chrinide / quantwo

second quantization program

Quantwo

The program transforms an operator expression into an expression in terms of integrals and amplitudes.

Complilation

The program can be compiled using the following commands:

tar -xvf quantwo.tgz
cd quantwo
make

It generates an executable quantwo

make commands

• make : compiles the program
• make clean : removes all object files and executables
• make veryclean : removes all object files, executables, *.pdf, *.aux, *.dvi, *.ps, *.log, *~
• make equation : generates equation.pdf from input.q2
• make depend : analyzes dependencies in src
• make base : creates working environment (see the next paragraph)

Rational numbers can be used instead of double precision numbers by activating the following line in Makefile

CFLAGS := $(CFLAGS) -D _RATIONAL

Execution

To run the program:

quantwo <input-file> [<output-file>]

If no <output-file> is given, the output will be written to <input-file without extension>.tex.

The output file can be transformed to PDF using latex:

pdflatex equation.tex

NOTE: the output filename has to be input.tex or should be changed in file equation.tex. The equation of course can also be copied to some external tex file. The definitions can be found in definitions.tex.

You can create a working environment by creating a symbolic link to the Makefile and running make base:

mkdir PATH_TO_DIR/WORKING_DIR
cd PATH_TO_DIR/WORKING_DIR
ln -s PATH_TO_Q2_DIR/Makefile .
make base

It creates symbolic links to all needed files from quantwo directory.

With make equation you can then create equation.pdf from input.q2-file.

With make equation out=<filename> you can create <filename>.pdf from input.q2-file.

With make equation in=<filename> you can create equation.pdf from <filename>.q2-file.

Command-line parameters

--help (-h) print help message
--verbose [level] (-v [level]) be chatty - print more 

Options

Various options can be changed in a config file params.reg (including a definition of simple new commands (set="newcommand"))

Changing options in input

One can change options in input-file using

<set>,<name>=<value>[,<name>=<value>[,...]]

e.g.,

syntax,beq="\beqa",eeq="\eeqa" 

Input file

The input file has a Latex form.

The equation has to be surrounded by \begin{equation} (or \beq) and \end{equation} (or \eeq) (syntax:beq,eeq).

To insert a comment use % (syntax:comment).

At this stage the following mathematical operators and latex commands are recognized:

(, [, ), ], {, }, +, -, <number>, \frac{<number>}{<number>}, \half, < <bra> |, | <ket> >, \op <operator>, \mu_<integer>,
\dagger, \dg, \tnsr <tensor>, \sum_{<excitation>[,<excitation>[,...]]}, \newcommand{}{}, \Perm{<from>}{<to>} 

Following latex commands will be ignored:

\left, \right, \lk, \rk, &, \\ (syntax:skipop)

All not recognized symbols will be ignored and a warning will be written.

newcommand

A simple version of \newcommand is implemented, which can be used from input file (one per line):

\newcommand{}{}

e.g.,

% replace all \U by (\op T_1 + \op T_2)
\newcommand{\U}{(\op T_1 + \op T_2)} 

newoperator

With \newoperator (or \newop) command one can define custom operators:

\newop{}{}

e.g.,

% Hamiltonian
\newop{H}{\op F + \op W}
% T operator for CCSD
\newop{T}{\op T_1 + \op T_2} 

operators

1. Parts of Hamiltonian: \op H, \op F, \op W and \op X (1-electron perturbation) (hamilton:fock hamilton:flucpot hamilton:perturbation).
2. Bare excitation operators: \op \tau_{<excitation>} (syntax:bexcop), where <excitation> can be, e.g., \mu_1, \nu_3, \mu_2^{-1} (non-conserving), etc. (syntax:excitation).
3. Cluster operators and intermediates: \op <name of operator>_<excitation class>^<addition to name and/or \dagger>, e.g., \op T_2, \op U_3, \op S_4^{X}, \op Z^{bla}_1
   • <name of operator> must not match name of a part of Hamiltonian (see point 1)!
   • Deexcitation operators have to be marked with ^\dagger (or ^\dg), e.g., \op T_2^\dg, \op T_4^{X\dagger}.
4. Non-conserving operators: \op <name of operator>_<excitation class>^<addition to name and/or \dagger and change in particle number>, e.g., \op T_2^{+1} (2 occ.orbs and 3 virt.orbs), \op T_1^{-1} (1 occ.orbs and 0 virt. orbs).
5. (De)Excitation operators with explicitly given types of orbitals: \op <name of operator>_<excitation class_<orbitals to annihilate>^<orbitals to create>>^<addition to name and/or \dagger>, e.g., \op T_{2_{ii}^{aa}} (usual excitation operator), \op T_{2_{iu}^{ua}} (annihilate electrons in closed-shell and active space and create in active and virtual space).

