## The VSCF programs (VSCF)

`VSCF`

,*options* [vscf]

The `VSCF`

program is exclusively based on the Watson Hamiltonian
\begin{align}
\hat{H} = \frac{1}{2} \sum_{\alpha\beta} ( \hat{J}_\alpha - \hat{\pi}_\alpha) \mu_{\alpha\beta}
(\hat{J}_\beta - \hat{\pi}_\beta)
-\frac{1}{8}\sum_\alpha \mu_{\alpha\alpha} -\frac{1}{2}\sum_i \frac{\partial^2}{\partial q_i^2} + V(q_1,\dots,q_{3N-6})
\label{eq:1}
\end{align}
in which the potential energy surfaces, $V(q_1,\dots,q_{3N-6})$, are provided by the `SURF`

module. The Watson correction term and the 0D term of the vibrational angular momentum terms are by default (`VAM=2`

) included. Within the grid-based version of the program the one-dimensional Schrödinger equation is solved by the DVR procedure of Hamilton and Light. Note that, the number of basis functions (e.g. distributed Gaussians) is determined by the grid points of the potential and cannot be increased without changing the PES grid representation. In contrast to that, the number of basis functions can be modified without restrictions in the version based on an analytical representation of the potential (polynomials, B-splines, Gaussians). As VSCF calculations are extremely fast, these calculations cannot be restarted. For details see:

J. Meisner, P.P. Hallmen, J. Kästner, G. Rauhut, *Vibrational analysis of methyl cation - rare gas atom complexes: CH$_3^+$-Rg (Rg=He, Ne, Ar, Kr)*, J. Chem. Phys. **150**, 084306 (2019).

G. Rauhut, T. Hrenar, *A Combined Variational and Perturbational Study on the Vibrational Spectrum of P$_2$F$_4$*, Chem. Phys. **346**, 160 (2008).

The following *options* are available:

By default state-specific VSCF calculations will be performed.`AVERAGE`

=*n*`AVERAGE=1`

allows for configuration averaged VSCF calculations, i.e. CAVSCF. The averaging will be perfomed for those states being defined by the`VIBSTATE`

program. The vibrational ground state will alway be excluded. An identical weight factor will be used for all states, i.e. the inverse of the number of states.`BASIS`

=*variable*`BASIS=DGB`

(default) defines a mode-specific basis of distributed Gaussians and distributes the Gaussians in a way, that the overlap integral between two functions is always the same (controlled by`THRBASOVLP`

). This guarantees that an increasing number of basis functions will always lead to an improvement.`BASIS=HO`

defines a harmonic oscillator basis. Using this basis together with`MAXITER=0`

and`VAM=0`

provides a simple harmonic oscillator basis to be used in all subsequent programs, e.g. in the VCI program.`BASIS=SIN`

uses a basis of sine functions. This is not a fully implemented feature, but primarily available for experimental purposes.By default the`COMBI`

=*n*`VSCF`

program calculates the fundamental modes of the molecule only. However, choosing`COMBI=`

$n$ allows for the calculation of the vibrational overtones and combination bands. The value of $n$ controls the excitation level, i.e. the number of states to be computed increases very rapidly for large values of $n$. Therefore, by default the upper limit is set to 5000 cm$^{-1}$, but this cutoff can be changed by the option`UBOUND`

. See also the`VIBSTATE`

program (section the DAT2GR program (DAT2GR)) for even more possibilities of defining vibrational states.`DIPOLE`

=*n*`DIPOLE=1`

allows for the calculation of infrared intensities. Calculation of infrared intensities requires the calculation of dipole surfaces within the`SURF`

program. By default the intensities will be computed on the basis of Hartree-Fock dipole surfaces.The initial guess for the VSCF programs is by default generated from the uncoupled one-dimensional potentials, i.e.`GUESS`

=*n*`GUESS=1`

. Alternatively, one may start within the calculation of excited vibrational states from the solution of the vibrational ground state,`GUESS=2`

