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Subsections

6.9 RASSI — A RAS State Interaction Program

Program RASSI (RAS State Interaction) computes matrix elements of the Hamiltonian and other operators in a wave function basis, which consists of individually optimized CI expansions from the RASSCF program. Also, it solves the Schrödinger equation within the space of these wave functions. There are many possible applications for such type of calculations. The first important consideration to have into account is that RASSI computes the interaction among RASSCF states expanding the same set of configurations, that is, having the same active space size and number of electrons.

The RASSI program is routinely used to compute electronic transition moments, as it is shown in the Advanced Examples in the calculation of transition dipole moments for the excited states of the thiophene molecule using CASSCF-type wave functions. By default the program will compute the matrix elements and expectation values of all the operators for which SEWARD has computed the integrals and has stored them in the ONEINT file.

RASSCF (or CASSCF) individually optimized states are interacting and non-orthogonal. It is imperative when the states involved have different symmetry to transform the states to a common eigenstate basis in such a way that the wave function remains unchanged. The State Interaction calculation gives an unambiguous set of non-interacting and orthonormal eigenstates to the projected Schrödinger equation and also the overlaps between the original RASSCF wave functions and the eigenstates. The analysis of the original states in terms of RASSI eigenstates is very useful to identify spurious local minima and also to inspect the wave functions obtained in different single-root RASSCF calculations, which can be mixed and be of no help to compare the states.

Finally, the RASSI program can be applied in situations when there are two strongly interacting states and there are two very different MCSCF solutions. This is a typical situation in transition metal chemistry when there are many close states associated each one to a configuration of the transition metal atom. It is also the case when there are two close quasi-equivalent localized and delocalized solutions. RASSI can provide with a single set of orbitals able to represent, for instance, avoided crossings. RASSI will produce a number of files containing the natural orbitals for each one of the desired eigenstates to be used in subsequent calculations.

RASSI requires as input files the ONEINT and ORDINT integral files and the JOBIPH files from the RASSCF program containing the states which are going to be computed. The JOBIPH files have to be named consecutively as JOB001, JOB002, etc. The input for the RASSI module has to contain at least the definition of the number of states available in each of the input JOBIPH files. Figure 6.11 lists the input file for the RASSI program in a calculation including two JOBIPH files (2 in the first line), the first one including three roots (3 in the first line) and the second five roots (5 in the first line). Each one of the following lines lists the number of these states within each JOBIPH file. Also in the input, keyword NATOrb indicates that three files (named sequentially NAT001, NAT002, and NAT003) will be created for the three lowest eigenstates.

**Figure 6.11:** Sample input requesting the RASSI module to calculate the matrix elements and expectation values for eight interacting RASSCF states
$\begin{figure}{---------------------------------------------------}\end{figure}$

&RASSI NROFjobiph= 2 3 5; 1 2 3; 1 2 3 4 5 NATOrb= 3

6.9.1 `RASSI` Output

The RASSI section of the MOLCAS output is basically divided in three parts. Initially, the program prints the information about the JOBIPH files and input file, optionally prints the wave functions, and checks that all the configuration spaces are the same in all the input states. In second place RASSI prints the expectation values of the one-electron operators, the Hamiltonian matrix, the overlap matrix, and the matrix elements of the one-electron operators, all for the basis of input RASSCF states. The third part starts with the eigenvectors and eigenvalues for the states computed in the new eigenbasis, as well as the overlap of the computed eigenstates with the input RASSCF states. After that, the expectation values and matrix elements of the one-electron operators are repeated on the basis of the new energy eigenstates. A final section informs about the occupation numbers of the natural orbitals computed by RASSI, if any.

In the Advanced Examples a detailed example of how to interpret the matrix elements output section for the thiophene molecule is displayed. The rest of the output is self-explanatory. It has to be remembered that to change the default origins for the one electron operators (the dipole moment operator uses the nuclear charge centroid and the higher order operators the center of the nuclear mass) keyword CENTer in GATEWAY must be used. Also, if multipoles higher than order two are required, the option MULTipole has to be used in GATEWAY.

The program RASSI can also be used to compute a spin-orbit Hamiltonian for the input CASSCF wave functions as defined above. The keyword AMFI has to be used in SEWARD to ensure that the corresponding integrals are available.

