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triat_u3 User's Guide


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triat_u3 User's Guide
Chapter 1 - Introduction


1.1 What's triat_u3?

The Fortran and Perl programs in the triat_u3 suite perform calculations using the two-dimensional limit of the Vibron Model. This approach is used to model the bending dynamics of linear, quasilinear and bent molecules, using a model based on a bosonic U(3) Lie algebra. This suite includes several programs, in particular it includes a program to fit a fourteen parameter Hamiltonian to experimental bending energy data. This Hamiltonian is the most general one-, two-, three-, and four-body Hamiltonian operator in the model. The package, includes the optimization program tri_min_T, as well as other utilities to work with the two-dimensional limit of the vibron model.


1.2 Copyright statement

Copyright (2007-) Francisco Perez-Bernal

All Rights Reserved

This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version.

The full text of the GPL is given in /usr/share/common-licenses/GPL or in the FSF website GPL license definition.


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triat_u3 User's Guide
Chapter 2 - Installation guide


The program files can be found as a tgz file (triat_U3_2.X.tgz, where X stands for the version number). The first step is to unpack the file:

     $ tar xzf triat_U3_2.2.1.tgz 
     $ ls
     triat_U3_2.2.1  triat_U3_2.2.1.tgz
     $ cd triat_U3_2.2.1/
     triat_U3_2.2.1$ ls
     bin/  doc/  examples/  NEWS  README  src/

The directory structure is the following

  1. bin: Executable files.

  1. src: Fortran source files and compilation Makefile.

  1. doc: Program documentation (this file) in several formats.

  1. examples: Different directories with examples of application of the programs.


2.1 Program compilation

The present section details the programs compilation procedure. There is also a version of the program compiled statically in the bin directory (files ending with the suffix stat). If you run into trouble during any program compilation you can move forward to Getting Started, Chapter 3 using the provided static version of the programs.

The program compilation in Debian systems should be quite smooth if the libraries BLAS and LAPACK are installed.

These libraries are FORTRAN libraries for numerical linear algebra and can be found in most Linux Distributions. In the Debian (Lenny or Squeeze) distribution they are associated with packages libblas-dev and liblapack-dev or other (atlas library). IMHO, the best option is to install the atlas package. More information about BLAS and LAPACK can be found in the Netlib website. <-- Compiled with minuit subroutine

The third library (only needed for the minimization program) is part of the CERN scientific routines, providing the Minuit minimization tool (See References, Chapter 4). This library in Debian can be installed with the packages libpacklib1-dev and libkernlib1-dev, though it is safer to install the cernlib-base-dev package. -->

Once the libraries are installed the compilation is quite direct.

     triat_U3_2.2.1$ cd src
     triat_U3_2.2.1/src$ ls
     assign_gen.f           enrgy_modelH_U3_tri.o  minuit-cern.f
     avalavec_gen_U3.f      fcn_sub_gen.f          minuit_u3.f
     avalavec_modelH_U3.f   hbldu3_gen.f           readenerg_gfort.f
     change_basis.f         hbldu3_modelH.f        scaleh_gen.f
     check_phase.f          hbldu3_modelH_tri.f    selectvl.f
     chisqre_u3_min_gen.f   hbldu3_modelH_tri.o    so3casimirW2.f
     cpexp.f                intrac.f90             so3casimirW2WB2.f
     disdata.f              ir_intensity.f90       so3casimirW4.f
     enrgy_gen_U3.f         Makefile
     enrgy_modelH_U3_tri.f  maxc.f

As can be seen in the directory Makefile, if the compilation is successful, the executable files are located under the triat_U3_2.2.1/bin directory.

List of executable files:

  1. tri_min_T: Optimization code. Look for the best fit to a set of experimental data.

  1. en_U3_gen_T: Compute general Hamiltonian eigenvalues given a set of parameters.

  1. avec_U3_gen_T: Compute general Hamiltonian eigenvalues and eigenstates given a set of parameters.

  1. en_U3_mh: Compute simplified model Hamiltonian eigenvalues given a set of parameters.

  1. avec_U3_mh: Compute simplified model Hamiltonian eigenvalues and eigenstates given a set of parameters.

  1. minuit_U3_genH: Perl script to launch the minimization program. Help on this script can be obtained executing minuit_U3_genH -h.

  1. expected_val_n: Perl script to compute the expectation value of the number of tau boson. Help on this script can be obtained executing expected_val_n -h.

  1. intensity_IR: Perl script to compute the squared expectation value of the infrared transition operator. Help on this script can be obtained executing intensity_IR -h.

There are also sections in the Makefile (named triat_min_Tstat, energenh_Tstat, and avecgenh_Tstat, etc.), not included in the all label, used to compile a static version of the programs. In case you want compile the program statically, type, for example, make triat_min_Tstat. Please, notice that in this case it is convenient to make a backup of the provided statically linked programs (tri_min_Tstat and en_U3_gen_Tstat) because you will delete them. If you need to compile all the static versions available then run make stats.

