Effective geometries

The (ro)vibrationally averaged geometries can be calculated from a knowledge of part of the cubic force field

$$\left<r_i\right> = r_{e,i} - \frac{1}{4\omega_i^2}\sum_{j=1}^{3N-6}\frac{V^{\left(3\right)}_{ijj}}{\omega_j}$$ (16.1)

where the summation runs over all normal modes in the molecule and where $\omega_i$ is the harmonic frequency of normal mode $i$ and $V^{\left(3\right)}_{ijj}$ is the cubic force field. A typical input for determining (ro)vibrationally averaged Hartree-Fock geometries for different water isotopomers will look like

**DALTON INPUT
.WALK
*WALK
.ANHARM
.DISPLACEMENT
0.001
.TEMPERATURES
 4
 0.0 300.0 500.0 1000.0
**WAVE FUNCTIONS
.HF
*SCF INPUT
.THRESH
 1.0D-10
**START
*RESPONS
.THRESH
 1.0D-5
**EACH STEP
*RESPONS
.THRESH
 1.0D-5
**PROPERTIES
.VIBANA
*RESPONS
.THRESH
 1.0D-5
*VIBANA
.ISOTOP
 3 3
 1 2 1
 1 2 2
 2 1 1
**END OF DALTON INPUT

The calculation of (ro)vibrationally averaged geometries are invoked be the keyword .ANHARM in the *WALK input module. In this example, the full cubic force field will be determined as first derivatives of analytical molecular Hessians. This will be done in Cartesian coordinates, and the calculation will therefore require the evaluation of $6K + 1$ analytical Hessians, where $K$ is the number of atoms in the molecules. Although expensive, it allows (ro)vibrational corrections to be calculated for any isotopic species, in the above example for H$_2\;^{16}$O, HD$\;^{16}$O, D$_2\;^{16}$O, H$_2\;^{18}$O. This is directed by the keyword .ISOTOP. We note that the most abundant isotope will always be calculated, and is therefore not included in the list above.

We have requested that rovibrationally averaged geometries be calculated for 5 different temperatures. By default, these geometries will include centrifugal distortions [16]. This can be turned by using the keyword .NO CENT in the *WALK input module.

By default, the numerical differentiation will use a step length of 0.0001 bohr. Experience show this to be too short [14], and we have therefore changed this to be 0.001 bohr in the example above by the use of the keyword .DISPLACMENT in the *WALK input module.

If only one (or a few) isotopic species are of interest, we can significantly speed up the calculation of the (ro)vibrationally averaged geometries by doing the numerical differentiation in the normal coordinates of the isotopic species of interest. This can be requested through the keyword .NORMAL. The relevant part of the cubic force field is then calculated as numerical second derivatives of analytical gradients. We note that the suggested step length in this case should be set to 0.0075 [14]. We note that we will still need to calculate one analytical Hessian in order to determine the normal coordinates.

The default maximum number of iterations is 20. However, DALTON will automatically reset the maximum number of iterations to 6$K$+1 in case of vibrational averaging calculations. The maximum number of iterations can also be set explicitly by using the keyword .MAX IT in the **DALTON INPUT module.