Contractions and vibrational basis set

Duo uses a contraction scheme to construct the rovibronic basis set used for the solution of the coupled problem. As a first step the J=0 vibration problem is solved for each electronic state, in which the corresponding Schroedinger equation is solved in the grid representation of npoints. Then a certain number of the resulted vibrational eigenfunctions |v\rangle with 0 \le v\le vmax and \tilde{E} \le EnerMax is selected to form the vibrational part of the basis set.

There are currently two contraction schemes supported by Duo: vibrational vib and Omega (diabatic).

The contraction type is defined in the section CONTRACTION (aliases: vibrationalbasis and vibrations) by the keywords vib or omega.

Vibrational contraction

This contraction uses a spin-free, fully uncoupled J=0 solution of the vibrational Schrödinger equation obtained independently for each electronic state as the vibrational basis. The rovibronic basis set is then form from the Lamda-Sigma wavefunctions:

| J \Omega S \Sigma \Lambda v \rangle = | J \Omega \rangle | S \Sigma \rangle | \Lambda \rangle | v \rangle

where | J \Omega \rangle and | S \Sigma \rangle are the rigid rotor functions and | \Lambda \rangle are the electronic wavefunctions implicitly taken from the ab initio calculations.

Omega (diabatic) contraction

This contraction is based on a solution of vibronically coupled J=0 problems for each value of \Omega=\Lambda+\Sigma. This contraction consists of two steps.

  1. For each grid value of r_i the electronic-orbital-spin-spin-orbit coupling is diagonalised on the Sigma/Lambda basis

|S\Sigma\rangle|\Lambda\rangle for each values of \Omega=\Lambda+\Sigma independently to form diabatic PECs.

  1. Vibrational (J=0) Schrödinger equations are solved for each diabatic PEC curve to obtain a Omega-vibrational basis set

|v,\Omega,n^{\Omega}\rangle (n^{\Omega} is a manyfold count within the same value of \Omega).

The rovibronic basis set in the Omega representation is given by

| J \Omega n v \rangle = | J \Omega \rangle | v,\Omega,n^{\Omega} \rangle

where | J \Omega \rangle are the rigid rotor functions.

Example 1:

  nmax 30
  enermax 25000

Example 2:

  nmax  30  10 10


  • vib and omega: contraction types
  • nmax

(alias: vmax, vibmax) specifies the value of the maximum vibrational functions to be computed and kept for the solution of the coupled problem. For example

nmax 15

specifies to compute for each PEC the lowest-energy 15 vibrational levels; it is also possible to specify different values of texttt{vmax} for each PEC, in which case the values must be given as a list; for example

nmax 10 15 8

specifies that for the PEC identified as poten 1 Duo should take 10 lowest vibrational states nmax=10, for poten 2, nmax=15 and for poten 3, nmax=8. If there are more PEC (poten 4 etc.) they will use for nmax the last value specified (nmax=8 in this example).

  • enermax

Alternatively or complementary to nmax one can select the vibrational energy levels to compute by specifying an upper energy threshold (in cm-1). Similarly to nmax, one can specify a different value of enermax for each PEC by writing a list of values; for example

enermax 30000.0 25000.0

selects a threshold of 30000 cm-1 for poten 1 and one of 25000 cm-1 for poten 2 and any other potential present. Note that by default Duo will shift the PECs so that the lowest point of the lowest-lying PEC has zero energy, and that the energy used for the enermax threshold are total vibrational energies including the zero point energy. One can prevent Duo from shifting the PECs by writing in the input (anywhere but not within an input section) the option do_not_shift_pecs.

If both enermax and vmax are specified only levels which satisfy both criteria are kept for the solution of the coupled problem. If neither of them is specified (or the vibrationalbasis input section is missing altogether) then vmax is taken equal to npoints for all PECs and there is a hard-coded limit of 10 8 cm-1 for enermax.