If the programs works correctly in your system you can also, running as root user, install them, by default under the directory /usr/local/bin. The associated Perl scripts will also be installed in this directory. The programs are installed in this directory with a suffix indicating the version installed and symbolic links to the original name are also built. It is very convenient to add to the default $PATH variable the /usr/local/bin in case it is not yet in it. This greatly facilitates the minimization program's execution instructions explained in Program Execution, Section 3.1.1.4 and Examples, Section 3.1.2, because in this case the explicit paths provided are not necessary. In order to install it you should do it as superuser or using sudo

     triat_U3_2.2.1/src$ sudo make install
     [sudo] password for sudouser: 
     triat_U3_2.2.1/src$ ls  /usr/local/bin/
     ...
     avec_U3_gen_T
     avec_U3_gen_T_2.2.1
     avec_U3_gen_Tstat
     avec_U3_gen_Tstat_2.2.1
     avec_U3_mh           
     avec_U3_mh_2.2.1
     ...

If you want to remove these files you can also do it executing sudo make uninstall.


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triat_u3 User's Guide
Chapter 3 - Getting Started


3.1 Minimization code tri_min_T

Given a set of experimental bending energies and assignments, the program tri_min_T and the Perl launcher minuit_U3_genH optimize the Hamiltonian parameters to reproduce the experimental information. The optimization is carried out using the minuit-CERN code (See References, Chapter 4).

Under the directory triat_U3_2.2.1/examples two examples of use of this program can be found, one for a linear molecule (fulminic acid fifth normal vibrational degree of freedom) and one for a bent molecule (magnesium hydroxide). We explain in detail the first case and chiefly highlight the differences that arise in the second case.


3.1.1 Input Files and Program Execution

We proceed now to explain the different input files and the way the provided programs are executed. There are three necessary input files. A file with the experimental energy files to be fit, a second file with general information for the program and a third file with the information on the Hamiltonian parameters needed by minuit to accomplish the minimization.


3.1.1.1 Experimental Energy File

The experimental energies have to be included in a file. This file starts with a first line which is an integer equals to the number of available experimental levels. Thereafter the experimental information is given, using the following format:

     energy  error   n_quanta  l_value

The fields n_quanta and l_value indicate the number of quanta of vibration and the vibrational angular momentum that characterize the state. The number of quanta could correspond to the labeling for linear or bent molecules, as indicated later in the input file.

If the experimental error is fixed to zero, then the level in question is included in the display and compared to the calculated values but the result of this comparison is not included in the calculation of the chi-square and standard deviation, thus making no effect in the fitting procedure.


3.1.1.2 General Input File

The general information not referring to Hamiltonian parameters is given in an input file with a NAMELIST structure. It provides the information needed to run the program apart from the information directly given to Minuit for the optimization.

We proceed to check the meaning of the variables are defined in this general input file. A template of it is as follows:

     #
     ## GENERAL INPUT
     #
      &INP0 BENT=.F., DTFL='expdata.dat' / 
      &INP1 N2=140, LMAX=5, VMAX=5, EMINL=.F. /
     
     #
      &INP2 IPRINT=0 /

Variables defined:

  1. BENT: logical type. If .T.(.F.) the molecule is bent(linear) and experimental data are given accordingly.

  1. DTFL: character type. Name of the file with experimental data.

  1. N2: integer type. N value for the totally symmetric U(3) representation. As version 2.1 the maximum possible N value is 4000.

  1. LMAX: integer type. Maximum value of the vibrational angular momentum (SO(2) quantum number) considered.

  1. VMAX: integer type. Highest experimental overtone included in the fit.

  1. EMINL: logical type. If this variable is .T. the spectrum energies are referred to the first eigenvalue for each vibrational angular momentum block. The default behavior is for EMINL=.F., that implies that all the eigenvalues are referred to the energy of the ground state for vibrational angular momentum zero.

  1. IPRINT: integer type. Program output verbosity. Mainly for debugging purposes. Possible values in the interval 0-5.


3.1.1.3 Minuit Parameter Input File

The information provided to Minuit for the algebraic spectroscopic parameter optimization is given in a file. The structure of this file is a three column list including the parameter labels, its initial values and initial minimization steps for the fourteen possible Hamiltonian parameters. This is followed by a statement using the Minuit command fix to indicate what parameter should be kept constant at the initial value. The file ends with the Minuit commands for the optimization. More information can be found in the Fortran Minuit Manual and in the Minuit homepage (See References, Chapter 4).

     SET TITLE
     'MINUIT MINIMIZATION. BENDING DYNAMICS'
     PARAMETERS
     1     'P11 '  6.129D+02         0.1D-02 
     2     'P21 '  9.9D+00           0.1D-02
     3     'P22 '  0.30D+00          0.1D-02
     4     'P23 '  -1.03D+00         0.1D-02 
     5     'P31 '  0.0D+00           0.0D+00
     6     'P32 '  0.0D+00           0.0D+00
     7     'P33 '  0.0D+00           0.0D+00 
     8     'P41 '  0.0D+00           0.0D+00 
     9     'P42 '  0.0D+00           0.0D+00 
     10    'P43 '  0.0D+00           0.0D+00 
     11    'P44 '  0.0D+00           0.0D+00 
     12    'P45 '  0.0D+00           0.0D+00 
     13    'P46 '  0.0D+00           0.0D+00 
     14    'P47 '  0.0D+00           0.0D+00 
     
     fix  5 7 8 9 10 11 12 13 14
     #set err 1.0D-05
     minimize 3000
     call 3
     exit

The parameters have labels Pnm, where n = 1,2,3,4 indicates that the operator is n-th body, and m is an index to distinguish between the different operators of the same order. From the Hamiltonian building subroutine hbldu3_gen.f:

       H = P11 n + 
           P21 n^2 + P22 l^2 + P23 W^2 +  
           P31 n^3 + P32 n·l^2 + P33 (n·W^2 + W^2·n) +
           P41 n^4 + P42 n^2·l^2 + P43 l^4 + P44 l^2·W^2 + 
           P45 (n^2·W^2 + W^2·n^2) + P46 W^4 + 
           P47 (W^2·Wbar^2 + Wbar^2·W^2)/2

where n is the U(2) number operator, l is the vibrational angular momentum, W^2 is the second order Casimir operator associated so SO(3), and Wbar^2 is the Casimir operator of the SObar(3) subalgebra.