.`INFO`

=*n*`INFO=1`

provides a list of the values of all relevant program parameters (options).By default VSCF calculations will be performed for non-rotating molecules, i.e. J=0. Rovibrational levels can be computed for arbitrary numbers of J$=n$. This will perform a purely rotational calculation (RCI). To obtain approximate rovibrational energies, vibrational energies have to be added.`JMAX`

=*n*This key sets the maximum number of iterations to be performed in the VSCF program.`MAXITER`

=*n*Plots all $\mu$-tensor surfaces up to`MUPLOT`

=*n**n*D and a corresponding Gnuplot script in a separate subdirectory (`plots`

). This option works only in combination with`POT=POLY`

. The`VAM`

option has to be set accordingly.The number of basis functions (distributed Gaussians) to be used for solving the VSCF equations can be controlled by`NBAS`

=*n*`NBAS`

=*n*. The default is`NBAS=20`

. This option is only active once an analytical representation of the potential has been chosen, see the option`POT`

and the`POLY`

program.The expansion of the potential in the`NDIM`

=*n*`VSCF`

calculation can differ from the expansion in the`SURF`

calculation. However, only values less or equal to the one used in the surface calculation can be used.Term after which the $n$-body expansions of the dipole surfaces shall be truncated. The default is set to 3. Note that`NDIMDIP`

=*n*`NDIMDIP`

has to be lower or equal to`NDIM`

.Term after which the $n$-body expansions of the polarizability tensor surfaces are truncated. The default is set to 0. Note that`NDIMPOL`

=*n*`NDIMPOL`

has to be lower or equal to`NDIM`

.By default the expansion of the $\mu$-tensor for calculating the vibrationally averaged rotational constants is truncated after the 2nd order terms, i.e.`NVARC`

=*n*`NVARC=2`

. This may be altered by the`NVARC`

keyword.Determines the type of orthogonalization within the VSCF program.`ORTHO`

=*n*`ORTHO=1`

invokes a symmetrical orthogonalization,`ORTHO=2`

a canonical one and`ORTHO=3`

uses a canonical one together with an elimination of linear dependencies (see also keyword`THRLINDEP`

. The default is`ORTHO=1`

.`POLAR`

=*n*`POLAR=1`

allows to compute Raman intensities in addition to infrared intensities, but of course requires polarizability tensor surfaces from the`SURF`

program. By default Raman intensities are switched off.VSCF solutions can be obtained using a potential in grid representation, i.e.`POT`

=*variable*`POT=GRID`

, or in an analytical representation,`POT=POLY`

,`POT=BSPLINE`

,`POT=GAUSS`

. In the latter case the`POLY`

program needs to be called prior to the`VSCF`

program in order to transform the potential.This option provides an extended output.`PRINT`

=*n*`PRINT=1`

prints the vibrationally averaged rotational constants for all computed states and the associated vibration-rotation constants $\alpha$. Moreover, the temperature dependence of bond lengths will also be printed, when the potential is represented by a linear combination of basis functions.`PRINT=2`

prints the effective 1D polynomials in case that the potential is represented in terms of polynomials, see the option`POT=POLY`

and the`POLY`

program. In addition the generalized VSCF property integrals, i.e. $\left < VSCF \left | q_i^r \right | VSCF \right >$ are printed. These integrals allow for the calculation of arbitrary vibrationally averaged properties once the property surfaces are available. Default:`PRINT=0`

.By default, i.e.`SADDLE`

=*n*`SADDLE=0`

, the`VSCF`

program assumes, that the reference point of the potential belongs to a local minimum. Once the PES calculation has been started from a transition state, this information must be provided to the`VSCF`

program by using`SADDLE=1`

. Currently, the`VSCF`

program can only handle symmetrical double-minimum potentials.For solving the one-dimensional Schrödinger equation within a grid representation two different algorithms can be used. The default, i.e.`SOLVER`