**Figure 6.12:** Sample input requesting the RASSI module to calculate and diagonalize the spin-orbit Hamiltonian the ground and triplet excited state in water.
$\begin{figure}{---------------------------------------------------}\end{figure}$

&RASSI NROFjobiph= 2 1 1; 1; 1 Spinorbit Ejob

The first JOBMIX file contains the wave function for the ground state and the second file the ³B₂ state discussed above. The keyword Ejob makes the RASSI program use the CASPT2 energies which have been written on the JOBMIX files in the diagonal of the spin-orbit Hamiltonian. The output of this calculation will give four spin-orbit states and the corresponding transition properties, which can for example be used to compute the radiative lifetime of the triplet state.

6.9.2 RASSI - Basic and Most Common Keywords

Keyword	Meaning
NROFjob	Number of input files, number of roots, and roots for each file
EJOB/HDIAG	Read energies from input file / inline
SPIN	Compute spin-orbit matrix elements for spin properties

6.9.3 CASVB — A non-orthogonal MCSCF program

CASVB is a program for carrying out quite general types of non-orthogonal MCSCF calculations, offering, for example, all the advantages associated with working within a valence bond formalism.

Warning: as for any general MCSCF program, one may experience convergence problems, (e.g., due to redundant parameters), and the non-orthogonal optimization of orbitals can furthermore give linear dependency problems. Several options in CASVB can help overcoming these difficulties.

This program can be used in two basic modes:
a) fully variational optimization

b) representation of CASSCF wavefunctions using overlap- (relatively inexpensive) or energy-based criteria.

CASVB executes the following logical steps: Setup of wavefunction information, starting guess generation, one, or several, optimization steps, various types of analysis of the converged solution.

6.9.4 `CASVB` input

CASVB attempts to define defaults for as many input quantities as possible, so that in the simplest case no input to the CASVB module is required. Sample input for a CASVB calculation on the lowest singlet state of CH₂:
&GATEWAY coord 3 ch2 molecule C 0.000000 0.000000 0.000000 H 0.000000 0.892226 0.708554 H 0.000000 -0.892226 0.708554 group= x y; basis= sto-3g &SEWARD &SCF &RASSCF nactel= 6 0 0; inactive= 1 0 0 0; ras2= 3 1 2 0 lumorb &CASVB

6.9.5 `CASVB` output

The amount of output in CASVB depends heavily on the setting of the PRINT levels. In case of problems with convergence behaviour it is recommended to increase these from their rather terse default values.

In the following the main features of the output are outlined, exemplified by the job in the input above. Initially, all relevant information from the previous RASSCF calculation is recovered from the JOBIPH interface file, after which the valence bond wavefunction information is summarized, as shown below. Since spatial configurations have not been specified explicitly in this example, a single covalent configuration is chosen as default. This gives 5 spin-adapted VB structures.

 Number of active electrons :   6
           active orbitals  :   6
           Total spin       : 0.0
           State symmetry   :   1

 Spatial VB configurations
 -------------------------
     Conf. =>   Orbitals
       1   =>    1  2  3  4  5  6

 Number of VB configurations :     1
           VB structures     :     5
           VB determinants   :    20

The output from the following optimization steps summarizes only the most relevant quantities and convergence information at the default print level. For the last optimization step, for example, The output below thus states that the VB wavefunction was found by maximizing the overlap with a previously optimized CASSCF wavefunction (output by the RASSCF program), and that the spin adaptation was done using the Yamanuchi-Kotani scheme. Convergence was reached in 7 iterations.

 -- Starting optimization - step  3 --------

 Overlap-based optimization (Svb).

 Optimization algorithm:            dFletch
 Maximum number of iterations:           50
 Spin basis:                         Kotani

 -------------------------------------------
 Optimization entering local region.
 Converged ... maximum update to coefficient:  0.59051924E-06
 Final Svb :    0.9978782695
 Number of iterations used:   7

Finally in the output below the converged solution is printed; orbital coefficients (in terms of the active CASSCF MOs) and structure coefficients. The overlap between orbitals are generally of interest, and, as also the structures are non-orthogonal, the structure weights in the total wavefunction. The total VB wavefunction is not symmetry-adapted explicitly (although one may ensure the correct symmetry by imposing constraints on orbitals and structure coefficients), so its components in the various irreducible representations can serve to check that it is physically plausible (a well behaved solution generally has just one non-vanishing component).