3.1.1.4 Program Execution

The Perl script minuit_U3_genH, under the directory triat_U3_2.2.1/bin, is provided as a convenient interface to launch the Fortran optimization program. It creates a temporary directory where the program runs and, in this way, several instances of the program can be executed simultaneously.

In the following

If the script is executed with no argument it displays a brief help message.

     triat_U3_2.2.1/examples/XCNO$ ../../bin/minuit_U3_genH 
     
     Usage: minuit_U3_genH [-vtS] minpar_file input_file output_file
            minuit_U3_genH [-vt] [-e prgfile] minpar_file input_file output_file
      Options:
               t :  do not remove temporary directory
               e :  minimization program
               S :  execute the statically linked program
               v :  verbose output

The important options are -e and -t. The first one implies that the user will provide the name (and path) of the executable program file to be launched, while the second option indicates -mainly for debugging purposes- that the temporary directory tmpdir_XXXX is not going to be removed once the program execution finishes. The option,-s, implies the use of the statically linked program tri_min_Tstat. Finally, the option -v forces a verbose output. If the option -e is absent, the script tries to locate the program and execute it.

The script is invoked with three arguments. The first, minpar_file is the file with the parameter input for Minuit (see Minuit Parameter Input File, Section 3.1.1.3). The second argument, input_file, is the general input (see General Input File, Section 3.1.1.2). The last argument is output_file, the name of the file with the program output. See Examples, Section 3.1.2 to check the application of the program to two cases, one corresponding to a linear molecule and the other to a bent molecule.


3.1.2 Examples


3.1.2.1 Linear Molecule Example: Fulminic Acid Large Amplitude Bending Mode

The files necessary to run this example are located under the directory examples/XCNO

     triat_U3_2.1$ cd examples/XCNO/
     triat_U3_2.1/examples/XCNO$ ls   
     exp_FulminicD_Feb03.dat   minuit_genpar_FulminicD  
     exp_FulminicH_Feb03.dat   minuit_genpar_FulminicH  
     input_U3_minuit_FulminicD output_FulminicD.orig
     input_U3_minuit_FulminicH output_FulminicH.orig

The files exp_FulminicH_Feb03.dat and exp_FulminicD_Feb03.dat contain the available experimental information for the large amplitude bending vibration of fulminic acid and its deuterated isotopologue, and they conform to the standard described in Experimental Energy File, Section 3.1.1.1. The general input file (see General Input File, Section 3.1.1.2) in this case is called input_U3_minuit_FulminicH (and input_U3_minuit_FulminicD in the deuterated molecule case).

The information for the algebraic spectroscopic parameter optimization is provided in files minuit_par_FulminicH and minuit_par_FulminicD for HCNO and DCNO, respectively.

The program is executed invoking the Perl script with the already mentioned three arguments.

The verbose execution of the program for the fulminic acid case for both species is launched with the following commands:

     triat_U3_2.2.1/examples/XCNO$ ../../bin/minuit_U3_genH -v \ 
     > -e ~/triat_U3_2.2.1/bin/tri_min_T \
     > minuit_genpar_FulminicD input_U3_minuit_FulminicD output_FulminicD
     
     Minuit CHISQRE minimization. U(3) General Hamiltonian. 
     Executable program: ~/triat_U3_2.2.1/bin/tri_min_T
     Making temporary directory ... Done.
     Experimental energy file name is: exp_FulminicD_Feb03.dat
     Copying files to temporary directory ... Done.
     Running optimization program ... Done.
     Copying output file from temporary directory ... Done.
     Removing temporary directory ... Done.
     triat_U3_2.2.1/examples/XCNO$
     triat_U3_2.2.1/examples/XCNO$ ../../bin/minuit_U3_genH -vS
     minuit_genpar_FulminicD \
     > input_U3_minuit_FulminicH output_FulminicH
     
     Minuit CHISQRE minimization. U(3) General Hamiltonian. 
     Executable program: ../../bin/tri_min_Tstat 
     Making temporary directory ... Done.
     Experimental energy file name is: exp_FulminicH_Feb03.dat
     Copying files to temporary directory ... Done.
     Running optimization program ... Done.
     Copying output file from temporary directory ... Done.
     Removing temporary directory ... Done.

In the HCNO case the static version of the program is employed. The final part of the output in these two cases, once convergence is reached, is included in files output_Fulminic_HCNO.orig and output_Fulminic_DCNO.orig.

Notice that the quality of the fit is given in terms of the standard deviation and not the rms. To compute the rms of the fit it is necessary to multiply the standard deviation by the square root of the total number of experimental data and divide it between the square root of the total number of experimental data minus the number of free parameters.


