selection based on similarity with the source language test - Translation Model: 150.000 best sentences from the global training set (Europarl+UNO+New commentaries) - Languge Model: 1700000 best sentences from the global training set - English corpus: de-contraction “havn’t” -> “have not” - Punctuations marks not included in the training corpus - Post-processing: Numbers and dates formatting - Background vocabulary for no translated words | no | Considering EU+UNO+NC corpus |
21.7 |
21.7 |
20.9 |
20.9 |
0.692 |
|||
| PROMT DeepHybrid (Details) | Alexander Molchanov PROMT LLC |
PROMT DeepHybrid is the first version of PROMT hybrid system. The system is based on PROMT RBMT-engine combined with statistical approach. Trained on Europarliament + NewsCommentary. | no | PROMT DeepHybrid engine (hybrid approach based on PROMT RBMT-engine) |
failed |
28.7 |
failed |
27.7 |
0.603 |
|||
| UCH-UPV-2011 (Details) | pakozm Universidad CEU-Cardenal Herrera and Universitat Politécnica de València |
Trained on Europarl+NewsCommentary+News+UnitedNations. Combination of independent phrase-tables over each corpus, plus Neural Network Language Models. NO LDC DATA | no | Smoothing phrase tables by counting values (similar to resampling), plus NNLMs. CONTRASTIVE SUBMISSION |
failed |
31.4 |
failed |
29.3 |
0.580 |
|||
| uedin-wmt11-en-es (Details) | Edinburgh University of Edinburgh |
no |
failed |
failed |


Check out items
Borrow or renew items for three weeks, and view your checkouts.
You must get a library card to check out items. Fill out the nearby paper form. New cards should be available within a week. Your card will be stored in the filing box.
Find items
Browse and search for items in the public catalog.
Return items
Place items in the slot of the nearby Book Return box. Do not return items to the shelves. Library helpers will check in the items and return them to the shelves.
Help
- Ask a library helper on Sundays (look for a person wearing a badge).
- Write a note and leave it at the library.
- Email the library at<EMAIL_ADDRESS>- Learn more about the library on the church website.


Before the LDM system is started, it needs to be told what data to get, from whom, and how to process it. See LDM Reference: Configuring an LDM Installation for instructions.


Update a Hotel Group
ID of the Hotel Group you want to update
Number of Hotels
Street
Street Number
Postal Code
City


MetaData
Table metadata includes the correspondence between the plain
table name and the encrypted table name because each encrypted
table name is randomly generated. Moreover, for each column
of the original plain table it also includes a column metadata
parameter containing the name and the data type of the corresponding
plain column (e.g., integer, string, timestamp).


Thomas Macaulay . 3 hours ago
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VENTURE CAPITAL
Education, +3
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UD_Tagalog-TRG
|
UD_Tagalog-Ugnayan
|
Tokenization and Word Segmentation
|
Tokenization and Word Segmentation
|
- This corpus contains 128 sentences and 734 tokens.
|
- This corpus contains 94 sentences, 1011 tokens and 1097 syntactic words.
|
- This corpus contains 133 tokens (18%) that are not followed by a space.
|
- This corpus contains 154 tokens (15%) that are not followed by a space.
|
- This corpus does not contain words with spaces.
|
- This corpus does not contain words with spaces.
|
- This corpus does not contain words that contain both letters and punctuation.
|
- This corpus contains 9 types of words that contain both letters and punctuation. Examples: -ng, 't, 'y, Isa-isa, Kapaki-pakinabang, Paulit-ulit, nag-aalaga, nag-isip, nag-party
|
|
|
- This corpus contains 86 multi-word tokens. On average, one multi-word token consists of 2.00 syntactic words.
- There are 49 types of multi-word tokens. Examples: maraming, isang, niyang, ilang, ibang, kakaibang, koryenteng, akong, aleng, dalang, kaniyang, lawang, pang, pinakamahabang, siyang, taong, Anong, Hilagang, Mukhang, Nagmukhang, Puting, Sandaling, bansang, bolang, daang, ding, estudyanteng, inyong, itong, kabutihang, kaming, kasamang, kilalang, limang, mahabang, malaking, malalaking, mapigilang, mong, natitirang, ngayong, ninyong, nitong, pangunahing, payasong, pinakamalaking, probinsyang, sinabing, turistang.
|
Morphology
Tags
- This corpus uses 13 UPOS tags out of 17 possible: ADJ, ADP, ADV, AUX, DET, INTJ, NOUN, PART, PRON, PROPN, PUNCT, SCONJ, VERB
- This corpus does not use the following tags: NUM, CCONJ, SYM, X
|
Morphology
Tags
- This corpus uses 14 UPOS tags out of 17 possible: ADJ, ADP, ADV, CCONJ, DET, INTJ, NOUN, NUM, PART, PRON, PROPN, PUNCT, SCONJ, VERB
- This corpus does not use the following tags: AUX, SYM, X
|
- This corpus contains 7 word types tagged as particles (PART): ano, ba, bang, daw, hindi, kaya, sana
|
- This corpus contains 10 word types tagged as particles (PART): 'y, -ng, ay, ba, e, hindi, na, nang, ng, po
|
- This corpus contains 9 lemmas tagged as pronouns (PRON): ako, ikaw, ito, iyan, kayo, sarili, sila, sino, siya
|
- This corpus contains 13 lemmas tagged as pronouns (PRON): akin, ako, ano, ikaw, inyo, ito, kami, kayo, ko, lahat, sila, siya, tayo
|
- This corpus contains 3 lemmas tagged as determiners (DET): lahat, mga, mismo
|
- This corpus contains 4 lemmas tagged as determiners (DET): bawat, ilan, marami, mga
|
|
|
|
- This corpus contains 1 lemmas tagged as auxiliaries (AUX): huwag
|
- This corpus contains 0 lemmas tagged as auxiliaries (AUX):
|
|
|
|
- This corpus does not use the VerbForm feature.
|
- This corpus does not use the VerbForm feature.
|
Nominal Features
|
Nominal Features
|
|
|
|
- Fem
- ADJ: Komika
- NOUN: Biyuda, maestra
- PROPN: Linda, Maria, Mary, Rosa
|
|
- Masc
- DET: mismo
- NOUN: Biyudo, maestro
- PROPN: Juan, Pedro, John, Bill
|
|
|
|
|
|
|
|
|
|
|
- Plur
- DET: mga
- PRON: nila, sila, kayong, natin
|
|
- Sing
- PRON: ka, ko, niya, siya, kaniyang, kong, siyang, ako, Iyan, ito
|
|
|
|
|
- Dat
- ADP: sa, kay
- PRON: kaniyang, kaniya
|
|
- Gen
- ADP: ng, ni
- PRON: ko, niya, kong, nila, mo, natin, niyang
|
|
- Nom
- ADP: ang, si
- PRON: ka, siya, siyang, ako, sila, Iyan, Sino, ito, kayong, kita
|
|
|
|
|
|
|
|
Degree and Polarity
|
Degree and Polarity
|
|
|
|
- Pos
- ADJ: Mabuti, Malapit, bago, Interesante, Maganda, Masagwa, Matalino, Matamis, Napakataas, Pagod
|
|
|
|
|
- Neg
- AUX: Huwag
- PART: hindi
- VERB: Wala, Walang
|
|
- Pos
- INTJ: Hindi, Oo
- VERB: Mayroon, May, Mayroong
|
|
|
|
|
Verbal Features
|
Verbal Features
|
|
|
|
- Hab
- VERB: Gusto, Kailangan, Ayaw
|
|
- Imp
- VERB: nagluluto, nagtatrabaho, Umuulan, nagiisa, pumapasok, sumasayaw, tumatanda, Binabasa, Bumabasa, Dumarating
|
|
- Perf
- VERB: nagluto, nakita, Inahit, Nagatubili, dumating, tumanggap, yumaman, Binalikan, Binigyan, Binili
|
|
- Prog
- VERB: darating, Aalisan, Aalisin, Ipagaalis, Magaalis, Susulat, Susulatin
|
|
|
|
|
|
|
|
- Imp
- AUX: Huwag
- VERB: Walisan, Bigyan, Magbigay
|
|
- Ind
- VERB: nagluluto, nagluto, nagtatrabaho, darating, nakita, Inahit, Nagatubili, Umuulan, dumating, nagiisa
|
|
|
|
|
|
|
|
|
|
|
- Act
- VERB: nagluluto, nagluto, nagtatrabaho, darating, Nagatubili, Umuulan, dumating, nagiisa, pumapasok, sumasayaw
|
|
|
|
|
|
|
|
- Lfoc
- VERB: Aalisan, Bigyan, Binalikan, Binigyan, Sinalpok, Tinakasan
|
|
- Pass
- VERB: Gusto, nakita, Inahit, Walisan, Aalisin, Ayaw, Binabasa, Binili, Binisita, Ginising
|
|
|
|
|
Pronouns, Determiners, Quantifiers
|
Pronouns, Determiners, Quantifiers
|
|
|
|
- Dem
- ADV: na, roon
- PRON: Iyan, ito
|
|
|
|
|
|
|
|
- Int
- ADJ: Napakaano
- ADV: nasaan, Bakit
- PRON: Sino
- VERB: Naano, Nagano
|
|
- Prs
- PRON: ka, ko, niya, siya, kaniyang, sarili, kong, nila, siyang, ako
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
- 1
- PRON: ko, kong, ako, kita, natin
|
|
|
|
|
- 3
- PRON: niya, siya, kaniyang, nila, siyang, sila, kaniya, niyang
|
|
|
|
|
|
|
|
|
|
|
Other Features
|
Other Features
|
|
|
|
|
|
|
|
|
|
- Link
- Yes
- ADJ: Bagong
- NOUN: batang, diyaryong, lalaking
- PART: bang
- PRON: kaniyang, kong, siyang, kayong, niyang, sariling
- PROPN: Juan
- VERB: Mayroong, Walang
|
|
- PartType
- Des
- Int
- PART: ba, ano, kaya, bang
- Nfh
|
|
Syntax
Auxiliary Verbs and Copula
- This corpus does not contain copulas.
|
Syntax
Auxiliary Verbs and Copula
- This corpus does not contain copulas.
|
- This corpus uses 1 lemmas as auxiliaries (aux). Examples: huwag.
|
- This corpus does not contain auxiliaries.
|
Core Arguments, Oblique Arguments and Adjuncts
Here we consider only relations between verbs (parent) and nouns or pronouns (child).
- nsubj
- VERB--NOUN (1)
- VERB--NOUN-ADP(ang) (30)
- VERB--PRON-ADP(ang) (1)
- VERB--PRON-Gen (3)
- VERB--PRON-Nom (14)
|
Core Arguments, Oblique Arguments and Adjuncts
Here we consider only relations between verbs (parent) and nouns or pronouns (child).
