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# Calculation Of Vibrationally Averaged Properties

Vibrationally averaged properties can be computed within an anharmonic force-field calculation (keywords ANHARM=VIBROT, ANHARM=CUBIC, or ANHARM=QUARTIC, for details see the corresponding section on the calculation of anharmonic force fields) based on a Taylor expansion of the expectation value of the property of interest with respect to the normal coordinates. The corresponding formula for a property {$A$} is:

{$\langle A \rangle =A_e + \sum_r \frac{\partial A}{\partial Q_r} \langle Q_r \rangle + \frac{1}{2} \sum_{r,s} \frac{\partial^2 A}{\partial Q_r \partial Q_s} \langle Q_r Q_s \rangle + ...$}

with

{$\langle Q_r\rangle = - \frac{\hbar}{4 \omega_r^2} \sum_s \frac{k_{rss}}{\omega_s}$}

and

{$\langle Q_r Q_s \rangle = \delta_{rs} \frac{\hbar}{2\omega_r}$}

with {$\omega_r$} as the corresponding harmonic vibrational frequency and {$k_{rss}$} as the cubic force constants in terms of normal coordinates.

Computationally, this means that the cubic and semidiagoncal force fields need to be calculated as well as the corresponding property derivatives.

The actual calculations thus consists of

(a) a harmonic force-field calculation at the equilibrium geometry in order to get harmonic frequencies and the normal coordinates;
(b) a property calculation at the equilibrium geometry in order to get the equilibrium value contribution;
(c) generation of input files for all displacements along the normal coordinates;
(d) harmonic force-field calculations at the displaced geometries;
(e) property calculations at the displaced geometries;
(f) final analysis of the data including carrying out the finite differentiation as well as calculation of the spectropopic properties and the vibrationally averaged properties.

The available properties are currently dipole moment (do not need additional calculations as already determined in the harmonic force-field calculation), polarizability tensor, NMR shielding tensors, nuclear spin-rotation tensors (as a byproduct in calculating vibrationally averaged NMR chemical shieldings), indirect spin-spin coupling constants, dipolar spin-spin coupling tensors, electric field-gradient tensors, quadrupole moment, magnetizability tensor and rotational g tensor.

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CFOUR is partially supported by the U.S. National Science Foundation.