3.1.2.2 Bent Molecule Example: Methinophosphide A Excited Electronic State Bending Mode

The files necessary to run this example are located under the directory HCP_A.

     triat_U3_2.2.1$ cd examples/HCP_A/
     triat_U3_2.2.1/examples/HCP_A$ ls   
     exp_HCP_A_bending.dat  minuit_genpar_HCP_A  
     input_U3_minuit_HCP_A  output_HCP_A_orig

The program is executed in the same way than in the linear case

     triat_U3_2.2.1/examples/HCP_A$ ../../bin/minuit_U3_genH -v minuit_genpar_HCP_A \
     > input_U3_minuit_HCP_A output_HCP_A
     
     Minuit CHISQRE minimization. U(3) General Hamiltonian. 
     Executable program: ../../../bin/tri_min_T 
     Making temporary directory ... Done.
     Experimental energy file name is: exp_HCP_A_bending.dat
     Copying files to temporary directory ... Done.
     Running optimization program ... Done.
     Copying output file from temporary directory ... Done.
     Removing temporary directory ... Done.

Last part of the output is saved on file output_HCP_A_orig in order that the use can compare the results.


3.2 Eigenvalue calculation codes

There are three possible codes for the calculation of the Hamiltonian eigenvalues. The first one, en_U3_gen_T is valid for the general, fourteen-parameter, Hamiltonian, while the other two, en_U3_mh and en_U3_mh_trid, are valid for a simplified model Hamiltonian, with only two parameters: the pairing and the number operator.


3.2.1 Eigenvalue calculation code en_U3_gen_T

Under the directory triat_U3_2.2.1/examples/Eigenvalues one example of use of this program can be found.


3.2.1.1 Input Files and Program Execution

The input file for this program is rather simple. An example input file with all parameters equal to zero called input_file is provided. A possible input file is

      10		# N      INPUT FILE 
      0  		# l
      2  		# iopts
      0.0d0	 	# P11
      0.0d0		# P21
      0.0d0  	# P22
      -10.0d0  	# P23
      0.0d0  	# P31
      0.0d0  	# P32
      0.0d0  	# P33
      0.0d0  	# P41
      0.0d0  	# P42
      0.0d0  	# P43
      0.0d0  	# P44
      0.0d0  	# P45
      0.0d0  	# P46
      0.0d0  	# P47

The parameter N is the value of the totally symmetric U(3) representation used to model the system. As version 2.1 the maximum possible N value is 6000. The vibrational angular momentum is given by the l value.

The parameter iopts controls the output of the eigenvalues. Possible values are

For debugging purposes, in the source file enrgy_gen_U3.f there is a variable fixed at zero, called IPRINT. Nonzero positive values increase the level of verbosity of the program.

This Fortran program is directly launched, providing the described input file as the standard input. If we execute the program with the input file input_file_doc given as an example the result is

     triat_U3_2.2.1/examples/Eigenvalues$ ../../bin/en_U3_gen_T < input_file_doc 
        0.0000000000000000     
        380.00000000000011     
        680.00000000000023     
        900.00000000000023     
        1040.0000000000002     
        1100.0000000000002

3.2.1.2 Examples

The files necessary to run this example are located under the directory examples/Eigenvalues

     triat_U3_2.2.1/examples/Eigenvalues$ ls
     input_file  input_file_doc  input_file_HCP

The obtention of the eigenvalues that correspond to the parameters obtained in the fit to the HCP A state are, for zero vibrational angular momentum

     triat_U3_2.2.1/examples/Eigenvalues$ ../../bin/en_U3_gen_T < input_file_HCP 
        0.0000000000000000     
        566.85562373229914     
        1028.5555120308779     
        1484.2632344381639     
        2046.9932672140753     
        2689.7360152596357     
        3382.0393187789541     
        4103.8525588498451     
        4838.4084093582642     
        5569.7433116564898     
        6281.1719539978449     
        6953.5491948507433     
        7562.3703503418556     
        8078.6701738697175     
        8529.0156142688957

3.2.2 Eigenvalue calculation code en_U3_mh

Under the directory triat_U3_2.2.1/examples/Eigenvalues one example of use of this program can be found. The LAPACK subroutine used for diagonalization is DSYEV.


3.2.2.1 Input Files and Program Execution

The input file for this program is rather simple. An example input file with all parameters equal to zero is provided (file input_file_mh). A possible input file is

      10		# N      INPUT FILE 
      0  		# l
      2  		# iopts
      1.0d0	 	# scale
      0.5d0		# xi

The parameter N is the value of the totally symmetric U(3) representation used to model the system. As version 2.2.1 the maximum possible N value is 6000. The vibrational angular momentum is given by the l value.

The parameter iopts controls the output of the eigenvalues. Possible values are

For debugging purposes, in the source file enrgy_modelH_U3_tri.f there is a variable fixed at zero, called IPRINT. Nonzero positive values increase the program verbosity.