- nsubj
- VERB--NOUN (1)
- VERB--NOUN-ADP(ang) (24)
- VERB--PRON (25)
|
- obj
- VERB--NOUN (5)
- VERB--NOUN-ADP(ng) (21)
- VERB--PRON-ADP(ang) (1)
- VERB--PRON-Gen (1)
- VERB--PRON-Nom (1)
|
- obj
- VERB--NOUN (13)
- VERB--NOUN-ADP(ng) (17)
- VERB--PRON (19)
|
|
|
|
|
|
|
|
|
|
Verbs with Reflexive Core Objects
- This corpus contains 1 lemmas that occur at least once with a reflexive core object (obj or iobj). Examples: ahit sarili
|
|
Relations Overview
- This corpus uses 9 relation subtypes: acl:relcl, compound:redup, iobj:patient, nmod:poss, nsubj:bfoc, nsubj:ifoc, nsubj:lfoc, nsubj:pass, obj:agent
- The following 2 main types are not used alone, they are always subtyped: acl, compound
- The following 18 relation types are not used in this corpus at all: vocative, expl, dislocated, discourse, cop, appos, nummod, amod, clf, conj, cc, fixed, list, parataxis, orphan, goeswith, reparandum, dep
|
Relations Overview
- This corpus does not use relation subtypes.
- The following 13 relation types are not used in this corpus at all: iobj, csubj, expl, dislocated, aux, cop, appos, clf, list, orphan, goeswith, reparandum, dep
|


Meetings
2017
-
Elder Meeting Discusses the LGTBQ Event
Jun 21, 2017
2016
-
Practical Tips for Responding to Those Affected by Human-Trafficking
Apr 19, 2016
-
Pastor Vernon's Elder Prayer
Mar 17, 2016
-
How to Study the Bible
Jan 20, 2016
2015
-
Sabbatical Report
Aug 19, 2015
-
Dechurched Discussion
Aug 5, 2015


public class GameClient
extends java.lang.Object
| Modifier and Type | Class and Description |
|---|---|
class |
GameClient.AlreadyInRoomException
Exception thrown when trying to join a room while already in a room.
|
class |
GameClient.NoSuchRoomException
Exception thrown when trying to access a room that doesn't exist.
|
class |
GameClient.RoomFullException
Exception thrown when trying to join a full room.
|
| Constructor and Description |
|---|
GameClient(processing.core.PApplet parent,
java.lang.String host,
int port) |
| Modifier and Type | Method and Description |
|---|---|
RoomInfo |
autojoinRoom(int capacity)
Join any room that isn't full, or create a new room if all rooms are full.
|
void |
broadcastMessage(processing.data.JSONObject message)
Send a message to all clients in the same room as this client, including this client.
|
boolean |
connected()
Check if this client is currently connected to the server.
|
int |
createRoom(int capacity)
Create a new room.
|
void |
disconnect()
Let the server know that we're disconnecting, and disconnect.
|
void |
dispose()
Disconnect if we haven't already.
|
Message[] |
getMessages()
Get all messages received by this client.
|
Message |
getNextMessage()
Get the next message received by this client from the queue.
|
RoomInfo |
getRoomInfo(int roomId)
Get info about a room.
|
RoomInfo[] |
getRoomsInfo()
Get info about all rooms on the server.
|
void |
getServerAttributes()
Get the attributes for the server as a JSONObject.
|
int |
id()
Get the unique identifier for this client.
|
RoomInfo |
joinRoom(int roomId)
Join an existing room.
|
void |
leaveRoom()
Leave the room that this client is currently in.
|
void |
putRoomAttribute(int roomId,
java.lang.String key,
boolean value)
Add a single attribute to a room.
|
void |
putRoomAttribute(int roomId,
java.lang.String key,
double value)
Add a single attribute to a room.
|
void |
putRoomAttribute(int roomId,
java.lang.String key,
float value)
Add a single attribute to a room.
|
void |
putRoomAttribute(int roomId,
java.lang.String key,
int value)
Add a single attribute to a room.
|
void |
putRoomAttribute(int roomId,
java.lang.String key,
processing.data.JSONArray value)
Add a single attribute to a room.
|
void |
putRoomAttribute(int roomId,
java.lang.String key,
processing.data.JSONObject value)
Add a single attribute to a room.
|
void |
putRoomAttribute(int roomId,
java.lang.String key,
java.lang.Long value)
Add a single attribute to a room.
|
void |
putRoomAttribute(int roomId,
java.lang.String key,
java.lang.String value)
Add a single attribute to a room.
|
java.lang.Integer |
roomId()
Get the id of the room that this client is currently in, or null if this client is not in a room.
|
void |
sendMessage(int[] recipientIds,
processing.data.JSONObject message)
Send a message to multiple clients.
|
void |
sendMessage(int recipientId,
processing.data.JSONObject message)
Send a message to a single client.
|
void |
setRoomAttributes(int roomId,
processing.data.JSONObject attributes)
Set the attributes for a room with a new JSONObject.
|
void |
setServerAttributes(processing.data.JSONObject attributes)
Set the attributes for the server with a new JSONObject.
|
public GameClient(processing.core.PApplet parent, java.lang.String host, int port)
parent
- the current sketch (this)host
- the hostname of the serverport
- the port to transfer data overpublic void dispose()
public void disconnect()
public boolean connected()
public int id()
public java.lang.Integer roomId()
public int createRoom(int capacity)
capacity
- the maximum number of clients allowed in the roompublic RoomInfo joinRoom(int roomId)
roomId
- the unique id of the room to joinjava.util.NoSuchElementException
- if no room exists with the given idGameClient.RoomFullException
- if the room is already fullGameClient.AlreadyInRoomException
- if this client is currently in a roompublic void leaveRoom()
public RoomInfo autojoinRoom(int capacity)
capacity
- the maximum number of clients allowed in a new room, if one is createdGameClient.AlreadyInRoomException
- if this client is currently in a roompublic RoomInfo getRoomInfo(int roomId)
roomId
- the unique id of the room to look upjava.util.NoSuchElementException
- if no room exists with the given idpublic RoomInfo[] getRoomsInfo()
public void setRoomAttributes(int roomId, processing.data.JSONObject attributes)
roomId
- the unique id of the room to set attributes forattributes
- the object containing the attributes to set for the roomjava.util.NoSuchElementException
- if no room exists with the given idpublic void putRoomAttribute(int roomId, java.lang.String key, java.lang.String value)
roomId
- the unique id of the room to add an attribute tokey
- the key or name of the attributevalue
- the value of the attributejava.util.NoSuchElementException
- if no room exists with the given idpublic void putRoomAttribute(int roomId, java.lang.String key, int value)
roomId
- the unique id of the room to add an attribute tokey
- the key or name of the attributevalue
- the value of the attributejava.util.NoSuchElementException
- if no room exists with the given idpublic void putRoomAttribute(int roomId, java.lang.String key, boolean value)
roomId
- the unique id of the room to add an attribute tokey
- the key or name of the attributevalue
- the value of the attributejava.util.NoSuchElementException
- if no room exists with the given idpublic void putRoomAttribute(int roomId, java.lang.String key, processing.data.JSONObject value)
roomId
- the unique id of the room to add an attribute tokey
- the key or name of the attributevalue
- the value of the attributejava.util.NoSuchElementException
- if no room exists with the given idpublic void putRoomAttribute(int roomId, java.lang.String key, processing.data.JSONArray value)
roomId
- the unique id of the room to add an attribute tokey
- the key or name of the attributevalue
- the value of the attributeGameClient.NoSuchRoomException
- if no room exists with the given idpublic void putRoomAttribute(int roomId, java.lang.String key, float value)
roomId
- the unique id of the room to add an attribute tokey
- the key or name of the attributevalue
- the value of the attributeGameClient.NoSuchRoomException
- if no room exists with the given idpublic void putRoomAttribute(int roomId, java.lang.String key, double value)
roomId
- the unique id of the room to add an attribute tokey
- the key or name of the attributevalue
- the value of the attributejava.util.NoSuchElementException
- if no room exists with the given idpublic void putRoomAttribute(int roomId, java.lang.String key, java.lang.Long value)
roomId
- the unique id of the room to add an attribute tokey
- the key or name of the attributevalue
- the value of the attributejava.util.NoSuchElementException
- if no room exists with the given idpublic void setServerAttributes(processing.data.JSONObject attributes)
attributes
- the object containing the attributes to set for the roompublic void getServerAttributes()
public void sendMessage(int recipientId, processing.data.JSONObject message)
recipientId
- the unique id of the message recipientmessage
- the body of the messagepublic void sendMessage(int[] recipientIds, processing.data.JSONObject message)
recipientIds
- an array of the unique ids of all message recipientsmessage
- the body of the messagepublic void broadcastMessage(processing.data.JSONObject message)
message
- the body of the messagepublic Message getNextMessage()
public Message[] getMessages()


vignettes/using_embeddings_for_machine_learning_on_clinical_text.Rmd
using_embeddings_for_machine_learning_on_clinical_text.Rmd
library(tidyverse)
library(tidymodels)
#> Warning: package 'tidymodels' was built under R version 3.6.3
#> -- Attaching packages ----------------------------------------------------------------------------------------- tidymodels 0.1.1 --
#> v broom 0.7.0 v recipes 0.1.13
#> v dials 0.0.9 v rsample 0.0.8
#> v infer 0.5.3 v tune 0.1.1
#> v modeldata 0.0.2 v workflows 0.2.0
#> v parsnip 0.1.3 v yardstick 0.0.7
#> Warning: package 'broom' was built under R version 3.6.3
#> Warning: package 'dials' was built under R version 3.6.3
#> Warning: package 'scales' was built under R version 3.6.3
#> Warning: package 'infer' was built under R version 3.6.3
#> Warning: package 'modeldata' was built under R version 3.6.3
#> Warning: package 'parsnip' was built under R version 3.6.3
#> Warning: package 'recipes' was built under R version 3.6.3
#> Warning: package 'rsample' was built under R version 3.6.3
#> Warning: package 'tune' was built under R version 3.6.3
#> Warning: package 'workflows' was built under R version 3.6.3
#> Warning: package 'yardstick' was built under R version 3.6.3
#> -- Conflicts -------------------------------------------------------------------------------------------- tidymodels_conflicts() --
#> x scales::discard() masks purrr::discard()
#> x dplyr::filter() masks stats::filter()
#> x recipes::fixed() masks stringr::fixed()
#> x dplyr::lag() masks stats::lag()
#> x yardstick::spec() masks readr::spec()
#> x recipes::step() masks stats::step()
library(clinspacy)
#> Welcome to clinspacy.
#> By default, this package will install and use miniconda and create a "clinspacy" conda environment.
