VIBCAL is a utility program allowing the calculation of vibrationally averaged properties.

Suppose that we have optimized the geometry of our molecule and have carried out a harmonic vibrational analysis. For a selected normal mode we expand the electronic energy \(E\) as function of the associated normal coordinate \(q\) about the equilibrium position \(q=0\).

\[E(q) = E^{[0]} + E^{[1]}q + \frac{1}{2}E^{[2]}q^2 + \frac{1}{6}E^{[3]}q^3 + \ldots;\quad
E^{[n]} = \left.\frac{\partial^n E}{\partial q^n}\right|_{q=0}\]

We note that \(q=0\) implies \(E^{[1]}=0\) and that the second derivative \(E^{[2]}\) corresponds to the force constant \(k\). We furthermore expand some property \(P\equiv P(q)\) in the same manner.

From perturbation theory we now obtain an expression for the expectation value of the property \(P\) for vibrational level \(\nu\) of the selected normal mode

\[\begin{split}P_{\nu}=\left<\nu\left|P(q)\right|\nu\right>_q = P^{[0]}
+ \frac{1}{2}P^{[2]}\left(\frac{\hbar}{\mu\omega_e}\right)\left(\nu+\frac{1}{2}\right)
-\frac{1}{2}P^{[1]}E^{[3]}\left(\frac{\hbar}{\mu\omega_e}\right)^2\frac{\left(\nu+\frac{1}{2}\right)}{\hbar\omega_e}+\ldots\end{split}\]

where appears the reduced mass \(\mu\) and harmonic frequency \(\omega_e=\sqrt{\frac{k}{\mu}}\) of the selected normal mode.

Experimentally one can observe the change in this expectation value between two vibrational levels, given by

\[P_{\nu'} - P_{\nu} = \frac{1}{2}\frac{\hbar}{\mu\omega_e}\left[P^{[2]}-\frac{1}{\mu\omega_e^2}P^{[1]}E^{[3]}\right]\Delta\nu;\quad \Delta\nu=\nu' - \nu\]

We note that the purely electronic contribution \(P^{[0]}\) does not contribute to the vibrational shift. Furthermore, we may note that for a purely harmonic mode (\(E^{[3]}=0\)) and a property purely linear in the normal coordinate (\(P^{[2]}=0\)) the vibrational shift is strictly zero.

VIBCAL will read a formatted file of distances, energies and property values. The distances may be in Angstroms, but all other quantities have to be in atomic units. VIBCAL will next6, through a dialogue, with the user generate the necessary derivatives for the calculation of vibrational shifts of the property.

We start from a set of CCSD(T) energies and B3LYP parity violation energies calculated along the normal coordinate \(Q\) associated with the C-F stretch of the chiral CHFClBr:

```
-0.5000000000000000 -0.61012399429E+03 2.633089925380E-17
-0.2000000000000000 -0.61097769875E+03 9.078015651573E-18
-0.1000000000000000 -0.61102238183E+03 4.933686545396E-18
0.0000000000000000 -0.61103293760E+03 1.544047622021E-18
0.1000000000000000 -0.61102641387E+03 -1.026099838285E-18
0.2000000000000000 -0.61101190951E+03 -2.755550508570E-18
0.5000000000000000 -0.61095789506E+03 -1.870247865432E-18
```

The coordinates are in this case given in Angstroms. We fire up VIBCAL:

```
$vibcal.x
*******************************************
********** VIBCAL for properties **********
*******************************************
Set print level [default:0]:
0. Basic print level
1. Print additional information
Reduced mass (in Daltons)
```

Here we chose the default print level. For the reduced mass you have to check with the preceeding vibrational analysis. We give:

```
9.7032
Name of the input file (A30) with potential/property curve
```

It is a good idea to have a telling name of the input file

```
CCpot_B3LYPprp
Select one of the following:
1. Bond lengths in Angstroms.
2. Bond lengths in atomic units.
(Note that all other quantities only in atomic units !)
1
```

Here we choose Angstroms for the normal coordinate.

```
Select one of the following:
1. Numerical derivatives (assumes fixed step length).
2. Polynomial fits
2
```

In this example we do not have a fixed step length for our distances, so we select the second option which uses polynomials fits.

```
We have 7 points
Give order of polynomial:
5
Select point for the calculation of derivatives
0.0
```

The distances refer to the normal coordinate associated with the C–F stretch, so we set \(Q=0.0\) since this refers to the equilibrium structure.

`Results have been written to CCpot_B3LYPprp.vibcal`

Looking into this file we find

```
******************************
**** SOME INFORMATION ****
******************************
Reduced mass in Daltons: 9.7032000000000007
2nd potential derivative: 0.42167678827523780
3rd potential derivative: -1.6807622640206681
4th potential derivative: 9.1374786165355175
0th property derivative: 1.55000030970252530E-018
1st property derivative: -1.58073567198751495E-017
2nd property derivative: 2.23189000229251784E-017
******************************
******** RESULTS ********
******************************
Difference : -0.24366752E-18 au = -1.6 mHz
PV shift: -3.207 mHz
Harmonic model:
Difference : 0.12921573E-18 au = 0.9 mHz
PV shift: 1.700 mHz
```