3.2.2.2 Examples

The recommended way of computing eigenvalues is making use of the Perl script alg_U3_energy described below in Model and General Hamiltonian Eigenvalues, Section 3.4.3. This Fortran program can also be directly launched, providing the described input file as the standard input. If we execute the program with the provided input file input_file_mh_doc, located in the directory examples/Eigenvalues, the result is

     triat_U3_2.2.1/examples/Eigenvalues$ ../../bin/en_U3_mh < input_file_mh_doc 
        0.0000000000000000     
        2.0025471824580259     
        3.4348964743602903     
        4.7423306229038324     
        6.5330201878030874     
        8.7380263091129393

3.2.3 Eigenvalue calculation code en_U3_mh_trid

Under the directory triat_U3_2.2.1/examples/Eigenvalues one example of use of this program can be found.


3.2.3.1 Input Files and Program Execution

As in the previous case, the input file for this program is quite simple. An example input file with all parameters equal to zero is provided (file input_file_mh). A possible input file is

      10		# N      INPUT FILE 
      0  		# l
      2  		# iopts
      1.0d0	 	# scale
      0.5d0		# xi

Again, the parameter N is the value of the totally symmetric U(3) representation used to model the system. The present program uses for the matrix diagonalization a LAPACK subroutine that takes advantage of the fact that the matrix is tridiagonal and symmetric (subroutine DSTEVX). The computing time is larger compared to the en_U3_mh program, but it is more stable numerically and it permits to reach larger N values. It also allows for the calculation of a limited subset of eigenvalues. As version 2.2.1 the maximum possible N value in this case is 40000. The vibrational angular momentum is given by the l value.

The parameter iopts controls the output of the eigenvalues. Possible values are

For debugging purposes, in the source file enrgy_modelH_U3_tri.f there is a variable fixed at zero, called IPRINT. Nonzero positive values increase the program verbosity.


3.2.3.2 Examples

This Fortran program can be directly launched, providing the described input file as the standard input. It is recommended to use the Perl script alg_U3_energy, decribed in Model and General Hamiltonian Eigenvalues, Section 3.4.3, to compute system eigenvalues. If we execute the program with the provided input file input_file_mh_doc, located in the directory examples/Eigenvalues, the result is

     triat_U3_2.2.1/examples/Eigenvalues$ ../../bin/en_U3_mh_trid < input_file_mh_doc 
        0.0000000000000000     
        2.0025471824580254     
        3.4348964743602899     
        4.7423306229038316     
        6.5330201878030865     
        8.7380263091129429

3.3 Eigenvalues and eigenstates calculation code avec_U3_gen_T

Under the directory triat_U3_2.2.1/examples/Eigenstates examples of use of this program can be found.


3.3.1 Input Files and Program Execution

The input file for this program is the same that for the previous program en_U3_gen_T. An example input file with all parameters equal to zero is provided

      10		# N      INPUT FILE 
      0  		# l
      2  		# iopts
      0.0d0	 	# P11
      0.0d0		# P21
      0.0d0  	# P22
      0.0d0  	# P23
      0.0d0  	# P31
      0.0d0  	# P32
      0.0d0  	# P33
      0.0d0  	# P41
      0.0d0  	# P42
      0.0d0  	# P43
      0.0d0  	# P44
      0.0d0  	# P45
      0.0d0  	# P46
      0.0d0  	# P47

See parameter explanation in Eigenvalue calculation code en_U3_gen_T, Section 3.2.1.

In the source file avalavec_gen_U3.f a variable named TOL, whose default value is TOL = 1.0D-05 that makes equal to zero the square of the components whose absolute value is less than TOL.

For debugging purposes, in the source file avec_gen_U3.f there is a variable fixed at zero, called IPRINT. Nonzero positive values increase the level of verbosity of the program.

This Fortran program is directly launched, providing the described input file as the standard input. Each energy is followed by the corresponding eigenstate expressed in the U(2) and SO(3) basis. The first column is the component, and the second the component squared, followed by the basis state quantum labels. If we execute the program with the input file found above the result is

     triat_U3_2.2.1/examples/Eigenstates$ ../../bin/avec_U3_gen_T \
     > < input_file_doc 
     
      N =    10 , l =   0 , DIM =      6
         Energy =           0.0000    
             Component U(2)         |  n ,  l  >        Component SO(3)        |  v ,  K  >
           0.36932E-01  0.13640E-02 | 10 ,   0 >      0.53532E-17   0.0000     |  5 ,   0 >
          -0.26115      0.68198E-01 |  8 ,   0 >      0.84737E-17   0.0000     |  4 ,   0 >
           0.60309      0.36372     |  6 ,   0 >      0.76209E-16   0.0000     |  3 ,   0 >
          -0.66066      0.43647     |  4 ,   0 >      0.75201E-16   0.0000     |  2 ,   0 >
           0.35314      0.12471     |  2 ,   0 >     -0.65821E-15   0.0000     |  1 ,   0 >
          -0.74448E-01  0.55424E-02 |  0 ,   0 >      -1.0000       1.0000     |  0 ,   0 >
     
         Energy =           380.00    
             Component U(2)         |  n ,  l  >        Component SO(3)        |  v ,  K  >
           0.11964      0.14313E-01 | 10 ,   0 >      0.30352E-15   0.0000     |  5 ,   0 >
          -0.52450      0.27510     |  8 ,   0 >     -0.10342E-15   0.0000     |  4 ,   0 >
           0.46888      0.21985     |  6 ,   0 >     -0.13730E-15   0.0000     |  3 ,   0 >
           0.29962      0.89772E-01 |  4 ,   0 >     -0.25502E-15   0.0000     |  2 ,   0 >
          -0.59486      0.35385     |  2 ,   0 >      -1.0000       1.0000     |  1 ,   0 >
           0.21705      0.47111E-01 |  0 ,   0 >      0.32605E-15   0.0000     |  0 ,   0 >
     
               .................................................................
     