#> If you want to override this behavior, use clinspacy_init(miniconda = FALSE) and specify an alternative environment using reticulate::use_python() or reticulate::use_conda().
library(runway)
mtsamples = dataset_mtsamples()
Here, we are aiming to predict which of the descriptions refer to ‘Cardiovascular / Pulmonary’ notes so we will convert the outcome into a binary outcome. We will remove all of the predictor variables other than
mtsamples_embeddings =
mtsamples %>%
clinspacy(df_col = 'description',
return_scispacy_embeddings = TRUE,
verbose = FALSE) %>%
bind_clinspacy_embeddings(mtsamples) %>%
mutate(is_cardiology_note =
if_else(medical_specialty == 'Cardiovascular / Pulmonary',
'Yes',
'No')) %>%
select(contains('emb_'), is_cardiology_note) %>%
na.omit()
set.seed(1)
logreg_workflow = workflow() %>%
add_model(logistic_reg() %>% set_engine('glm')) %>%
add_recipe((recipe(is_cardiology_note~., data = mtsamples_embeddings)))
logreg_result =
fit_resamples(logreg_workflow,
resamples = validation_split(data = mtsamples_embeddings, prop = 2/3),
metrics = metric_set(roc_auc, pr_auc),
control = control_resamples(save_pred = TRUE))
logreg_result %>% collect_metrics()
#> # A tibble: 2 x 5
#> .metric .estimator mean n std_err
#> <chr> <chr> <dbl> <int> <dbl>
#> 1 pr_auc binary 0.986 1 NA
#> 2 roc_auc binary 0.870 1 NA
set.seed(1)
rf_workflow = workflow() %>%
add_model(rand_forest(mode = 'classification', trees = 1000) %>%
set_engine('ranger')) %>%
add_recipe((recipe(is_cardiology_note~., data = mtsamples_embeddings)))
rf_result =
fit_resamples(rf_workflow,
resamples = validation_split(data = mtsamples_embeddings, prop = 2/3),
metrics = metric_set(roc_auc, pr_auc),
control = control_resamples(save_pred = TRUE))
rf_result %>% collect_metrics()
#> # A tibble: 2 x 5
#> .metric .estimator mean n std_err
#> <chr> <chr> <dbl> <int> <dbl>
#> 1 pr_auc binary 0.984 1 NA
#> 2 roc_auc binary 0.837 1 NA
combined_predictions =
bind_rows(
logreg_result %>% collect_predictions() %>% mutate(model_name = 'Logistic regression'),
rf_result %>% collect_predictions() %>% mutate(model_name = 'Random forest')
)
combined_predictions %>%
mutate(is_cardiology_note = if_else(is_cardiology_note == 'Yes', 1, 0)) %>%
threshperf_plot_multi(outcome = 'is_cardiology_note',
prediction = '.pred_Yes',
model = 'model_name')
#> Warning: No fonts detected on your system.


Hartree-Fock (HF) theory is an algorithm for finding an approximative expression for the ground state of a given Hamiltonian. The basic ingredients are
We will show that the Hartree-Fock Hamiltonian \( \hat{h}^{\mathrm{HF}} \) equals our definition of the operator \( \hat{f} \) discussed in connection with the new definition of the normal-ordered Hamiltonian (see later lectures), that is we have, for a specific matrix element $$ \langle p |\hat{h}^{\mathrm{HF}}| q \rangle =\langle p |\hat{f}| q \rangle=\langle p|\hat{t}+\hat{u}_{\mathrm{ext}}|q \rangle +\sum_{i\le F} \langle pi | \hat{V} | qi\rangle_{AS}, $$ meaning that $$ \langle p|\hat{u}^{\mathrm{HF}}|q\rangle = \sum_{i\le F} \langle pi | \hat{V} | qi\rangle_{AS}. $$ The so-called Hartree-Fock potential \( \hat{u}^{\mathrm{HF}} \) brings an explicit medium dependence due to the summation over all single-particle states below the Fermi level \( F \). It brings also in an explicit dependence on the two-body interaction (in nuclear physics we can also have complicated three- or higher-body forces). The two-body interaction, with its contribution from the other bystanding fermions, creates an effective mean field in which a given fermion moves, in addition to the external potential \( \hat{u}_{\mathrm{ext}} \) which confines the motion of the fermion. For systems like nuclei, there is no external confining potential. Nuclei are examples of self-bound systems, where the binding arises due to the intrinsic nature of the strong force. For nuclear systems thus, there would be no external one-body potential in the Hartree-Fock Hamiltonian.
An operator is defined as \( \hat{O} \) throughout. Unless otherwise specified the number of particles is always \( N \) and \( d \) is the dimension of the system. In nuclear physics we normally define the total number of particles to be \( A=N+Z \), where \( N \) is total number of neutrons and \( Z \) the total number of protons. In case of other baryons such isobars \( \Delta \) or various hyperons such as \( \Lambda \) or \( \Sigma \), one needs to add their definitions. Hereafter, \( N \) is reserved for the total number of particles, unless otherwise specificied.
The quantum numbers of a single-particle state in coordinate space are defined by the variable $$ x=({\bf r},\sigma), $$ where $$ {\bf r}\in {\mathbb{R}}^{d}, $$ with \( d=1,2,3 \) represents the spatial coordinates and \( \sigma \) is the eigenspin of the particle. For fermions with eigenspin \( 1/2 \) this means that $$ x\in {\mathbb{R}}^{d}\oplus (\frac{1}{2}), $$ and the integral \( \int dx = \sum_{\sigma}\int d^dr = \sum_{\sigma}\int d{\bf r} \), and $$ \int d^Nx= \int dx_1\int dx_2\dots\int dx_N. $$
The quantum mechanical wave function of a given state with quantum numbers \( \lambda \) (encompassing all quantum numbers needed to specify the system), ignoring time, is $$ \Psi_{\lambda}=\Psi_{\lambda}(x_1,x_2,\dots,x_N), $$ with \( x_i=({\bf r}_i,\sigma_i) \) and the projection of \( \sigma_i \) takes the values \( \{-1/2,+1/2\} \) for particles with spin \( 1/2 \). We will hereafter always refer to \( \Psi_{\lambda} \) as the exact wave function, and if the ground state is not degenerate we label it as $$ \Psi_0=\Psi_0(x_1,x_2,\dots,x_N). $$
Since the solution \( \Psi_{\lambda} \) seldomly can be found in closed form, approximations are sought. Here we define an approximative wave function or an ansatz to the exact wave function as $$ \Phi_{\lambda}=\Phi_{\lambda}(x_1,x_2,\dots,x_N), $$ with $$ \Phi_0=\Phi_0(x_1,x_2,\dots,x_N), $$ being the ansatz to the ground state.
The wave function \( \Psi_{\lambda} \) is sought in the Hilbert space of either symmetric or anti-symmetric \( N \)-body functions, namely $$ \Psi_{\lambda}\in {\cal H}_N:= {\cal H}_1\oplus{\cal H}_1\oplus\dots\oplus{\cal H}_1, $$ where the single-particle Hilbert space \( \hat{H}_1 \) is the space of square integrable functions over \( \in {\mathbb{R}}^{d}\oplus (\sigma) \) resulting in $$ {\cal H}_1:= L^2(\mathbb{R}^{d}\oplus (\sigma)). $$
Our Hamiltonian is invariant under the permutation (interchange) of two particles. Since we deal with fermions however, the total wave function is antisymmetric. Let \( \hat{P} \) be an operator which interchanges two particles. Due to the symmetries we have ascribed to our Hamiltonian, this operator commutes with the total Hamiltonian, $$ [\hat{H},\hat{P}] = 0, $$ meaning that \( \Psi_{\lambda}(x_1, x_2, \dots , x_N) \) is an eigenfunction of \( \hat{P} \) as well, that is $$ \hat{P}_{ij}\Psi_{\lambda}(x_1, x_2, \dots,x_i,\dots,x_j,\dots,x_N)= \beta\Psi_{\lambda}(x_1, x_2, \dots,x_j,\dots,x_i,\dots,x_N), $$ where \( \beta \) is the eigenvalue of \( \hat{P} \). We have introduced the suffix \( ij \) in order to indicate that we permute particles \( i \) and \( j \). The Pauli principle tells us that the total wave function for a system of fermions has to be antisymmetric, resulting in the eigenvalue \( \beta = -1 \).
The Schrodinger equation reads $$ \begin{equation} \hat{H}(x_1, x_2, \dots , x_N) \Psi_{\lambda}(x_1, x_2, \dots , x_N) = E_\lambda \Psi_\lambda(x_1, x_2, \dots , x_N), \label{eq:basicSE1} \end{equation} $$ where the vector \( x_i \) represents the coordinates (spatial and spin) of particle \( i \), \( \lambda \) stands for all the quantum numbers needed to classify a given \( N \)-particle state and \( \Psi_{\lambda} \) is the pertaining eigenfunction. Throughout this course, \( \Psi \) refers to the exact eigenfunction, unless otherwise stated.
We write the Hamilton operator, or Hamiltonian, in a generic way $$ \hat{H} = \hat{T} + \hat{V} $$ where \( \hat{T} \) represents the kinetic energy of the system $$ \hat{T} = \sum_{i=1}^N \frac{\mathbf{p}_i^2}{2m_i} = \sum_{i=1}^N \left( -\frac{\hbar^2}{2m_i} \mathbf{\nabla_i}^2 \right) = \sum_{i=1}^N t(x_i) $$ while the operator \( \hat{V} \) for the potential energy is given by $$ \begin{equation} \hat{V} = \sum_{i=1}^N \hat{u}_{\mathrm{ext}}(x_i) + \sum_{j < i=1}^N v(x_i,x_j)+\sum_{i < j < k=1}^Nv(x_i,x_j,x_k)+\dots \label{eq:firstv} \end{equation} $$ Hereafter we use natural units, viz. \( \hbar=c=e=1 \), with \( e \) the elementary charge and \( c \) the speed of light. This means that momenta and masses have dimension energy.
If one does quantum chemistry, after having introduced the Born-Oppenheimer approximation which effectively freezes out the nucleonic degrees of freedom, the Hamiltonian for \( N=n_e \) electrons takes the following form $$ \hat{H} = \sum_{i=1}^{n_e} t(x_i) - \sum_{i=1}^{n_e} k\frac{Z}{r_i} + \sum_{i < j}^{n_e} \frac{k}{r_{ij}}, $$ with \( k=1.44 \) eVnm
We can rewrite this as $$ \begin{equation} \hat{H} = \hat{H}_0 + \hat{H}_I = \sum_{i=1}^{n_e}\hat{h}_0(x_i) + \sum_{i < j}^{n_e}\frac{1}{r_{ij}}, \label{H1H2} \end{equation} $$ where we have defined $$ r_{ij}=| {\bf r}_i-{\bf r}_j|, $$ and $$ \begin{equation} \hat{h}_0(x_i) = \hat{t}(x_i) - \frac{Z}{x_i}. \label{hi} \end{equation} $$ The first term of Eq. \eqref{H1H2}, \( H_0 \), is the sum of the \( N \) one-body Hamiltonians \( \hat{h}_0 \). Each individual Hamiltonian \( \hat{h}_0 \) contains the kinetic energy operator of an electron and its potential energy due to the attraction of the nucleus. The second term, \( H_I \), is the sum of the \( n_e(n_e-1)/2 \) two-body interactions between each pair of electrons. Note that the double sum carries a restriction \( i < j \).
