**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**.

**Input description for calculating vibrationally averaged properties**

**Implementational details for calculating vibrationally averaged properties**

**Examples for calculating vibrationally averaged properties**

**Recommendations for calculating vibrationally averaged properties**