^{1,a)}and Ove Christiansen

^{1,b)}

### Abstract

An automatic and general procedure for the calculation of geometrical derivatives of the energy and general propertysurfaces for molecular systems is developed and implemented. General expressions for an -mode representation are derived, where the -mode representation includes only the couplings between or less degrees of freedom. The general expressions are specialized to derivative force fields and propertysurfaces, and a scheme for calculation of the numerical derivatives is implemented. The implementation is interfaced to electronic structure programs and may be used for both ground and excited electronic states. The implementation is done in the context of a vibrational structure program and can be used in combination with vibrational self-consistent field (VSCF), vibrational configuration interaction (VCI), vibrational Møller-Plesset, and vibrational coupled cluster calculations of anharmonic wave functions and calculation of vibrational averaged properties at the VSCF and VCI levels. Sample calculations are presented for fundamental vibrational energies and vibrationally averaged dipole moments and frequency dependent polarizabilities and hyperpolarizabilities of water and formaldehyde.

This work has been supported by the Danish Research Council, the Danish National Research Foundation, and Danish Center for Super Computing (DCSC).

I. INTRODUCTION

II. THEORY

A. Restricted mode coupling

B. Mode combination and expanded representation of operators

C. Polynomial expansions and restricted mode potentials

D. Derivatives of the potential

1. One-dimensional numerical derivatives

2. Derivatives for functions of many variables

3. Propertysurfaces

E. Second quantization and the interface to vibrational wave functiontheory

F. The calculation of vibrational averaged properties

III. IMPLEMENTATION

IV. COMPUTATIONAL DETAILS

V. RESULTS AND DISCUSSION

A. Molecular geometries

B. Fundamental vibrations

C. The dipole moments

D. The (hyper)polarizabilities

VI. SUMMARY AND OUTLOOK

### Key Topics

- Polarizability
- 27.0
- Wave functions
- 20.0
- General molecular properties
- 19.0
- Potential energy surfaces
- 17.0
- Electric dipole moments
- 12.0

## Tables

Electronic ground state geometry (in angstroms and degrees) for and .

Electronic ground state geometry (in angstroms and degrees) for and .

Fundamental vibrational frequencies for . The number of HO one-mode basis functions is 16. Results are given in . HO refers to the harmonic oscillator approximation.

Fundamental vibrational frequencies for . The number of HO one-mode basis functions is 16. Results are given in . HO refers to the harmonic oscillator approximation.

Fundamental vibrational frequencies for . The number of HO one-mode basis functions is 16 and in the correlated vibrational structure calculations the number of modals included is 7. Results are given in . HO refers to the harmonic oscillator approximation. : CH symmetric stretch, : CO stretch, : HCH bend, : out-of-plane bend, : CH antisymmetric stretch, and : rock.

Fundamental vibrational frequencies for . The number of HO one-mode basis functions is 16 and in the correlated vibrational structure calculations the number of modals included is 7. Results are given in . HO refers to the harmonic oscillator approximation. : CH symmetric stretch, : CO stretch, : HCH bend, : out-of-plane bend, : CH antisymmetric stretch, and : rock.

Electronic and zero-point vibrational VSCF and FVCI contributions to the dipole moment for and . The number of one-mode basis functions is 16 and for the number of modals included in the correlated vibrational structure calculations is 7. Results are given in Debye. Results for water are based on and for formaldehyde we have used . Experimental values (including vibrational corrections) are water: (see Ref. 74) and formaldehyde: (see Ref. 56).

Electronic and zero-point vibrational VSCF and FVCI contributions to the dipole moment for and . The number of one-mode basis functions is 16 and for the number of modals included in the correlated vibrational structure calculations is 7. Results are given in Debye. Results for water are based on and for formaldehyde we have used . Experimental values (including vibrational corrections) are water: (see Ref. 74) and formaldehyde: (see Ref. 56).

Electronic and zero-point vibrational FVCI contributions to the static polarizability and static hyperpolarizability tensors for . The number of one-mode basis functions is 16. The electronic structure calculations are CCSD/d-aug-cc-pVTZ. Results are given in a.u.

Electronic and zero-point vibrational FVCI contributions to the static polarizability and static hyperpolarizability tensors for . The number of one-mode basis functions is 16. The electronic structure calculations are CCSD/d-aug-cc-pVTZ. Results are given in a.u.

Electronic and zero-point vibrational FVCI contributions to the static polarizability and static hyperpolarizability tensors for . The number of one-mode basis functions is 16 and the number of modals included in the FVCI calculations is 7. The electronic structure calculations are at the level of CCSD/aug-cc-pVTZ. Results are given in a.u.

Electronic and zero-point vibrational FVCI contributions to the static polarizability and static hyperpolarizability tensors for . The number of one-mode basis functions is 16 and the number of modals included in the FVCI calculations is 7. The electronic structure calculations are at the level of CCSD/aug-cc-pVTZ. Results are given in a.u.

Electronic and zero-point vibrational VSCF and FVCI contributions to the dynamic polarizability tensor for and . The number of one-mode basis functions is 16 and for the number of modals included in the FVCI calculations is 7. The electronic structure calculations are at the level of CCSD/d-aug-cc-pVTZ for and CCSD/aug-cc-pVTZ for . Results are given in a.u.

Electronic and zero-point vibrational VSCF and FVCI contributions to the dynamic polarizability tensor for and . The number of one-mode basis functions is 16 and for the number of modals included in the FVCI calculations is 7. The electronic structure calculations are at the level of CCSD/d-aug-cc-pVTZ for and CCSD/aug-cc-pVTZ for . Results are given in a.u.

Electronic and zero-point vibrational VSCF and FVCI contributions to the dynamic second-harmonic-generation first hyperpolarizability tensor for and . The number of one-mode basis functions is 16 and for the number of modals included in the FVCI calculations is 7. The electronic structure calculations are at the level of CCSD/d-aug-cc-pVTZ for and CCSD/aug-cc-pVTZ for . Results are given in a.u.

Electronic and zero-point vibrational VSCF and FVCI contributions to the dynamic second-harmonic-generation first hyperpolarizability tensor for and . The number of one-mode basis functions is 16 and for the number of modals included in the FVCI calculations is 7. The electronic structure calculations are at the level of CCSD/d-aug-cc-pVTZ for and CCSD/aug-cc-pVTZ for . Results are given in a.u.

Comparison of the present and literature values of the ZPVCs to the static (hyper)polarizabilities. Results indicated with PT are obtained using perturbational expressions. For details on the “present work” calculations see the text. Results are in a.u.

Comparison of the present and literature values of the ZPVCs to the static (hyper)polarizabilities. Results indicated with PT are obtained using perturbational expressions. For details on the “present work” calculations see the text. Results are in a.u.

Parameters for representing and SHG as function of frequency. For definitions see the text. Results are given in a.u.

Parameters for representing and SHG as function of frequency. For definitions see the text. Results are given in a.u.

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