The potential energy term due to the attraction of the nucleus defines the onebody field \( u_i=u_{\mathrm{ext}}(x_i) \) of Eq. \eqref{eq:firstv}. We have moved this term into the \( \hat{H}_0 \) part of the Hamiltonian, instead of keeping it in \( \hat{V} \) as in Eq. \eqref{eq:firstv}. The reason is that we will hereafter treat \( \hat{H}_0 \) as our non-interacting Hamiltonian. For a many-body wavefunction \( \Phi_{\lambda} \) defined by an appropriate single-particle basis, we may solve exactly the non-interacting eigenvalue problem $$ \hat{H}_0\Phi_{\lambda}= w_{\lambda}\Phi_{\lambda}, $$ with \( w_{\lambda} \) being the non-interacting energy. This energy is defined by the sum over single-particle energies to be defined below. For atoms the single-particle energies could be the hydrogen-like single-particle energies corrected for the charge \( Z \). For nuclei and quantum dots, these energies could be given by the harmonic oscillator in three and two dimensions, respectively.
We will assume that the interacting part of the Hamiltonian can be approximated by a two-body interaction. This means that our Hamiltonian is written as $$ \begin{equation} \hat{H} = \hat{H}_0 + \hat{H}_I = \sum_{i=1}^N \hat{h}_0(x_i) + \sum_{i < j}^N V(r_{ij}), \label{Hnuclei} \end{equation} $$ with $$ \begin{equation} H_0=\sum_{i=1}^N \hat{h}_0(x_i) = \sum_{i=1}^N\left(\hat{t}(x_i) + \hat{u}_{\mathrm{ext}}(x_i)\right). \label{hinuclei} \end{equation} $$ The onebody part \( u_{\mathrm{ext}}(x_i) \) is normally approximated by a harmonic oscillator potential or the Coulomb interaction an electron feels from the nucleus. However, other potentials are fully possible, such as one derived from the self-consistent solution of the Hartree-Fock equations.
Our Hamiltonian is invariant under the permutation (interchange) of two particles. % (exercise here, prove it) Since we deal with fermions however, the total wave function is antisymmetric. Let \( \hat{P} \) be an operator which interchanges two particles. Due to the symmetries we have ascribed to our Hamiltonian, this operator commutes with the total Hamiltonian, $$ [\hat{H},\hat{P}] = 0, $$ meaning that \( \Psi_{\lambda}(x_1, x_2, \dots , x_N) \) is an eigenfunction of \( \hat{P} \) as well, that is $$ \hat{P}_{ij}\Psi_{\lambda}(x_1, x_2, \dots,x_i,\dots,x_j,\dots,x_N)= \beta\Psi_{\lambda}(x_1, x_2, \dots,x_i,\dots,x_j,\dots,x_N), $$ where \( \beta \) is the eigenvalue of \( \hat{P} \). We have introduced the suffix \( ij \) in order to indicate that we permute particles \( i \) and \( j \). The Pauli principle tells us that the total wave function for a system of fermions has to be antisymmetric, resulting in the eigenvalue \( \beta = -1 \).
In our case we assume that we can approximate the exact eigenfunction with a Slater determinant $$ \begin{equation} \Phi(x_1, x_2,\dots ,x_N,\alpha,\beta,\dots, \sigma)=\frac{1}{\sqrt{N!}} \left| \begin{array}{ccccc} \psi_{\alpha}(x_1)& \psi_{\alpha}(x_2)& \dots & \dots & \psi_{\alpha}(x_N)\\ \psi_{\beta}(x_1)&\psi_{\beta}(x_2)& \dots & \dots & \psi_{\beta}(x_N)\\ \dots & \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots \\ \psi_{\sigma}(x_1)&\psi_{\sigma}(x_2)& \dots & \dots & \psi_{\sigma}(x_N)\end{array} \right|, \label{eq:HartreeFockDet} \end{equation} $$ where \( x_i \) stand for the coordinates and spin values of a particle \( i \) and \( \alpha,\beta,\dots, \gamma \) are quantum numbers needed to describe remaining quantum numbers.
The single-particle function \( \psi_{\alpha}(x_i) \) are eigenfunctions of the onebody Hamiltonian \( h_i \), that is $$ \hat{h}_0(x_i)=\hat{t}(x_i) + \hat{u}_{\mathrm{ext}}(x_i), $$ with eigenvalues $$ \hat{h}_0(x_i) \psi_{\alpha}(x_i)=\left(\hat{t}(x_i) + \hat{u}_{\mathrm{ext}}(x_i)\right)\psi_{\alpha}(x_i)=\varepsilon_{\alpha}\psi_{\alpha}(x_i). $$ The energies \( \varepsilon_{\alpha} \) are the so-called non-interacting single-particle energies, or unperturbed energies. The total energy is in this case the sum over all single-particle energies, if no two-body or more complicated many-body interactions are present.
Let us denote the ground state energy by \( E_0 \). According to the variational principle we have $$ E_0 \le E[\Phi] = \int \Phi^*\hat{H}\Phi d\mathbf{\tau} $$ where \( \Phi \) is a trial function which we assume to be normalized $$ \int \Phi^*\Phi d\mathbf{\tau} = 1, $$ where we have used the shorthand \( d\mathbf{\tau}=d\mathbf{r}_1d\mathbf{r}_2\dots d\mathbf{r}_N \).
Before we proceed with a more compact representation of a Slater determinant, we would like to repeat some linear algebra properties which will be useful for our derivations of the energy as function of a Slater determinant, Hartree-Fock theory and later the nuclear shell model.
The inverse of a matrix is defined by $$ \mathbf{A}^{-1} \cdot \mathbf{A} = I $$ A unitary matrix \( \mathbf{A} \) is one whose inverse is its adjoint $$ \mathbf{A}^{-1}=\mathbf{A}^{\dagger} $$ A real unitary matrix is called orthogonal and its inverse is equal to its transpose. A hermitian matrix is its own self-adjoint, that is $$ \mathbf{A}=\mathbf{A}^{\dagger}. $$
| Relations | Name | matrix elements |
|---|---|---|
| \( A = A^{T} \) | symmetric | \( a_{ij} = a_{ji} \) |
| \( A = \left (A^{T} \right )^{-1} \) | real orthogonal | \( \sum_k a_{ik} a_{jk} = \sum_k a_{ki} a_{kj} = \delta_{ij} \) |
| \( A = A^{ * } \) | real matrix | \( a_{ij} = a_{ij}^{ * } \) |
| \( A = A^{\dagger} \) | hermitian | \( a_{ij} = a_{ji}^{ * } \) |
| \( A = \left (A^{\dagger} \right )^{-1} \) | unitary | \( \sum_k a_{ik} a_{jk}^{ * } = \sum_k a_{ki}^{ * } a_{kj} = \delta_{ij} \) |
Since we will deal with Fermions (identical and indistinguishable particles) we will form an ansatz for a given state in terms of so-called Slater determinants determined by a chosen basis of single-particle functions.
For a given \( n\times n \) matrix \( \mathbf{A} \) we can write its determinant $$ det(\mathbf{A})=|\mathbf{A}|= \left| \begin{array}{ccccc} a_{11}& a_{12}& \dots & \dots & a_{1n}\\ a_{21}&a_{22}& \dots & \dots & a_{2n}\\ \dots & \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots \\ a_{n1}& a_{n2}& \dots & \dots & a_{nn}\end{array} \right|, $$ in a more compact form as $$ |\mathbf{A}|= \sum_{i=1}^{n!}(-1)^{p_i}\hat{P}_i a_{11}a_{22}\dots a_{nn}, $$ where \( \hat{P}_i \) is a permutation operator which permutes the column indices \( 1,2,3,\dots,n \) and the sum runs over all \( n! \) permutations. The quantity \( p_i \) represents the number of transpositions of column indices that are needed in order to bring a given permutation back to its initial ordering, in our case given by \( a_{11}a_{22}\dots a_{nn} \) here.
A simple \( 2\times 2 \) determinant illustrates this. We have $$ det(\mathbf{A})= \left| \begin{array}{cc} a_{11}& a_{12}\\ a_{21}&a_{22}\end{array} \right|= (-1)^0a_{11}a_{22}+(-1)^1a_{12}a_{21}, $$ where in the last term we have interchanged the column indices \( 1 \) and \( 2 \). The natural ordering we have chosen is \( a_{11}a_{22} \).
With the above we can rewrite our Slater determinant in a more compact form. In the Hartree-Fock method the trial function is the Slater determinant of Eq. \eqref{eq:HartreeFockDet} which can be rewritten as $$ \Phi(x_1,x_2,\dots,x_N,\alpha,\beta,\dots,\nu) = \frac{1}{\sqrt{N!}}\sum_{P} (-)^P\hat{P}\psi_{\alpha}(x_1) \psi_{\beta}(x_2)\dots\psi_{\nu}(x_N)=\sqrt{N!}\hat{A}\Phi_H, $$ where we have introduced the antisymmetrization operator \( \hat{A} \) defined by the summation over all possible permutations of two particles.
It is defined as $$ \begin{equation} \hat{A} = \frac{1}{N!}\sum_{p} (-)^p\hat{P}, \label{antiSymmetryOperator} \end{equation} $$ with \( p \) standing for the number of permutations. We have introduced for later use the so-called Hartree-function, defined by the simple product of all possible single-particle functions $$ \Phi_H(x_1,x_2,\dots,x_N,\alpha,\beta,\dots,\nu) = \psi_{\alpha}(x_1) \psi_{\beta}(x_2)\dots\psi_{\nu}(x_N). $$
Both \( \hat{H}_0 \) and \( \hat{H}_I \) are invariant under all possible permutations of any two particles and hence commute with \( \hat{A} \) $$ \begin{equation} [H_0,\hat{A}] = [H_I,\hat{A}] = 0. \label{commutionAntiSym} \end{equation} $$ Furthermore, \( \hat{A} \) satisfies $$ \begin{equation} \hat{A}^2 = \hat{A}, \label{AntiSymSquared} \end{equation} $$ since every permutation of the Slater determinant reproduces it.
The expectation value of \( \hat{H}_0 \) $$ \int \Phi^*\hat{H}_0\Phi d\mathbf{\tau} = N! \int \Phi_H^*\hat{A}\hat{H}_0\hat{A}\Phi_H d\mathbf{\tau} $$ is readily reduced to $$ \int \Phi^*\hat{H}_0\Phi d\mathbf{\tau} = N! \int \Phi_H^*\hat{H}_0\hat{A}\Phi_H d\mathbf{\tau}, $$ where we have used Eqs. \eqref{commutionAntiSym} and \eqref{AntiSymSquared}. The next step is to replace the antisymmetrization operator by its definition and to replace \( \hat{H}_0 \) with the sum of one-body operators $$ \int \Phi^*\hat{H}_0\Phi d\mathbf{\tau} = \sum_{i=1}^N \sum_{p} (-)^p\int \Phi_H^*\hat{h}_0\hat{P}\Phi_H d\mathbf{\tau}. $$
The integral vanishes if two or more particles are permuted in only one of the Hartree-functions \( \Phi_H \) because the individual single-particle wave functions are orthogonal. We obtain then $$ \int \Phi^*\hat{H}_0\Phi d\mathbf{\tau}= \sum_{i=1}^N \int \Phi_H^*\hat{h}_0\Phi_H d\mathbf{\tau}. $$ Orthogonality of the single-particle functions allows us to further simplify the integral, and we arrive at the following expression for the expectation values of the sum of one-body Hamiltonians $$ \begin{equation} \int \Phi^*\hat{H}_0\Phi d\mathbf{\tau} = \sum_{\mu=1}^N \int \psi_{\mu}^*(\mathbf{r})\hat{h}_0\psi_{\mu}(\mathbf{r}) d\mathbf{r}. \label{H1Expectation} \end{equation} $$
We introduce the following shorthand for the above integral $$ \langle \mu | \hat{h}_0 | \mu \rangle = \int \psi_{\mu}^*(\mathbf{r})\hat{h}_0\psi_{\mu}(\mathbf{r}) d\mathbf{r}, $$ and rewrite Eq. \eqref{H1Expectation} as $$ \begin{equation} \int \Phi^*\hat{H}_0\Phi d\mathbf{\tau} = \sum_{\mu=1}^N \langle \mu | \hat{h}_0 | \mu \rangle. \label{H1Expectation1} \end{equation} $$
The expectation value of the two-body part of the Hamiltonian is obtained in a similar manner. We have $$ \int \Phi^*\hat{H}_I\Phi d\mathbf{\tau} = N! \int \Phi_H^*\hat{A}\hat{H}_I\hat{A}\Phi_H d\mathbf{\tau}, $$ which reduces to $$ \int \Phi^*\hat{H}_I\Phi d\mathbf{\tau} = \sum_{i\le j=1}^N \sum_{p} (-)^p\int \Phi_H^*V(r_{ij})\hat{P}\Phi_H d\mathbf{\tau}, $$ by following the same arguments as for the one-body Hamiltonian.