Next follows the one-electron density with natural-orbital analysis, again with quantities printed in the basis of the active CASSCF MOs.

 Orbital coefficients :
 ----------------------
           1           2           3           4           5           6
   1  0.43397359 -0.43397359 -0.79451779 -0.68987187 -0.79451780 -0.68987186
   2 -0.80889967  0.80889967 -0.05986171 -0.05516284 -0.05986171 -0.05516284
   3  0.00005587 -0.00005587  0.20401015 -0.20582094  0.20401016 -0.20582095
   4  0.39667145  0.39667145  0.00000000  0.00000000  0.00000000  0.00000000
   5 -0.00000001 -0.00000001 -0.53361427 -0.65931951  0.53361425  0.65931952
   6  0.00000000  0.00000000  0.19696124 -0.20968879 -0.19696124  0.20968879

 Overlap between orbitals :
 --------------------------
           1           2           3           4           5           6
   1  1.00000000 -0.68530352 -0.29636622 -0.25477647 -0.29636623 -0.25477647
   2 -0.68530352  1.00000000  0.29636622  0.25477647  0.29636623  0.25477646
   3 -0.29636622  0.29636622  1.00000000  0.81994979  0.35292419  0.19890631
   4 -0.25477647  0.25477647  0.81994979  1.00000000  0.19890634  0.04265679
   5 -0.29636623  0.29636623  0.35292419  0.19890634  1.00000000  0.81994978
   6 -0.25477647  0.25477646  0.19890631  0.04265679  0.81994978  1.00000000

 Structure coefficients :
 ------------------------
      0.00000000  0.00000001  0.09455957  0.00000000 -0.99551921
 
 Saving VB wavefunction to file VBWFN.
 
 Saving VB CI vector to file JOBIPH.
 
 Svb :          0.9978782695
 Evb :        -38.4265149062

 Chirgwin-Coulson weights of structures :
 ----------------------------------------
 VB spin+space (norm   1.00000000) :
      0.00000000  0.00000000 -0.00211737  0.00000000  1.00211737
 VB spin only  (norm   0.38213666) :
      0.00000000  0.00000000  0.00894151  0.00000000  0.99105849

 Symmetry contributions to total VB wavefunction :
 -------------------------------------------------
 Irreps 1 to 4 :  0.10000000E+01  0.15118834E-17  0.17653074E-17  0.49309519E-17

 Energies for components > 1d-10 :
 ---------------------------------
 Irreps 1 to 4 : -0.38426515E+02  0.00000000E+00  0.00000000E+00  0.00000000E+00

 One-electron density :
 ----------------------
           1           2           3           4           5           6
   1  1.98488829 -0.00021330  0.00011757  0.00000000  0.00000000  0.00000000
   2 -0.00021330  1.90209222 -0.00006927  0.00000000  0.00000000  0.00000000
   3  0.00011757 -0.00006927  0.02068155  0.00000000  0.00000000  0.00000000
   4  0.00000000  0.00000000  0.00000000  0.09447774  0.00000000  0.00000000
   5  0.00000000  0.00000000  0.00000000  0.00000000  1.97572540 -0.00030574
   6  0.00000000  0.00000000  0.00000000  0.00000000 -0.00030574  0.02213479

 Natural orbitals :
 ------------------
           1           2           3           4           5           6
   1 -0.99999668  0.00000000  0.00257629  0.00000000  0.00000000  0.00005985
   2  0.00257628  0.00000000  0.99999668  0.00000000  0.00000000 -0.00003681
   3 -0.00005995  0.00000000 -0.00003666  0.00000000 -0.00000001 -1.00000000
   4  0.00000000  0.00000000  0.00000000  1.00000000  0.00000001  0.00000000
   5  0.00000000  0.99999999  0.00000000  0.00000000  0.00015650  0.00000000
   6  0.00000000 -0.00015650  0.00000000 -0.00000001  0.99999999 -0.00000001

 Occupation numbers :
 --------------------
           1           2           3           4           5           6
   1  1.98488885  1.97572545  1.90209167  0.09447774  0.02213475  0.02068154

6.9.6 Viewing and plotting VB orbitals

In many cases it can be helpful to view the shape of the converged valence bond orbitals. MOLCAS therefore provides two facilities for doing this. For the Molden program, an interface file is generated at the end of each CASVB run (see also Section