=*n*`SOLVER=1`

, calls the discrete variable representation (DVR) as proposed by Hamilton and Light. Alternatively, the collocation algorithm of Young and Peet can be used (`SOLVER=2`

).`THERMO`

=*n*`THERMO=1`

allows for the improved calculation of thermodynamical quantities (compare the`THERMO`

keyword in combination with a harmonic frequency calculation). However, the approach used here is an approximation: While the harmonic approximation is still retained in the equation for the partition functions, the actual values of the frequencies entering into these functions are the anharmonic values derived from the`VSCF`

calculation. Default:`THERMO=0`

.Overlap between two Gaussian basis functions, once`THRBASOVLP`

=*value*`BASIS=DGB`

has been chosen. The default is 0.75.Threshold for eliminating linear dependencies in the VSCF procedure (see keyword`THRLINDEP`

=*value*`BASIS=DGB`

). The default is`THRLINDEP=1e-8`

.Once overtones and combination bands shall be computed, the upper energy limit is controlled by the keyword`UBOUND`

=*n*`UBOUND`

, i.e. states, for which the harmonic estimate is larger than $n$, will not be computed. the default is set to $n$=5000 cm$^{-1}$.Once vibrational states have been defined with the`USERMODE`

=*n*`VIBSTATE`

program (section the DAT2GR program (DAT2GR)), the VSCF program can be forced to compute just these states by the option`USERMODE=1`

. Note that the vibrational ground state will always be computed and needs not to be specified explicitly.The 0D terms of the vibrational angular momentum terms, i.e. $\frac{1}{2} \sum_{\alpha\beta} \hat{\pi}_\alpha\mu_{\alpha\beta} \hat{\pi}_\beta$, and the Watson correction term are by default (`VAM`

=*n*`VAM=2`

) included.

`VAM=1`

adds the Watson correction term (see Eq. \eqref{eq:1}) as a pseudo-potential like contribution to the fine grid of the potential.

`VAM=2`

adds the `VAM`

operator using the approximation that the $\mu$ tensor is given as the inverse of the moment of inertia tensor at equilibrium geometry to the quasi Fock operator.

`VAM=3`

allows for the calculation of the integrals of the `VAM`

operator using the approximation that the $\mu$ tensor is given as the inverse of the moment of inertia tensor at equilibrium geometry.

When using `VAM=4`

the expansion of the effective moment of inertia tensor will be truncated after the 1D terms (rather than the 0D term in case of `VAM=2`

). Note that values higher than 2 are only active for non-linear molecules. `VAM=5`

truncates the series after the 2D term. In almost all cases `VAM=2`

is fully sufficient. Vibrational angular momentum terms are accounted for in a perturbational manner and do not affect the wavefunction.

The following input example for a grid based calculation of anharmonic frequencies and intensities on the `VSCF`

level (1) optimizes the geometry of water, (2) computes the harmonic frequencies,(3) generates a potential energy surface around the equilibrium structure, (4) transforms the grid points to polynomials and (5) computes the nuclear wave function and the infrared intensities at the `VSCF`

level. Vibrational angular momentum terms (`VAM`

) are included. Note, that it is recommended to perform a `VCI`

calculation after a `VSCF`

calculation. The details of the `VCI`

input are described in the next chapter the VCI program (VCI).

memory,20,m basis=vdz orient,mass geometry={ 3 Water O 0.0675762564 0.0000000000 -1.3259214590 H -0.4362118830 -0.7612267436 -1.7014971211 H -0.4362118830 0.7612267436 -1.7014971211 } mass,iso hf mp2 optg !(1) optimizes the geometry frequencies,symm=auto !(2) compute harmonic frequencies label1 {hf start,atden} {mp2 cphf,1} {xsurf,sym=auto !(3) generate potential energy surface intensity,dipole=2} poly !(4) transform to polynomials vscf,pot=poly !(5) do a VSCF calculation