Because of the dependence on the inter-particle distance \( r_{ij} \), permutations of any two particles no longer vanish, and we get $$ \int \Phi^*\hat{H}_I\Phi d\mathbf{\tau} = \sum_{i < j=1}^N \int \Phi_H^*V(r_{ij})(1-P_{ij})\Phi_H d\mathbf{\tau}. $$ where \( P_{ij} \) is the permutation operator that interchanges particle \( i \) and particle \( j \). Again we use the assumption that the single-particle wave functions are orthogonal.
We obtain $$ \begin{align} \int \Phi^*\hat{H}_I\Phi d\mathbf{\tau} = \frac{1}{2}\sum_{\mu=1}^N\sum_{\nu=1}^N &\left[ \int \psi_{\mu}^*(x_i)\psi_{\nu}^*(x_j)V(r_{ij})\psi_{\mu}(x_i)\psi_{\nu}(x_j) dx_idx_j \right. \label{_auto1}\\ &\left. - \int \psi_{\mu}^*(x_i)\psi_{\nu}^*(x_j) V(r_{ij})\psi_{\nu}(x_i)\psi_{\mu}(x_j) dx_idx_j \right]. \label{H2Expectation} \end{align} $$ The first term is the so-called direct term. It is frequently also called the Hartree term, while the second is due to the Pauli principle and is called the exchange term or just the Fock term. The factor \( 1/2 \) is introduced because we now run over all pairs twice.
The last equation allows us to introduce some further definitions. The single-particle wave functions \( \psi_{\mu}(x) \), defined by the quantum numbers \( \mu \) and \( x \) are defined as the overlap $$ \psi_{\alpha}(x) = \langle x | \alpha \rangle . $$
We introduce the following shorthands for the above two integrals $$ \langle \mu\nu|\hat{v}|\mu\nu\rangle = \int \psi_{\mu}^*(x_i)\psi_{\nu}^*(x_j)V(r_{ij})\psi_{\mu}(x_i)\psi_{\nu}(x_j) dx_idx_j, $$ and $$ \langle \mu\nu|\hat{v}|\nu\mu\rangle = \int \psi_{\mu}^*(x_i)\psi_{\nu}^*(x_j) V(r_{ij})\psi_{\nu}(x_i)\psi_{\mu}(x_j) dx_idx_j. $$
The direct and exchange matrix elements can be brought together if we define the antisymmetrized matrix element $$ \langle \mu\nu|\hat{v}|\mu\nu\rangle_{\mathrm{AS}}= \langle \mu\nu|\hat{v}|\mu\nu\rangle-\langle \mu\nu|\hat{v}|\nu\mu\rangle, $$ or for a general matrix element $$ \langle \mu\nu|\hat{v}|\sigma\tau\rangle_{\mathrm{AS}}= \langle \mu\nu|\hat{v}|\sigma\tau\rangle-\langle \mu\nu|\hat{v}|\tau\sigma\rangle. $$ It has the symmetry property $$ \langle \mu\nu|\hat{v}|\sigma\tau\rangle_{\mathrm{AS}}= -\langle \mu\nu|\hat{v}|\tau\sigma\rangle_{\mathrm{AS}}=-\langle \nu\mu|\hat{v}|\sigma\tau\rangle_{\mathrm{AS}}. $$
The antisymmetric matrix element is also hermitian, implying $$ \langle \mu\nu|\hat{v}|\sigma\tau\rangle_{\mathrm{AS}}= \langle \sigma\tau|\hat{v}|\mu\nu\rangle_{\mathrm{AS}}. $$ With these notations we rewrite Eq. \eqref{H2Expectation} as $$ \begin{equation} \int \Phi^*\hat{H}_I\Phi d\mathbf{\tau} = \frac{1}{2}\sum_{\mu=1}^N\sum_{\nu=1}^N \langle \mu\nu|\hat{v}|\mu\nu\rangle_{\mathrm{AS}}. \label{H2Expectation2} \end{equation} $$
Combining Eqs. \eqref{H1Expectation1} and \eqref{H2Expectation2} we obtain the energy functional $$ \begin{equation} E[\Phi] = \sum_{\mu=1}^N \langle \mu | \hat{h}_0 | \mu \rangle + \frac{1}{2}\sum_{{\mu}=1}^N\sum_{{\nu}=1}^N \langle \mu\nu|\hat{v}|\mu\nu\rangle_{\mathrm{AS}}. \label{FunctionalEPhi} \end{equation} $$ which we will use as our starting point for the Hartree-Fock calculations.
We write our Hamiltonian as a one-body part $$ \hat{H}_0=\sum_{i=1}^{N_e}\left(-{\frac{1}{2}}\nabla^2_{i}+\frac{ \omega^2}{2}r^2_{i} \right), $$ and an interacting part $$ \hat{V}=\sum_{i < j}^{N_e}\frac{1}{|{\bf r}_i-{\bf r}_j|}. $$ The unperturbed part of the Hamiltonian yields the single-particle energies $$ \epsilon_i = \omega\left(2n+|m| + 1\right), $$ where \( n = 0,1,2,3,.. \) and \( m = 0, \pm 1, \pm 2,.. \). The index \( i \) runs from \( 0,1,2,\dots \).
Set up the harmonic oscillator wave function in cartesian coordinates for an electron with \( n_x=n_y=0 \) and find the oscillator energy. Use this result to find the unperturbed energy $$ \int \Phi^*\hat{H_0}\Phi d\mathbf{\tau} = \sum_{\mu=1}^N \langle \mu | h | \mu \rangle. $$ for two electrons with the same quantum numbers.
The calculus of variations involves problems where the quantity to be minimized or maximized is an integral.
In the general case we have an integral of the type $$ E[\Phi]= \int_a^b f(\Phi(x),\frac{\partial \Phi}{\partial x},x)dx, $$ where \( E \) is the quantity which is sought minimized or maximized. The problem is that although \( f \) is a function of the variables \( \Phi \), \( \partial \Phi/\partial x \) and \( x \), the exact dependence of \( \Phi \) on \( x \) is not known. This means again that even though the integral has fixed limits \( a \) and \( b \), the path of integration is not known. In our case the unknown quantities are the single-particle wave functions and we wish to choose an integration path which makes the functional \( E[\Phi] \) stationary. This means that we want to find minima, or maxima or saddle points. In physics we search normally for minima. Our task is therefore to find the minimum of \( E[\Phi] \) so that its variation \( \delta E \) is zero subject to specific constraints. In our case the constraints appear as the integral which expresses the orthogonality of the single-particle wave functions. The constraints can be treated via the technique of Lagrangian multipliers
Let us specialize to the expectation value of the energy for one particle in three-dimensions. This expectation value reads $$ E=\int dxdydz \psi^*(x,y,z) \hat{H} \psi(x,y,z), $$ with the constraint $$ \int dxdydz \psi^*(x,y,z) \psi(x,y,z)=1, $$ and a Hamiltonian $$ \hat{H}=-\frac{1}{2}\nabla^2+V(x,y,z). $$ We will, for the sake of notational convenience, skip the variables \( x,y,z \) below, and write for example \( V(x,y,z)=V \).
The integral involving the kinetic energy can be written as, with the function \( \psi \) vanishing strongly for large values of \( x,y,z \) (given here by the limits \( a \) and \( b \)), $$ \int_a^b dxdydz \psi^* \left(-\frac{1}{2}\nabla^2\right) \psi dxdydz = \psi^*\nabla\psi|_a^b+\int_a^b dxdydz\frac{1}{2}\nabla\psi^*\nabla\psi. $$ We will drop the limits \( a \) and \( b \) in the remaining discussion. Inserting this expression into the expectation value for the energy and taking the variational minimum we obtain $$ \delta E = \delta \left\{\int dxdydz\left( \frac{1}{2}\nabla\psi^*\nabla\psi+V\psi^*\psi\right)\right\} = 0. $$
The constraint appears in integral form as $$ \int dxdydz \psi^* \psi=\mathrm{constant}, $$ and multiplying with a Lagrangian multiplier \( \lambda \) and taking the variational minimum we obtain the final variational equation $$ \delta \left\{\int dxdydz\left( \frac{1}{2}\nabla\psi^*\nabla\psi+V\psi^*\psi-\lambda\psi^*\psi\right)\right\} = 0. $$ We introduce the function \( f \) $$ f = \frac{1}{2}\nabla\psi^*\nabla\psi+V\psi^*\psi-\lambda\psi^*\psi= \frac{1}{2}(\psi^*_x\psi_x+\psi^*_y\psi_y+\psi^*_z\psi_z)+V\psi^*\psi-\lambda\psi^*\psi, $$ where we have skipped the dependence on \( x,y,z \) and introduced the shorthand \( \psi_x \), \( \psi_y \) and \( \psi_z \) for the various derivatives.
For \( \psi^* \) the Euler-Lagrange equations yield $$ \frac{\partial f}{\partial \psi^*}- \frac{\partial }{\partial x}\frac{\partial f}{\partial \psi^*_x}-\frac{\partial }{\partial y}\frac{\partial f}{\partial \psi^*_y}-\frac{\partial }{\partial z}\frac{\partial f}{\partial \psi^*_z}=0, $$ which results in $$ -\frac{1}{2}(\psi_{xx}+\psi_{yy}+\psi_{zz})+V\psi=\lambda \psi. $$ We can then identify the Lagrangian multiplier as the energy of the system. The last equation is nothing but the standard Schroedinger equation and the variational approach discussed here provides a powerful method for obtaining approximate solutions of the wave function.