         Energy =           1100.0    
             Component U(2)         |  n ,  l  >        Component SO(3)        |  v ,  K  >
           0.60779      0.36941     | 10 ,   0 >      -1.0000       1.0000     |  5 ,   0 >
           0.42977      0.18470     |  8 ,   0 >     -0.12420E-15   0.0000     |  4 ,   0 >
           0.37219      0.13853     |  6 ,   0 >      0.15755E-15   0.0000     |  3 ,   0 >
           0.33976      0.11544     |  4 ,   0 >      0.35379E-16   0.0000     |  2 ,   0 >
           0.31782      0.10101     |  2 ,   0 >      0.32559E-15   0.0000     |  1 ,   0 >
           0.30151      0.90909E-01 |  0 ,   0 >      0.55052E-16   0.0000     |  0 ,   0 >
     
     triat_U3_2.2.1/examples/Eigenstates$

3.3.2 Examples

The files necessary to run this example are located under the directory examples/Eigenstates

     triat_U3_2.2.1/examples/Eigenstates$ ls
     avec_HCP_A.dat input_file_doc  input_file_HCP

The obtention of the eigenvalues and eigenstates that correspond to the parameters obtained in the fit to the HCP A state are, for zero vibrational angular momentum, included in the file avec_HCP_A.dat. You can reproduce this calculation as follows

     triat_U3_2.2.1/examples/Eigenstates$ ../../bin/avec_U3_gen_T \
     > < input_file_HCP > avec_HCP_A_new.dat

3.4 Calculation of observables of interest

We now proceed to describe a set of Perl scripts provided to facilitate the calculation of different observables of interest. The sample and input files provided for these scripts are located in the directory triat_U3_2.2.1/examples/Observables.


3.4.1 Expectation value of the number of tau bosons operator

The Perl script expected_val_n computes the expectation value of the tau boson number operator with different options. A basic help can be obtained running the script with the -h option.

     triat_U3_2.2.1/examples/Observables$ ../../bin/expected_val_n -h
     
     	Perl script to calculate the normalized expected value of
     the U(2) Casimir n for a given eigenstate u (g.s. -> u = 1) and a xi
     value interval, for all the eigenstates given a xi value, or for the
     eigenstates of a general Hamiltonian.
     
     	The expected value is normalized by the number of bosons N (<n>/N).
     
      by Currix TM
     
     Usage: expected_val_n [-vhS] [-e progpath] -N Nval lval scale xi
            expected_val_n [-vhS] [-e progpath] -E Nval lval scale xi
            expected_val_n [-vhS] [-e progpath] Nval lval scale u ximin ximax xistep
            expected_val_n [-vhS] [-e progpath] [-n] -g inputfile
     
       Options:
         N : expected value of n as a function of v (0, 1, 2, ... ) for all states
         E : expected value of n as a function of energy 
             normalized by N for all states
         S : execute statically linked Fortran program
         v : verbose output
         h : help message
         g : general Hamiltonian (supply name of the parameter file)
         n : normalize energies by N in the general Hamiltonian case or 
             or v by N if -N
         e : provide the executable triat files

The script allows the user to compute the expected value of the number operator for all the eigenstates for a given control parameter xi value [1] For example, to compute the expected value of the number operator in the eigenstates of the zero angular momentum, N = 10, and xi = 0.4 Hamiltonian

     triat_U3_2.2.1/examples/Observables$ ../../bin/expected_val_n -E \
     10 0 1.0 0.4
     0.227411444586968 0.282433724484853
     0.375148937971884 0.209292031824226
     0.489249830006017 0.275091429272299
     0.63853518398931 0.506402309333926
     0.831820286396651 0.740247061857603
     1.06005653927139 0.986533443227093

If the dependence with the number of quanta, instead of the energy, is required, the -E option is replaced by -N.

The expected value for a particular eigenstate and evaluated in an interval of control parameter values can also be computed[2]. For example, to compute the expected value of n for the ground state of the zero angular momentum, N = 10, and 0.2 < xi < 0.4 Hamiltonian with a step of 0.05 in the control parameter

     triat_U3_2.2.1/examples/Observables$ ../../bin/expected_val_n \
      10 0 1.0 1 0.2 0.4 0.05
     0.2 0.0395832133992901
     0.25 0.094015233850396
     0.3 0.168639783188264
     0.35 0.233906937667085
     0.4 0.282433724484853

Finally, the last option is to compute the expectation value for the eigenstates of a given general Hamiltonian. To do so, a file with the parameter values of the Hamiltonian has to be provided. In the examples directory you can find the file input_file_HCP, with the Hamiltonian parameters of the HCP molecule. To compute the expectation value of n for these eigenstates

     triat_U3_2.2.1/examples/Observables$ ../../bin/expected_val_n \
      -ng input_file_HCP 
     0 0.252375640144704
     19.5468965517241 0.214629419159648
     35.4689655172414 0.175676105843901
     51.1827586206897 0.223634301458906
     70.5862068965517 0.293699978181552
     92.748275862069 0.354259363279492
     116.620689655172 0.411790067302586
     141.513793103448 0.468494422286793
     166.841379310345 0.525142319743586
     192.058620689655 0.582222581646207
     216.593103448276 0.640168981577777
     239.775862068966 0.699641599554915
     260.772413793103 0.761652148046633
     278.575862068966 0.817777007616914
     294.103448275862 0.820195218254436

These results can be checked against the contents of the file expected_n_HCP.out.


