Eigenfunctions and reduced density

: .. _Eigenfunctions and reduced density:

The computed eigenfunctions can be printed out into a sperate file (checkpoint). This option can be enabled via the section Checkpoint:

Checkpoint
 density save
 eigenvectors save
 Filename xxxxx
End

The keywords eigenvectors save are to stitch the corresponding checkpointing on.

The eigenfunction-checkpoints consist of two files, eigen_vectors.chk and eigen_vib.chk. The file eigen_vib.chk contains the vibrational part of the basis set in the grid representation in the following format (example):

   1          0.000000      1   0   A1Sigma+
   0.124132175316E-13
  -0.952336606315E-14
   0.982508543282E-14
......

where the each basis function is given in a block. The first line specifies the sate (number, energy, electronic state and vibrational quantum number) followed by the grid values.

The file eigen_vectors.chk contains the expansion coefficients of the eigenfunction in terms of the Duo ro-vibronic basis set functions using the following format (example):

Molecule = Ca-40           O-16
masses   =      39.962590600000     15.994914630000
Nroots   =       10
Nbasis   =       50
Nestates =        5
Npoints   =      501
range   =      1.0000000     4.0000000
X1Sigma+, Ap1Pi, a3Pi, b3Sigma+, A1Sigma+,    <- States
 |   # |    J | p |           Coeff.   | St vib Lambda  Sigma  Omega ivib|
     1      0.0  0   0.999999782551E+00   1   1   0      0.0      0.0    0
     .....

Here the first eight lines represent a signature of the spectroscopic model (atoms, masses, specification of the basis), the line 9 is a header followed by the records with the eigen-coefficients and corresponding quantum numbers and labels using the following format: running number within the J,parity(p)-block i, J, p, the coefficient C_i^{J,p}, State, v, \Lambda, \Sigma and vibrational basis set number (a combined number representing the contracted vibrational basis set function from for all electronic states combined).

The optional keyword Filename (alias Vector-Filename) is to change the checkpoint-prefix eigen to filename. The default name is eigen_vectors.chk.

The option density save is to compute the vibrational reduced density for all eigenfuncitons on a grid of bond length points, computed as follows

\rho_(r) = \sum_{v,v'} \sum_{{\rm State},\Lambda,\Sigma} [C_{v,{\rm State},\Lambda,\Sigma}^{i}]^* C_{v',{\rm State},\Lambda,\Sigma}^i \phi_{v}(r)^* \phi_{v'}(r) \Delta r

Writing the wave functions to disk

Both the vibrational (J=0, uncoupled) basis functions and the coefficients of the expansion of the final (J>0 and coupled) wave functions can be written to disk by including in the Duo input a section with the following structure:

checkpoint
  eigenfunc save
  filename CO
end

Two files will be produced, called in our example CO_vib.chk and CO_vectors.chk. The file CO_vib.chk contains the values of the vibrational basis functions at the grid points and has the following structure:

1          0.000000      1   0   A_1Sigma+
0.417251193034E-06
0.913182486541E-06
0.140429031525E-05
0.191466765349E-05
0.243955552609E-05
0.298913870277E-05
0.356440215967E-05
0.417282770822E-05
0.481737299860E-05
0.550475969611E-05
0.623909577848E-05

The first line describes the assignment of the vibrational basis function; the first number is a counter over all vibrational wave functions; the second is the energy in cm-1; the third is the state quantum number indicating the electronic state; the fourth is the v vibrational quantum number; finally, the label of the electronic state is reported. What follows is the value of the vibrational wave function at the grid points. The file ends with the line

End of contracted basis

The file CO_vectors.chk contains the values of the expansion coefficients of the final wave functions. The structure is as follows:

Molecule = C-12            O-16
masses   =      12.000000000000     15.994914504752
Nroots   =        3
Nbasis   =        0
Nestates =        1
Npoints   =      100
range   =      0.6500000     3.0000000
Morse_   <- States
     |   # |    J | p |           Coeff.   | St vib Lambda  Sigma  Omega|
        1      0.0  1   0.100000000000E+01   1   1   0      0.0      0.0
        1      0.0  1   0.000000000000E+00   1   2   0      0.0      0.0
        1      0.0  1   0.000000000000E+00   1   3   0      0.0      0.0
        2      0.0  1   0.000000000000E+00   1   1   0      0.0      0.0
        2      0.0  1   0.100000000000E+01   1   2   0      0.0      0.0
        2      0.0  1   0.000000000000E+00   1   3   0      0.0      0.0
        3      0.0  1   0.000000000000E+00   1   1   0      0.0      0.0
        3      0.0  1   0.000000000000E+00   1   2   0      0.0      0.0
        3      0.0  1   0.100000000000E+01   1   3   0      0.0      0.0
End of eigenvector

The first seven lines are a header containing the names of the atoms, the atomic masses, the number of wave functions computed, the total dimension of the J>0 or coupled Hamiltonian matrix, the number of electronic states in the calculations, the number of grid points and range of the grid (in AA). The numbers following are: # is a counter over the rovibronic wave functions; J is the total [2]

The density checkpoint file has the following structure:

0.545190480438E-08 ||      1.5  0       1
0.286121234769E-07 ||      1.5  0       1
0.134835397210E-06 ||      1.5  0       1
0.572802754694E-06 ||      1.5  0       1
0.220181930274E-05 ||      1.5  0       1
0.768598025530E-05 ||      1.5  0       1
0.244490197607E-04 ||      1.5  0       1

Where the first column represent the reduced density value on a grid point r_i, followed by a dilemeter ||, J, parity \tau and the state number as in the Duo output.

Footnotes

[2]Stricly speaking, \mathbf{J} = \mathbf{R} + \mathbf{L} + \mathbf{S} is the sum of the rotational and total electronic angular momenta; it is the total angular momentum only if the nuclear angular momentum \mathbf{I} is zero (or is neglected).} angular momentum; p is the total \pm parity (0 for + and 1 for -); Coeff. is the value of the coefficient in the expansion; following are the quantum number of the basis function (electronic, vibrational, \Lambda, \Sigma and \Omega).