In deriving the Hartree-Fock equations, we will expand the single-particle functions in a known basis and vary the coefficients, that is, the new single-particle wave function is written as a linear expansion in terms of a fixed chosen orthogonal basis (for example the well-known harmonic oscillator functions or the hydrogen-like functions etc). We define our new Hartree-Fock single-particle basis by performing a unitary transformation on our previous basis (labelled with greek indices) as $$ \begin{equation} \psi_p^{HF} = \sum_{\lambda} C_{p\lambda}\phi_{\lambda}. \label{eq:newbasis} \end{equation} $$ In this case we vary the coefficients \( C_{p\lambda} \). If the basis has infinitely many solutions, we need to truncate the above sum. We assume that the basis \( \phi_{\lambda} \) is orthogonal. A unitary transformation keeps the orthogonality, as discussed in exercise 1 below.
In the previous slide we stated that a unitary transformation keeps the orthogonality, as discussed in exercise 1 below. To see this consider first a basis of vectors \( \mathbf{v}_i \), $$ \mathbf{v}_i = \begin{bmatrix} v_{i1} \\ \dots \\ \dots \\v_{in} \end{bmatrix} $$ We assume that the basis is orthogonal, that is $$ \mathbf{v}_j^T\mathbf{v}_i = \delta_{ij}. $$ An orthogonal or unitary transformation $$ \mathbf{w}_i=\mathbf{U}\mathbf{v}_i, $$ preserves the dot product and orthogonality since $$ \mathbf{w}_j^T\mathbf{w}_i=(\mathbf{U}\mathbf{v}_j)^T\mathbf{U}\mathbf{v}_i=\mathbf{v}_j^T\mathbf{U}^T\mathbf{U}\mathbf{v}_i= \mathbf{v}_j^T\mathbf{v}_i = \delta_{ij}. $$
This means that if the coefficients \( C_{p\lambda} \) belong to a unitary or orthogonal trasformation (using the Dirac bra-ket notation) $$ \vert p\rangle = \sum_{\lambda} C_{p\lambda}\vert\lambda\rangle, $$ orthogonality is preserved, that is \( \langle \alpha \vert \beta\rangle = \delta_{\alpha\beta} \) and \( \langle p \vert q\rangle = \delta_{pq} \).
This propertry is extremely useful when we build up a basis of many-body Stater determinant based states.
Note also that although a basis \( \vert \alpha\rangle \) contains an infinity of states, for practical calculations we have always to make some truncations.
Before we develop the Hartree-Fock equations, there is another very useful property of determinants that we will use both in connection with Hartree-Fock calculations and later shell-model calculations.
Consider the following determinant $$ \left| \begin{array}{cc} \alpha_1b_{11}+\alpha_2sb_{12}& a_{12}\\ \alpha_1b_{21}+\alpha_2b_{22}&a_{22}\end{array} \right|=\alpha_1\left|\begin{array}{cc} b_{11}& a_{12}\\ b_{21}&a_{22}\end{array} \right|+\alpha_2\left| \begin{array}{cc} b_{12}& a_{12}\\b_{22}&a_{22}\end{array} \right| $$
We can generalize this to an \( n\times n \) matrix and have $$ \left| \begin{array}{cccccc} a_{11}& a_{12} & \dots & \sum_{k=1}^n c_k b_{1k} &\dots & a_{1n}\\ a_{21}& a_{22} & \dots & \sum_{k=1}^n c_k b_{2k} &\dots & a_{2n}\\ \dots & \dots & \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots & \dots \\ a_{n1}& a_{n2} & \dots & \sum_{k=1}^n c_k b_{nk} &\dots & a_{nn}\end{array} \right|= \sum_{k=1}^n c_k\left| \begin{array}{cccccc} a_{11}& a_{12} & \dots & b_{1k} &\dots & a_{1n}\\ a_{21}& a_{22} & \dots & b_{2k} &\dots & a_{2n}\\ \dots & \dots & \dots & \dots & \dots & \dots\\ \dots & \dots & \dots & \dots & \dots & \dots\\ a_{n1}& a_{n2} & \dots & b_{nk} &\dots & a_{nn}\end{array} \right| . $$ This is a property we will use in our Hartree-Fock discussions.
We can generalize the previous results, now with all elements \( a_{ij} \) being given as functions of linear combinations of various coefficients \( c \) and elements \( b_{ij} \), $$ \left| \begin{array}{cccccc} \sum_{k=1}^n b_{1k}c_{k1}& \sum_{k=1}^n b_{1k}c_{k2} & \dots & \sum_{k=1}^n b_{1k}c_{kj} &\dots & \sum_{k=1}^n b_{1k}c_{kn}\\ \sum_{k=1}^n b_{2k}c_{k1}& \sum_{k=1}^n b_{2k}c_{k2} & \dots & \sum_{k=1}^n b_{2k}c_{kj} &\dots & \sum_{k=1}^n b_{2k}c_{kn}\\ \dots & \dots & \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots &\dots \\ \sum_{k=1}^n b_{nk}c_{k1}& \sum_{k=1}^n b_{nk}c_{k2} & \dots & \sum_{k=1}^n b_{nk}c_{kj} &\dots & \sum_{k=1}^n b_{nk}c_{kn}\end{array} \right|=det(\mathbf{C})det(\mathbf{B}), $$ where \( det(\mathbf{C}) \) and \( det(\mathbf{B}) \) are the determinants of \( n\times n \) matrices with elements \( c_{ij} \) and \( b_{ij} \) respectively. This is a property we will use in our Hartree-Fock discussions. Convince yourself about the correctness of the above expression by setting \( n=2 \).
With our definition of the new basis in terms of an orthogonal basis we have $$ \psi_p(x) = \sum_{\lambda} C_{p\lambda}\phi_{\lambda}(x). $$ If the coefficients \( C_{p\lambda} \) belong to an orthogonal or unitary matrix, the new basis is also orthogonal. Our Slater determinant in the new basis \( \psi_p(x) \) is written as $$ \frac{1}{\sqrt{A!}} \left| \begin{array}{ccccc} \psi_{p}(x_1)& \psi_{p}(x_2)& \dots & \dots & \psi_{p}(x_A)\\ \psi_{q}(x_1)&\psi_{q}(x_2)& \dots & \dots & \psi_{q}(x_A)\\ \dots & \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots \\ \psi_{t}(x_1)&\psi_{t}(x_2)& \dots & \dots & \psi_{t}(x_A)\end{array} \right|=\frac{1}{\sqrt{A!}} \left| \begin{array}{ccccc} \sum_{\lambda} C_{p\lambda}\phi_{\lambda}(x_1)& \sum_{\lambda} C_{p\lambda}\phi_{\lambda}(x_2)& \dots & \dots & \sum_{\lambda} C_{p\lambda}\phi_{\lambda}(x_A)\\ \sum_{\lambda} C_{q\lambda}\phi_{\lambda}(x_1)&\sum_{\lambda} C_{q\lambda}\phi_{\lambda}(x_2)& \dots & \dots & \sum_{\lambda} C_{q\lambda}\phi_{\lambda}(x_A)\\ \dots & \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots \\ \sum_{\lambda} C_{t\lambda}\phi_{\lambda}(x_1)&\sum_{\lambda} C_{t\lambda}\phi_{\lambda}(x_2)& \dots & \dots & \sum_{\lambda} C_{t\lambda}\phi_{\lambda}(x_A)\end{array} \right|, $$ which is nothing but \( det(\mathbf{C})det(\Phi) \), with \( det(\Phi) \) being the determinant given by the basis functions \( \phi_{\lambda}(x) \).
It is normal to choose a single-particle basis defined as the eigenfunctions of parts of the full Hamiltonian. The typical situation consists of the solutions of the one-body part of the Hamiltonian, that is we have $$ \hat{h}_0\phi_{\lambda}=\epsilon_{\lambda}\phi_{\lambda}. $$ The single-particle wave functions \( \phi_{\lambda}({\bf r}) \), defined by the quantum numbers \( \lambda \) and \( {\bf r} \) are defined as the overlap $$ \phi_{\lambda}({\bf r}) = \langle {\bf r} | \lambda \rangle . $$
In our discussions hereafter we will use our definitions of single-particle states above and below the Fermi (\( F \)) level given by the labels \( ijkl\dots \le F \) for so-called single-hole states and \( abcd\dots > F \) for so-called particle states. For general single-particle states we employ the labels \( pqrs\dots \).
In Eq. \eqref{FunctionalEPhi}, restated here $$ E[\Phi] = \sum_{\mu=1}^A \langle \mu | h | \mu \rangle + \frac{1}{2}\sum_{{\mu}=1}^A\sum_{{\nu}=1}^A \langle \mu\nu|\hat{v}|\mu\nu\rangle_{AS}, $$ we found the expression for the energy functional in terms of the basis function \( \phi_{\lambda}({\bf r}) \). We then varied the above energy functional with respect to the basis functions \( |\mu \rangle \). Now we are interested in defining a new basis defined in terms of a chosen basis as defined in Eq. \eqref{eq:newbasis}. We can then rewrite the energy functional as $$ \begin{equation} E[\Phi^{HF}] = \sum_{i=1}^A \langle i | h | i \rangle + \frac{1}{2}\sum_{ij=1}^A\langle ij|\hat{v}|ij\rangle_{AS}, \label{FunctionalEPhi2} \end{equation} $$ where \( \Phi^{HF} \) is the new Slater determinant defined by the new basis of Eq. \eqref{eq:newbasis}.
Using Eq. \eqref{eq:newbasis} we can rewrite Eq. \eqref{FunctionalEPhi2} as $$ \begin{equation} E[\Psi] = \sum_{i=1}^A \sum_{\alpha\beta} C^*_{i\alpha}C_{i\beta}\langle \alpha | h | \beta \rangle + \frac{1}{2}\sum_{ij=1}^A\sum_{{\alpha\beta\gamma\delta}} C^*_{i\alpha}C^*_{j\beta}C_{i\gamma}C_{j\delta}\langle \alpha\beta|\hat{v}|\gamma\delta\rangle_{AS}. \label{FunctionalEPhi3} \end{equation} $$
We wish now to minimize the above functional. We introduce again a set of Lagrange multipliers, noting that since \( \langle i | j \rangle = \delta_{i,j} \) and \( \langle \alpha | \beta \rangle = \delta_{\alpha,\beta} \), the coefficients \( C_{i\gamma} \) obey the relation $$ \langle i | j \rangle=\delta_{i,j}=\sum_{\alpha\beta} C^*_{i\alpha}C_{i\beta}\langle \alpha | \beta \rangle= \sum_{\alpha} C^*_{i\alpha}C_{i\alpha}, $$ which allows us to define a functional to be minimized that reads $$ \begin{equation} F[\Phi^{HF}]=E[\Phi^{HF}] - \sum_{i=1}^A\epsilon_i\sum_{\alpha} C^*_{i\alpha}C_{i\alpha}. \label{_auto2} \end{equation} $$
Minimizing with respect to \( C^*_{i\alpha} \), remembering that the equations for \( C^*_{i\alpha} \) and \( C_{i\alpha} \) can be written as two independent equations, we obtain $$ \frac{d}{dC^*_{i\alpha}}\left[ E[\Phi^{HF}] - \sum_{j}\epsilon_j\sum_{\alpha} C^*_{j\alpha}C_{j\alpha}\right]=0, $$ which yields for every single-particle state \( i \) and index \( \alpha \) (recalling that the coefficients \( C_{i\alpha} \) are matrix elements of a unitary (or orthogonal for a real symmetric matrix) matrix) the following Hartree-Fock equations $$ \sum_{\beta} C_{i\beta}\langle \alpha | h | \beta \rangle+ \sum_{j=1}^A\sum_{\beta\gamma\delta} C^*_{j\beta}C_{j\delta}C_{i\gamma}\langle \alpha\beta|\hat{v}|\gamma\delta\rangle_{AS}=\epsilon_i^{HF}C_{i\alpha}. $$
We can rewrite this equation as (changing dummy variables) $$ \sum_{\beta} \left\{\langle \alpha | h | \beta \rangle+ \sum_{j}^A\sum_{\gamma\delta} C^*_{j\gamma}C_{j\delta}\langle \alpha\gamma|\hat{v}|\beta\delta\rangle_{AS}\right\}C_{i\beta}=\epsilon_i^{HF}C_{i\alpha}. $$ Note that the sums over greek indices run over the number of basis set functions (in principle an infinite number).
