# VIBCAL¶

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

## Theory¶

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 $$q=0$$ .

\begin{equation*} 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} \end{equation*}

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{equation*} 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{equation*}

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

\begin{equation*} 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 \end{equation*}

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.

## Usage¶

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.

## Example: Parity violation shift associated with the C-F stretch of CHFClBr¶

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

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