3.4.2 Expectation value of the IR transition operator

The Perl script intensity_IR computes the squared expectation value of the infrared transition operator with different options[3]. A basic help can be obtained running the script with the -h option.

     triat_U3_2.2.1/examples/Observables$ ../../bin/intensity_IR -h
     
     Perl script to calculate the infrared intensity for a dipole transition 
     between eigenstates {l_1}_{u_1} and {l_2}_{u_2} of the U(3) model Hamiltonian 
     in a given xi or N value interval or for a given xi value.
     
     Note that {l}_{u} stands for u-th eigenvector with l angular momentum (g.s. u=1).
     
     The computed matrix element is
     
     |<{l_2}_{u_2}|T_+|{l_1}_{u_1}>|^2 + |<{l_2}_{u_2}|T_-|{l_1}_{u_1}>|^2
     
     with the selection rule |l_1-l_2| = 1. 
     
     
          by Currix TM
     
     Usage: intensity_IR [-Sv] Nval l1 u1 l2 u2 scale xi
     
            intensity_IR [-Sv] -s Nval l1 u1 l2 u2 scale ximin ximax xistep
     
            intensity_IR [-Sv] -n Nmin Nmax Nstep l1 u1 l2 u2 scale xi
     
            intensity_IR [-h] 
     
       Options:
         S : execute statically linked Fortran program
         v : verbose output
         h : help message
         s : consider a series of control parameter values
         n : consider a series of  N values

The script allows the user to compute the squared expectation value of the transition operator for a given pair of eigenstates of the model Hamiltonian for a given control parameter xi value [4] For example, to compute the transition intensity between the ground state (zero angular momentum) and the first eigenstate with angular momentum equal to one, in the N = 10, and xi = 0.4 Hamiltonian we will execute

     triat_U3_2.2.1/examples/Observables$  ../../bin/intensity_IR \
     > 10 0 1 1 1 1.0 0.4
     
      0.4       9.6007861982152359788

The expectation value for a particular pair of eigenstates and evaluated in an interval of control parameter values can also be computed using the option -s. For example, to compute the transition intensity between the ground state (zero angular momentum) and the second eigenstate with angular momentum l=1, with N = 10, and 0.2 < xi < 0.4 Hamiltonian with a step of 0.05 in the control parameter

     triat_U3_2.2.1/examples/Observables$ ../../bin/intensity_IR \
     > -s 10 0 1 1 2 1.0 0.2 0.5 0.05
     
      0.2      1.03195951590279317806E-0002
      0.25      4.88985645132075223243E-0002
      0.3      0.10416957597335169859      
      0.35      0.11597972613072936657      
      0.4      9.46745961688594321492E-0002
      0.45      6.89270459035718042020E-0002
      0.5      4.81758165286241743362E-0002

Finally, the last option is to compute the expectation value for two eigenstates varying the system's size, for different N values, using the option -n. For example, to compute the transition intensity between the ground state (zero angular momentum) and the second eigenstate with angular momentum l=1 in the critical value of the control parameter (xi = 0.2) with N taking values between 10 and 110 with a step of 20 bosons

     triat_U3_2.2.1/examples/Observables$ ../../bin/intensity_IR \
     > -n 10 110 20 0 1 1 2 1.0 0.2
      10      1.03195951590279317806E-0002
      30      1.76649230478514733430E-0002
      50      2.11485562246530897512E-0002
      70      2.35353506851742359296E-0002
      90      2.53900792349369549193E-0002
      110      2.69266373464041480800E-0002

3.4.3 Model and General Hamiltonian Eigenvalues

The Perl script alg_U3_energy computes the Hamiltonian eigenvalues for the model and the general algebraic Hamiltonian. [5]. A basic help can be obtained running the script with the -h option.

     triat_U3_2.2.1/examples/Observables$ ../../bin/alg_U3_energy -h
     
     Perl script to calculate the either the ground state energy or 
     the full spectrum of a U(3) model or general Hamiltonian.
     
     In the model Hamiltonian case, the calculation can be limited to only a subset 
     of the eigenvalues (first nstates), and a program taking advantage of the 
     tridiagonal character of the matrix is provided (and default for N > 5000).
     
     The output can be adapted to the drawing of Birge-Sponer plots.
     
          by Currix TM
     
     Usage: alg_U3_energy [-STvbB] Nval lval scale xi
            alg_U3_energy [-STvbB] [-I nstates] Nval lval scale xi
            alg_U3_energy [-STvG] Nval lval scale xi
            alg_U3_energy [-SvbB] [-n] -g inputfile
            alg_U3_energy -h
     
     
       Options:
         b : Birge Sponer plot output.
         B : Birge Sponer plot output with energy dependence.
         v : verbose output.
         I : compute the energies for the first nstates (nstates > 2)
         T : use the program version with tridiagonal matrix diagonalization
         G : compute only the ground state energy
         g : general Hamiltonian (supply name of the parameter file).
         S : execute the statically linked program.
         n : normalize energies or quantum number by N in the general Hamiltonian case.
         h : help message.