Defining $$ h_{\alpha\beta}^{HF}=\langle \alpha | h | \beta \rangle+ \sum_{j=1}^A\sum_{\gamma\delta} C^*_{j\gamma}C_{j\delta}\langle \alpha\gamma|\hat{v}|\beta\delta\rangle_{AS}, $$ we can rewrite the new equations as $$ \begin{equation} \sum_{\gamma}h_{\alpha\beta}^{HF}C_{i\beta}=\epsilon_i^{HF}C_{i\alpha}. \label{eq:newhf} \end{equation} $$ The latter is nothing but a standard eigenvalue problem.
It suffices to tabulate the matrix elements \( \langle \alpha | h | \beta \rangle \) and \( \langle \alpha\gamma|\hat{v}|\beta\delta\rangle_{AS} \) once and for all. Successive iterations require thus only a look-up in tables over one-body and two-body matrix elements. These details will be discussed below when we solve the Hartree-Fock equations numerically.
Our Hartree-Fock matrix is thus $$ \hat{h}_{\alpha\beta}^{HF}=\langle \alpha | \hat{h}_0 | \beta \rangle+ \sum_{j=1}^A\sum_{\gamma\delta} C^*_{j\gamma}C_{j\delta}\langle \alpha\gamma|\hat{v}|\beta\delta\rangle_{AS}. $$ The Hartree-Fock equations are solved in an iterative waym starting with a guess for the coefficients \( C_{j\gamma}=\delta_{j,\gamma} \) and solving the equations by diagonalization till the new single-particle energies \( \epsilon_i^{\mathrm{HF}} \) do not change anymore by a prefixed quantity.
Normally we assume that the single-particle basis \( |\beta\rangle \) forms an eigenbasis for the operator \( \hat{h}_0 \), meaning that the Hartree-Fock matrix becomes $$ \hat{h}_{\alpha\beta}^{HF}=\epsilon_{\alpha}\delta_{\alpha,\beta}+ \sum_{j=1}^A\sum_{\gamma\delta} C^*_{j\gamma}C_{j\delta}\langle \alpha\gamma|\hat{v}|\beta\delta\rangle_{AS}. $$ The Hartree-Fock eigenvalue problem $$ \sum_{\beta}\hat{h}_{\alpha\beta}^{HF}C_{i\beta}=\epsilon_i^{\mathrm{HF}}C_{i\alpha}, $$ can be written out in a more compact form as $$ \hat{h}^{HF}\hat{C}=\epsilon^{\mathrm{HF}}\hat{C}. $$
The Hartree-Fock equations are, in their simplest form, solved in an iterative way, starting with a guess for the coefficients \( C_{i\alpha} \). We label the coefficients as \( C_{i\alpha}^{(n)} \), where the subscript \( n \) stands for iteration \( n \). To set up the algorithm we can proceed as follows:
Solving the Hartree-Fock eigenvalue problem yields then new eigenvectors \( C_{i\alpha}^{(1)} \) and eigenvalues \( \epsilon_i^{HF(1)} \).
We can rewrite the ground state energy by adding and subtracting \( \hat{u}^{HF}(x_i) \) $$ E_0^{HF} =\langle \Phi_0 | \hat{H} | \Phi_0\rangle = \sum_{i\le F}^A \langle i | \hat{h}_0 +\hat{u}^{HF}| j\rangle+ \frac{1}{2}\sum_{i\le F}^A\sum_{j \le F}^A\left[\langle ij |\hat{v}|ij \rangle-\langle ij|\hat{v}|ji\rangle\right]-\sum_{i\le F}^A \langle i |\hat{u}^{HF}| i\rangle, $$ which results in $$ E_0^{HF} = \sum_{i\le F}^A \varepsilon_i^{HF} + \frac{1}{2}\sum_{i\le F}^A\sum_{j \le F}^A\left[\langle ij |\hat{v}|ij \rangle-\langle ij|\hat{v}|ji\rangle\right]-\sum_{i\le F}^A \langle i |\hat{u}^{HF}| i\rangle. $$ Our single-particle states \( ijk\dots \) are now single-particle states obtained from the solution of the Hartree-Fock equations.
Using our definition of the Hartree-Fock single-particle energies we obtain then the following expression for the total ground-state energy $$ E_0^{HF} = \sum_{i\le F}^A \varepsilon_i - \frac{1}{2}\sum_{i\le F}^A\sum_{j \le F}^A\left[\langle ij |\hat{v}|ij \rangle-\langle ij|\hat{v}|ji\rangle\right]. $$ This form will be used in our discussion of Koopman's theorem.
We have $$ E[\Phi^{\mathrm{HF}}(N)] = \sum_{i=1}^H \langle i | \hat{h}_0 | i \rangle + \frac{1}{2}\sum_{ij=1}^N\langle ij|\hat{v}|ij\rangle_{AS}, $$ where \( \Phi^{\mathrm{HF}}(N) \) is the new Slater determinant defined by the new basis of Eq. \eqref{eq:newbasis} for \( N \) electrons (same \( Z \)). If we assume that the single-particle wave functions in the new basis do not change when we remove one electron or add one electron, we can then define the corresponding energy for the \( N-1 \) systems as $$ E[\Phi^{\mathrm{HF}}(N-1)] = \sum_{i=1; i\ne k}^N \langle i | \hat{h}_0 | i \rangle + \frac{1}{2}\sum_{ij=1;i,j\ne k}^N\langle ij|\hat{v}|ij\rangle_{AS}, $$ where we have removed a single-particle state \( k\le F \), that is a state below the Fermi level.
Calculating the difference $$ E[\Phi^{\mathrm{HF}}(N)]- E[\Phi^{\mathrm{HF}}(N-1)] = \langle k | \hat{h}_0 | k \rangle + \frac{1}{2}\sum_{i=1;i\ne k}^N\langle ik|\hat{v}|ik\rangle_{AS} \frac{1}{2}\sum_{j=1;j\ne k}^N\langle kj|\hat{v}|kj\rangle_{AS}, $$ we obtain $$ E[\Phi^{\mathrm{HF}}(N)]- E[\Phi^{\mathrm{HF}}(N-1)] = \langle k | \hat{h}_0 | k \rangle + \frac{1}{2}\sum_{j=1}^N\langle kj|\hat{v}|kj\rangle_{AS} $$ which is just our definition of the Hartree-Fock single-particle energy $$ E[\Phi^{\mathrm{HF}}(N)]- E[\Phi^{\mathrm{HF}}(N-1)] = \epsilon_k^{\mathrm{HF}} $$
Similarly, we can now compute the difference (we label the single-particle states above the Fermi level as \( abcd > F \)) $$ E[\Phi^{\mathrm{HF}}(N+1)]- E[\Phi^{\mathrm{HF}}(N)]= \epsilon_a^{\mathrm{HF}}. $$ These two equations can thus be used to the electron affinity or ionization energies, respectively. Koopman's theorem states that for example the ionization energy of a closed-shell system is given by the energy of the highest occupied single-particle state. If we assume that changing the number of electrons from \( N \) to \( N+1 \) does not change the Hartree-Fock single-particle energies and eigenfunctions, then Koopman's theorem simply states that the ionization energy of an atom is given by the single-particle energy of the last bound state. In a similar way, we can also define the electron affinities.
As an example, consider a simple model for atomic sodium, Na. Neutral sodium has eleven electrons, with the weakest bound one being confined the \( 3s \) single-particle quantum numbers. The energy needed to remove an electron from neutral sodium is rather small, 5.1391 eV, a feature which pertains to all alkali metals. Having performed a Hartree-Fock calculation for neutral sodium would then allows us to compute the ionization energy by using the single-particle energy for the \( 3s \) states, namely \( \epsilon_{3s}^{\mathrm{HF}} \).
From these considerations, we see that Hartree-Fock theory allows us to make a connection between experimental observables (here ionization and affinity energies) and the underlying interactions between particles. In this sense, we are now linking the dynamics and structure of a many-body system with the laws of motion which govern the system. Our approach is a reductionistic one, meaning that we want to understand the laws of motion in terms of the particles or degrees of freedom which we believe are the fundamental ones. Our Slater determinant, being constructed as the product of various single-particle functions, follows this philosophy.
With similar arguments as in atomic physics, we can now use Hartree-Fock theory to make a link between nuclear forces and separation energies. Changing to nuclear system, we define $$ E[\Phi^{\mathrm{HF}}(A)] = \sum_{i=1}^A \langle i | \hat{h}_0 | i \rangle + \frac{1}{2}\sum_{ij=1}^A\langle ij|\hat{v}|ij\rangle_{AS}, $$ where \( \Phi^{\mathrm{HF}}(A) \) is the new Slater determinant defined by the new basis of Eq. \eqref{eq:newbasis} for \( A \) nucleons, where \( A=N+Z \), with \( N \) now being the number of neutrons and \( Z \) th enumber of protons. If we assume again that the single-particle wave functions in the new basis do not change from a nucleus with \( A \) nucleons to a nucleus with \( A-1 \) nucleons, we can then define the corresponding energy for the \( A-1 \) systems as $$ E[\Phi^{\mathrm{HF}}(A-1)] = \sum_{i=1; i\ne k}^A \langle i | \hat{h}_0 | i \rangle + \frac{1}{2}\sum_{ij=1;i,j\ne k}^A\langle ij|\hat{v}|ij\rangle_{AS}, $$ where we have removed a single-particle state \( k\le F \), that is a state below the Fermi level.
