^{1,a)}, Ove Christiansen

^{1,b)}and Christof Hättig

^{2,c)}

### Abstract

Quadratic response functions are derived and implemented for a vibrational configuration interaction state. Combined electronic and vibrational quadratic response functions are derived using Born–Oppenheimer vibronic product wave functions. Computational tractable expressions are derived for determining the total quadratic response contribution as a sum of contributions involving both electronic and vibrational linear and quadratic response functions. In the general frequency-dependent case this includes a new and more troublesome type of electronic linear response function. Pilot calculations for the FH, , , and pyrrole molecules demonstrate the importance of vibrational contributions for accurate comparison to experiment and that the vibrational contributions in some cases can be very large. The calculation of transition properties between vibrational states is combined with sum-over-states expressions for analysis purposes. On the basis of this some simple analysis methods are suggested. Also, a preliminary study of the effect of finite lifetimes on quadratic response functions is presented.

We thank T. Helgaker and P. Sałek for providing us with a CAM-B3LYP implementation in DALTON prior to general release. This work was supported by the Lundbeck Foundation and the Danish Center for Scientific Computing (DCSC). O.C. acknowledges support from the Danish National Research Foundation, the Lundbeck Foundation, and the EUROHORCs through a EURYI award.

I. INTRODUCTION

II. THEORY

A. General aspects

B. Separation of electronic and vibrational contributions

C. Calculating the electronic asymmetric linear response function

D. Response functions for VCI wave functions

E. Frequency dispersion analysis

III. COMPUTATIONAL DETAILS

IV. SAMPLE CALCULATIONS

A. Mode excitation level

1. Water

2. Formaldehyde

B. Sum-over-states results

1. Convergence with respect to the number of vibrational states

2. Frequency dispersion

C. Frequency dispersion function

D. Total response

1. Hydrogen fluoride

2. Water

3. Formaldehyde

4. Pyrrole

V. SUMMARY

### Key Topics

- Speed of sound
- 28.0
- Wave functions
- 27.0
- Excitation energies
- 15.0
- Excited states
- 13.0
- General molecular properties
- 10.0

## Figures

Convergence of the part of the PV quadratic response function for water as a function of the number of states included. Based on a VCI[3] calculation using 3M4T potential and property surfaces.

Convergence of the part of the PV quadratic response function for water as a function of the number of states included. Based on a VCI[3] calculation using 3M4T potential and property surfaces.

SHG PV for water as a function of external frequency calculated for different damping factors.

SHG PV for water as a function of external frequency calculated for different damping factors.

SHG PV difference between exact SOS expression and the dispersion function estimated equivalent for water. See text for details.

SHG PV difference between exact SOS expression and the dispersion function estimated equivalent for water. See text for details.

The total vibrational contributions to for hydrogen fluoride calculated using either the exact or approximate electronic asymmetric linear response functions (see text for details) as a function of external frequency. The full curve represents the exact values and the broken curve the approximate results.

The total vibrational contributions to for hydrogen fluoride calculated using either the exact or approximate electronic asymmetric linear response functions (see text for details) as a function of external frequency. The full curve represents the exact values and the broken curve the approximate results.

## Tables

Individual parts of the PV quadratic response function (SHG) calculated for the water molecule as a function of the VCI mode excitation level at different optical frequencies and at the static limit using V3M4T/P3M4T potential and property surfaces, respectively.

Individual parts of the PV quadratic response function (SHG) calculated for the water molecule as a function of the VCI mode excitation level at different optical frequencies and at the static limit using V3M4T/P3M4T potential and property surfaces, respectively.

Individual parts of the PV quadratic response function calculated for formaldehyde as a function of the VCI mode excitation level at different optical frequencies and at the static limit using V4M4T/P4M4T potential and property surfaces, respectively.

Individual parts of the PV quadratic response function calculated for formaldehyde as a function of the VCI mode excitation level at different optical frequencies and at the static limit using V4M4T/P4M4T potential and property surfaces, respectively.

Excitation energies used in the approximation of electronic asymmetric linear response functions. The electronic structure models used for the individual molecules are the same as those used to construct the molecular property surfaces, see text for details. Values are in eV.

Excitation energies used in the approximation of electronic asymmetric linear response functions. The electronic structure models used for the individual molecules are the same as those used to construct the molecular property surfaces, see text for details. Values are in eV.

Total vibrational contributions to for hydrogen fluoride calculated at external frequencies of , 0.0656, 0.0720, 0.0885, and 0.0995 a.u. Values are in atomic units.

Total vibrational contributions to for hydrogen fluoride calculated at external frequencies of , 0.0656, 0.0720, 0.0885, and 0.0995 a.u. Values are in atomic units.

SHG first hyperpolarizability of FH at external frequency of 0.0720 a.u. Results are in atomic units.

SHG first hyperpolarizability of FH at external frequency of 0.0720 a.u. Results are in atomic units.

Total vibrational contributions to for water calculated at external frequencies of , 0.0428, and 0.0656 a.u. Values are in atomic units.

Total vibrational contributions to for water calculated at external frequencies of , 0.0428, and 0.0656 a.u. Values are in atomic units.

Total vibrational contributions to for formaldehyde calculated at external frequencies of , 0.0428, and 0.0656 a.u. Values are in atomic units.

Total vibrational contributions to for formaldehyde calculated at external frequencies of , 0.0428, and 0.0656 a.u. Values are in atomic units.

Total vibrational contributions to for pyrrole calculated at external frequencies of , 0.0428, and 0.0656 a.u. Values are in atomic units.

Total vibrational contributions to for pyrrole calculated at external frequencies of , 0.0428, and 0.0656 a.u. Values are in atomic units.

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