bra/ket

bra and ket can be

1. reference state: 0, \Phi_0 or HF (all have the same meaning) (syntax:ref),
2. excited determinant: <excitation> (like in excitation operators ), e.g., \mu_1, \nu_3, etc,
3. contravariant excited determinant: \bracs, \bracd, bract for single, double and triple (quasi-contravariant), respectively,
4. explicitly given excited determinant: \Phi^<set of indices>_<set of indices>, e.g. \Phi^{aa_1}_{ii_{1}}, \Phi^{ab}_{ij}, etc (syntax:csf).

tensors

1. Tensor with excitation indices: \tnsr <name of tensor>_<excitation>^<addition to name> (command:tensor) e.g., \tnsr \Lambda_{\mu_2}.
   <excitation> should match one of <excitation>s in bare excitation operators or excited determinant.
2. Number: \tnsr <name of tensor>^<addition to name> , e.g., \tnsr \omega^X.

sum

Sum over excitations:

\sum_{<excitation>[,<excitation>[,...]]} 

e.g., \sum_{\mu_2}.

<excitation> should match one of <excitation>s in bare excitation operators or excited determinant.

connections

Connection (disconnection) of an expression can be set using _C (_D). E.g.,

(\op T_2^\dg \op W \op T_2 \op T_1)_C 

means that the whole expression has to be connected.

(\op T_2^\dg \op W \op T_2 \op T_1)_D 

means that a part of the equation has to be disconnected. The following equality applies:

(...) = (...)_C + (...)_D

If connection sign is written after ket (|...>_C), the expression inside of bra/ket has to be connected (as in usual Coupled Cluster).

no normal ordering

To switch off the normal ordering set prog,wick=2,noorder=2

noorder=1: only Hamiltonian is not ordered.

In order to replace h_{pq} in the final expressions by f_{pq} set prog,usefock=1.

multireference

In order to produce internally contracted expressions with active orbitals set prog,multiref=1.

Contracted excitation operators can be activated by setting prog,contrexcop=1 (default now).

Examples

Energy expression:

\beq
<0|\op W (\op T_2+\frac{1}{2}\op T_1 \op T_1) |0>
\eeq

LCCD amplitude equation:

\beq
<\mu_2|(\op F + \op W) (1+\op T_2) |0>_C
\eeq

MP2 Lagrangian:

\beq
<0|\op W \op T_2 |0> + \sum_{\mu_2} \tnsr \Lambda_{\mu_2} <\mu_2|\op W +\op F \op T_2 |0>_C
\eeq

Output file

By default, closed-shell expressions in spatial orbitals are generated.

• Occupied indices: ijklmno (syntax:occorb)
• Virtual indices: abcdefgh (syntax:virorb)
• General indices: pqrs (syntax:genorb)
• Active indices: tuvwxyz (syntax:actorb)

For explicit-spin orbitals, lowercase letters are used for alpha-spin orbitals and uppercase letters for beta-spin orbitals.

4-index integrals are written in the chemical notation: (ai|bj) using \tnsr \intg command (command:integral). Indices of amplitudes are written as superscript (for annihilators) and subscript (for creators): \tnsr T^{ij}_{ab}. Amplitudes coming from deexcitation operators have superscript and subscript indices swapped, e.g. \tnsr T^{\snam{\dg}ab}_{ij}.

Permutations

\Perm{<from>}{<to>}

Indices in the expression have to be permuted, the first set of indices is "from", the second is "to", e.g.,

\Perm{ijk}{jki} T^{ijk}_{abc} => T^{jki}_{abc}

It is possible to divide all expressions by the sum of permutations using

act,divide=$(<sum of permutations>)$

e.g., in order to derive spin-summed perturbative triples equations:

act,divide=$(1 - \Perm{abc}{cab})$
\beq
\bract \op H (\op T_2 ) |0>_C
\eeq