Calculating the difference $$ E[\Phi^{\mathrm{HF}}(A)]- E[\Phi^{\mathrm{HF}}(A-1)] = \langle k | \hat{h}_0 | k \rangle + \frac{1}{2}\sum_{i=1;i\ne k}^A\langle ik|\hat{v}|ik\rangle_{AS}+ \frac{1}{2}\sum_{j=1;j\ne k}^A\langle kj|\hat{v}|kj\rangle_{AS}, $$ which becomes $$ E[\Phi^{\mathrm{HF}}(A)]- E[\Phi^{\mathrm{HF}}(A-1)] = \langle k | \hat{h}_0 | k \rangle + \frac{1}{2}\sum_{j=1}^A\langle kj|\hat{v}|kj\rangle_{AS} $$ which is just our definition of the Hartree-Fock single-particle energy $$ E[\Phi^{\mathrm{HF}}(A)]- E[\Phi^{\mathrm{HF}}(A-1)] = \epsilon_k^{\mathrm{HF}} $$
Similarly, we can now compute the difference (recall that the single-particle states \( abcd > F \)) $$ E[\Phi^{\mathrm{HF}}(A+1)]- E[\Phi^{\mathrm{HF}}(A)]= \epsilon_a^{\mathrm{HF}}. $$ If we then recall that the binding energy differences $$ BE(A)-BE(A-1) \hspace{0.5cm} \mathrm{and} \hspace{0.5cm} BE(A+1)-BE(A), $$ define the separation energies, we see that the Hartree-Fock single-particle energies can be used to define separation energies. We have thus our first link between nuclear forces (included in the potential energy term) and an observable quantity defined by differences in binding energies.
We have thus the following interpretations (if the single-particle field do not change) $$ BE(A)-BE(A-1)\approx E[\Phi^{\mathrm{HF}}(A)]- E[\Phi^{\mathrm{HF}}(A-1)] = \epsilon_k^{\mathrm{HF}}, $$ and $$ BE(A+1)-BE(A)\approx E[\Phi^{\mathrm{HF}}(A+1)]- E[\Phi^{\mathrm{HF}}(A)] = \epsilon_a^{\mathrm{HF}}. $$ If we use \( {}^{16}\mbox{O} \) as our closed-shell nucleus, we could then interpret the separation energy $$ BE(^{16}\mathrm{O})-BE(^{15}\mathrm{O})\approx \epsilon_{0p^{\nu}_{1/2}}^{\mathrm{HF}}, $$ and $$ BE(^{16}\mathrm{O})-BE(^{15}\mathrm{N})\approx \epsilon_{0p^{\pi}_{1/2}}^{\mathrm{HF}}. $$
Similalry, we could interpret $$ BE(^{17}\mathrm{O})-BE(^{16}\mathrm{O})\approx \epsilon_{0d^{\nu}_{5/2}}^{\mathrm{HF}}, $$ and $$ BE(^{17}\mathrm{F})-BE(^{16}\mathrm{O})\approx\epsilon_{0d^{\pi}_{5/2}}^{\mathrm{HF}}. $$ We can continue like this for all \( A\pm 1 \) nuclei where \( A \) is a good closed-shell (or subshell closure) nucleus. Examples are \( {}^{22}\mbox{O} \), \( {}^{24}\mbox{O} \), \( {}^{40}\mbox{Ca} \), \( {}^{48}\mbox{Ca} \), \( {}^{52}\mbox{Ca} \), \( {}^{54}\mbox{Ca} \), \( {}^{56}\mbox{Ni} \), \( {}^{68}\mbox{Ni} \), \( {}^{78}\mbox{Ni} \), \( {}^{90}\mbox{Zr} \), \( {}^{88}\mbox{Sr} \), \( {}^{100}\mbox{Sn} \), \( {}^{132}\mbox{Sn} \) and \( {}^{208}\mbox{Pb} \), to mention some possile cases.
We can thus make our first interpretation of the separation energies in terms of the simplest possible many-body theory. If we also recall that the so-called energy gap for neutrons (or protons) is defined as $$ \Delta S_n= 2BE(N,Z)-BE(N-1,Z)-BE(N+1,Z), $$ for neutrons and the corresponding gap for protons $$ \Delta S_p= 2BE(N,Z)-BE(N,Z-1)-BE(N,Z+1), $$ we can define the neutron and proton energy gaps for \( {}^{16}\mbox{O} \) as $$ \Delta S_{\nu}=\epsilon_{0d^{\nu}_{5/2}}^{\mathrm{HF}}-\epsilon_{0p^{\nu}_{1/2}}^{\mathrm{HF}}, $$ and $$ \Delta S_{\pi}=\epsilon_{0d^{\pi}_{5/2}}^{\mathrm{HF}}-\epsilon_{0p^{\pi}_{1/2}}^{\mathrm{HF}}. $$
Consider a Slater determinant built up of orthogonal single-particle orbitals \( \psi_{\lambda} \), with \( \lambda = 1,2,\dots,A \).
The unitary transformation $$ \psi_a = \sum_{\lambda} C_{a\lambda}\phi_{\lambda}, $$ brings us into the new basis. The new basis has quantum numbers \( a=1,2,\dots,A \).
a) Show that the new basis is orthogonal.
b) Show that the new Slater determinant constructed from the new single-particle wave functions can be written as the determinant based on the previous basis and the determinant of the matrix \( C \).
c) Show that the old and the new Slater determinants are equal up to a complex constant with absolute value unity.
Hint. Hint: \( C \) is a unitary matrix.
We will assume that we can build various Slater determinants using an orthogonal single-particle basis \( \psi_{\lambda} \), with \( \lambda = 1,2,\dots,A \).
The aim of this exercise is to set up specific matrix elements that will turn useful when we start our discussions of the nuclear shell model. In particular you will notice, depending on the character of the operator, that many matrix elements will actually be zero.
Consider three \( A \)-particle Slater determinants \( |\Phi_0 \), \( |\Phi_i^a\rangle \) and \( |\Phi_{ij}^{ab}\rangle \), where the notation means that Slater determinant \( |\Phi_i^a\rangle \) differs from \( |\Phi_0\rangle \) by one single-particle state, that is a single-particle state \( \psi_i \) is replaced by a single-particle state \( \psi_a \). It will later be interpreted as a so-called one-particle-one-hole excitation. Similarly, the Slater determinant \( |\Phi_{ij}^{ab}\rangle \) differs by two single-particle states from \( |\Phi_0\rangle \) and is normally thought of as a two-particle-two-hole excitation.
Define a general onebody operator \( \hat{F} = \sum_{i}^A\hat{f}(x_{i}) \) and a general twobody operator \( \hat{G}=\sum_{i>j}^A\hat{g}(x_{i},x_{j}) \) with \( g \) being invariant under the interchange of the coordinates of particles \( i \) and \( j \). You can use here the results from the second exercise set, exercise 3.
a) Calculate $$ \langle \Phi_0 \vert\hat{F}\vert\Phi_0\rangle, $$ and $$ \langle \Phi_0\vert\hat{G}|\Phi_0\rangle. $$
b) Find thereafter $$ \langle \Phi_0 |\hat{F}|\Phi_i^a\rangle, $$ and $$ \langle \Phi_0|\hat{G}|\Phi_i^a\rangle, $$
c) Finally, find $$ \langle \Phi_0 |\hat{F}|\Phi_{ij}^{ab}\rangle, $$ and $$ \langle \Phi_0|\hat{G}|\Phi_{ij}^{ab}\rangle. $$ What happens with the two-body operator if we have a transition probability of the type $$ \langle \Phi_0|\hat{G}|\Phi_{ijk}^{abc}\rangle, $$ where the Slater determinant to the right of the operator differs by more than two single-particle states?
d) With an orthogonal basis of Slater determinants \( \Phi_{\lambda} \), we can now construct an exact many-body state as a linear expansion of Slater determinants, that is, a given exact state $$ \Psi_i = \sum_{\lambda =0}^{\infty}C_{i\lambda}\Phi_{\lambda}. $$ In all practical calculations the infinity is replaced by a given truncation in the sum.
If you are to compute the expectation value of (at most) a two-body Hamiltonian for the above exact state $$ \langle \Psi_i \vert \hat{H} \vert \Psi_i\rangle, $$ based on the calculations above, which are the only elements which will contribute? (there is no need to perform any calculation here, use your results from exercises a), b), and c)).
These results simplify to a large extent shell-model calculations.
The Hamiltonian for a system of \( N \) electrons confined in a harmonic potential reads $$ \hat{H} = \sum_{i=1}^{N} \frac{\hat{p}_{i}^{2}}{2m}+\sum_{i=1}^{N} \frac{1}{2} m\omega {r}_{i}^{2}+\sum_{i < j} \hat{V}_{ij}, $$ with \( \hat{V}_{ij} \) is the two-body potential whose matrix elements are calculated on fly in your program. See expression below.
The Hartree-Fock algorithm can be broken down as follows. We recall that our Hartree-Fock matrix is $$ \hat{h}_{\alpha\beta}^{HF}=\langle \alpha \vert\hat{h}_0 \vert \beta \rangle+ \sum_{j=1}^N\sum_{\gamma\delta} C^*_{j\gamma}C_{j\delta}\langle \alpha\gamma|V|\beta\delta\rangle_{AS}. $$ Normally we assume that the single-particle basis \( \vert\beta\rangle \) forms an eigenbasis for the operator \( \hat{h}_0 \) (this is our case), meaning that the Hartree-Fock matrix becomes $$ \hat{h}_{\alpha\beta}^{HF}=\epsilon_{\alpha}\delta_{\alpha,\beta}+ \sum_{j=1}^N\sum_{\gamma\delta} C^*_{j\gamma}C_{j\delta}\langle \alpha\gamma|V|\beta\delta\rangle_{AS}. $$ The Hartree-Fock eigenvalue problem $$ \sum_{\beta}\hat{h}_{\alpha\beta}^{HF}C_{i\beta}=\epsilon_i^{\mathrm{HF}}C_{i\alpha}, $$ can be written out in a more compact form as $$ \hat{h}^{HF}\hat{C}=\epsilon^{\mathrm{HF}}\hat{C}. $$
The equations are often rewritten in terms of a so-called density matrix, which is defined as $$ \begin{equation} \rho_{\gamma\delta}=\sum_{i=1}^{N}\langle\gamma|i\rangle\langle i|\delta\rangle = \sum_{i=1}^{N}C_{i\gamma}C^*_{i\delta}. \label{_auto3} \end{equation} $$ It means that we can rewrite the Hartree-Fock Hamiltonian as $$ \hat{h}_{\alpha\beta}^{HF}=\epsilon_{\alpha}\delta_{\alpha,\beta}+ \sum_{\gamma\delta} \rho_{\gamma\delta}\langle \alpha\gamma|V|\beta\delta\rangle_{AS}. $$ It is convenient to use the density matrix since we can precalculate in every iteration the product of two eigenvector components \( C \).


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nmap as a service
This is an HTTP endpoint that informs the client if it has a port open to the world.
Examples:
Check port 22:
/ports{% if secret %}?secret={{ secret }}{% endif %}
Check an arbitrary port:
/ports?port=80{% if secret %}&secret={{ secret }}{% endif %}


our sales
adding new locations .
location name:
minHourlyCustomers:
maxHourlyCustomers:
averageCookies:


Hello, World!
Posted on January 24, 2018 at 13:18
Welcome to Eave-Guilds brand new website! Here you'll see updates, and other cool stuff.
Posted on January 24, 2018 at 13:18
Welcome to Eave-Guilds brand new website! Here you'll see updates, and other cool stuff.


By implementing the base drag and drop class, the manager class will keep track of the element's virtual position, but not actually move anything. You can click on any of the three squares and watch the various events that the Drag and Drop Manager fires in logger window.