For example, to compute the energy spectrum for the first three excited states with angular momentum l = 2, N = 1000, and scale and control parameter equal to 1.0 and 0.555, respectively.

     triat_U3_2.2.1/examples/Observables$ ../../bin/alg_U3_energy -I 4 1000 2 1.0 0.555
      0       200.08946405931044     
      1       202.26535936605771     
      2       204.43652907169133     
      3       206.60297075985744

To compute the ground state energy for N = 10000, l = 0, and scale and control parameter equal to 1.0 and 0.75, respectively.

     triat_U3_2.2.1/examples/Observables$ ../../bin/alg_U3_energy -G 10000 0 1.0 0.75
      0       1197.8541544435745

Finally, to compute the full energy spectrum for l = 2, N = 1000, and scale and control parameter equal to 1.0 and 0.555

     triat_U3_2.2.1/examples/Observables$ ../../bin/alg_U3_energy -I 1 1000 2 1.0 0.555
      0       200.08946405931044     
      1       202.26535936605774     
      2       204.43652907169133     
      3       206.60297075985744     
      4       208.76468198691873     
      5       210.92166028156083     
      ....
     
     
      495       992.09574571198141     
      496       994.26349601727406     
      497       996.43459207588285     
      498       998.60902996360642     
      499       1000.7868057766277

When the general Hamiltonian spectrum is computed it is necessary to provide the program with an input file, as the provided example input file input_file_HCP

     triat_U3_2.2.1/examples/Observables$ ../../bin/alg_U3_energy -g input_file_HCP 
      0       0.0000000000000000     
      1       566.85562373229914     
      2       1028.5555120308779     
      3       1484.2632344381639     
      4       2046.9932672140753     
      5       2689.7360152596357     
      6       3382.0393187789541     
      7       4103.8525588498442     
      8       4838.4084093582642     
      9       5569.7433116564898     
      10       6281.1719539978449     
      11       6953.5491948507442     
      12       7562.3703503418556     
      13       8078.6701738697193     
      14       8529.0156142688975

3.4.4 Excitation Energy Diagrams

The Perl script exc_energ_diag_U3_mh computes the excitation energies necessary to plot correlation energy diagrams for given control parameter values with different options[6]. A basic help can be obtained running the script with the -h option.

     triat_U3_2.2.1/examples/Observables$ ../../bin/exc_energ_diag_U3_mh -h
     	Perl script to calculate the u first eigenvalues (g.s. -> u = 1) of 
     the algebraic U(3) model Hamiltonian in a given xi value interval. 
     If  u = 0 all anergies are included.
     
     	The energy values are normalized by the number of bosons N (E/N).
     
      by Currix TM
     
     Usage: exc_energ_diag_U3_mh [-vhS] [-e progpath] Nval lval scale u ximin ximax xistep
            
     
       Options:
         S : execute statically linked Fortran program
         v : verbose output
         h : help message
         e : provide the executable triat files

For example, to compute the excitation energy for the first three excited states with zero angular momentum, with N = 10, and in the interval of xi starting at 0.1 and ending at 0.4 with a step of 0.05 we execute

     triat_U3_2.2.1/examples/Observables$ ../../bin/exc_energ_diag_U3_mh \
     10 0 1.0 4 0.1 0.4 0.05
     
     0.1 0.0969591084744126 0.24303900731891 0.410527294642692 0.595860630601058 
     0.15 0.141605046653722 0.261843236909951 0.416273614122525 0.596349388138778 
     0.2 0.180891112017846 0.279990992516115 0.423496605540945 0.599080754985964 
     0.25 0.210586829109494 0.300331396846074 0.433272340537003 0.604387056209837 
     0.3 0.226957621450917 0.325171141950419 0.446759291603703 0.612551514295007 
     0.35 0.2312738839248 0.351753480722973 0.465174292657181 0.623837735817851 
     0.4 0.227411444586968 0.375148937971884 0.489249830006017 0.63853518398931


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triat_u3 User's Guide
Chapter 4 - References


  1. Minuit minimization package: Comput. Phys. Commun. 10 (1975) 343-367.

  1. J. Chem. Phys. 104 (1996) 6956.

  1. Chem. Phys. Lett. 365 (2002) 57-68.

  1. Chem. Phys. Lett. 375 (2003) 309-320.

  1. Phys. Rev. A77 (2008) 032115.


triat_u3 User's Guide
Footnotes

1

This makes reference to the model Hamiltonian H = scale[(1-xi) n + xi/(N-1) P].

2

This makes reference to the model Hamiltonian H = scale[(1-xi) n + xi/(N-1) P].

3

The transition operator is defined as T = D+/sqrt(N) + D-/sqrt(N).

4

This makes reference to the model Hamiltonian H = scale[(1-xi) n + xi/(N-1) P].

5

This makes reference to the model Hamiltonian H = scale[(1-xi) n + xi/(N-1) P].

6

This makes reference to the model Hamiltonian H = scale[(1-xi) n + xi/(N-1) P].


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triat_u3 User's Guide

Version 2.2.1

Curro Perez Bernal francisco.perez@dfaie.uhu.es