^{1,a)}, Mikkel Bo Hansen

^{1}, Peter Seidler

^{1}and Ove Christiansen

^{1}

### Abstract

We report the theory and implementation of vibrational coupled cluster (VCC) damped response functions. From the imaginary part of the damped VCC response function the absorption as function of frequency can be obtained, requiring formally the solution of the now complex VCC response equations. The absorptionspectrum can in this formulation be seen as a matrix function of the characteristic VCC Jacobian response matrix. The asymmetric matrix version of the Lanczos method is used to generate a tridiagonal representation of the VCC response Jacobian. Solving the complex response equations in the relevant Lanczos space provides a method for calculating the VCC damped response functions and thereby subsequently the absorption spectra. The convergence behaviour of the algorithm is discussed theoretically and tested for different levels of completeness of the VCC expansion. Comparison is made with results from the recently reported [P. Seidler, M. B. Hansen, W. Györffy, D. Toffoli, and O. Christiansen, J. Chem. Phys.132, 164105 (2010)] vibrational configuration interaction damped response function calculated using a symmetric Lanczos algorithm. Calculations of IR spectra of oxazole, cyclopropene, and uracil illustrate the usefulness of the new VCC based method.

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

I. INTRODUCTION

II. THEORY

A. General aspects of damped response theory

B. Vibrational coupled cluster response theory

III. CALCULATING THE RESPONSE FUNCTIONS THROUGH A LANCZOS RECURSION CHAIN

A. The asymmetric Lanczos algorithm

B. Construction of the VCC response function using a Lanczos chain

C. Convergence

D. Analysis of the absorptionspectrum

IV. COMPUTATIONAL DETAILS

V. SAMPLE CALCULATIONS

A. Oxazole

1. Convergence with respect to chain length

2. Comparison to vibrational configuration interaction and experimental spectra

3. Analysis and assignment of the VCC[3] spectrum

B. Cyclopropene

1. Convergence with respect to the length of the Lanczos chain

2. Comparison to vibrational configuration interaction and experimental spectra

C. Uracil

1. Comparison to vibrational configuration interaction and experimental spectra

VI. SUMMARY AND OUTLOOK

### Key Topics

- Infrared spectra
- 49.0
- Eigenvalues
- 35.0
- Absorption spectra
- 15.0
- Wave functions
- 11.0
- Coupled cluster
- 9.0

## Figures

Oxazole with the atomic numbering used in Table I.

Oxazole with the atomic numbering used in Table I.

The convergence of the VCC[3] oxazole IR spectra with respect to chain length. The value of γ is 10 cm^{−1}. The spectra are amplified by a factor 10 in the interval 1750 cm^{−1}–4000 cm^{−1} for clarity. The spectra have been divided into five intervals marked by vertical lines. For each interval, Eq. (60) has been used for calculating the relative difference between the current and the preceding chain length.

The convergence of the VCC[3] oxazole IR spectra with respect to chain length. The value of γ is 10 cm^{−1}. The spectra are amplified by a factor 10 in the interval 1750 cm^{−1}–4000 cm^{−1} for clarity. The spectra have been divided into five intervals marked by vertical lines. For each interval, Eq. (60) has been used for calculating the relative difference between the current and the preceding chain length.

The convergence of the term contribution with respect to chain length for oxazole. The value of γ is 10 cm^{−1}. The contribution has been divided into five intervals, similar to Fig. 2. For each interval Eq. (60) has been used for calculating the relative difference between the current and the preceding chain length.

The convergence of the term contribution with respect to chain length for oxazole. The value of γ is 10 cm^{−1}. The contribution has been divided into five intervals, similar to Fig. 2. For each interval Eq. (60) has been used for calculating the relative difference between the current and the preceding chain length.

The IR spectra of oxazole computed via VCI[2], VCI[3], VCC[2], VCC[2pt3], and VCC[3] methods compared using chain length k = 2000. The value of γ is 10 cm^{−1}.

The IR spectra of oxazole computed via VCI[2], VCI[3], VCC[2], VCC[2pt3], and VCC[3] methods compared using chain length k = 2000. The value of γ is 10 cm^{−1}.

The calculated and experimental spectra of oxazole. The calculated spectrum of oxazole uses the VCC[3] approximation and a chain length k = 2000. The bars indicate the frequency ranges from Table II. The value of γ is 10 cm^{−1}. The experimental spectrum is taken from Ref. 59.

The calculated and experimental spectra of oxazole. The calculated spectrum of oxazole uses the VCC[3] approximation and a chain length k = 2000. The bars indicate the frequency ranges from Table II. The value of γ is 10 cm^{−1}. The experimental spectrum is taken from Ref. 59.

The convergence of the VCC[3] IR spectra of cyclopropene as a function of the length of the Lanczos chain. The value of γ is 10 cm^{−1}. Above 3600 cm^{−1}, the spectra are scaled by a factor of 10 for clarity. The spectra have been divided into five intervals marked by vertical lines. For each interval, Eq. (60) has been used for calculating the relative difference between the current and the preceding chain length.

The convergence of the VCC[3] IR spectra of cyclopropene as a function of the length of the Lanczos chain. The value of γ is 10 cm^{−1}. Above 3600 cm^{−1}, the spectra are scaled by a factor of 10 for clarity. The spectra have been divided into five intervals marked by vertical lines. For each interval, Eq. (60) has been used for calculating the relative difference between the current and the preceding chain length.

Spectra of cyclopropene generated by different excitation levels in VCI and VCC. The value of γ used is 10 cm^{−1}. All spectra are based on a chain length of 1000.

Spectra of cyclopropene generated by different excitation levels in VCI and VCC. The value of γ used is 10 cm^{−1}. All spectra are based on a chain length of 1000.

Spectra of cyclopropene for the VCI[4] and VCC[3] models, with bars denoting the different spectral peaks and the experimental spectra from Ref. 62. The value of γ used is 10 cm^{−1}. All spectra are based on a chain length of 1000.

Spectra of cyclopropene for the VCI[4] and VCC[3] models, with bars denoting the different spectral peaks and the experimental spectra from Ref. 62. The value of γ used is 10 cm^{−1}. All spectra are based on a chain length of 1000.

The IR spectra of uracil computed via VCI[3] and VCC[3], compared to the experimental spectra. The IR spectra from Refs. 64 and 65 are included for comparison, the lower spectra stemming from Ref. 64, while the upper stems from Ref. 65. The value of γ is 10 cm^{−1}.

The IR spectra of uracil computed via VCI[3] and VCC[3], compared to the experimental spectra. The IR spectra from Refs. 64 and 65 are included for comparison, the lower spectra stemming from Ref. 64, while the upper stems from Ref. 65. The value of γ is 10 cm^{−1}.

## Tables

The normal mode vibrations of oxazole and their interpretations. From the coordinate analysis at the equilibrium geometry, for details see Ref. 54.

The normal mode vibrations of oxazole and their interpretations. From the coordinate analysis at the equilibrium geometry, for details see Ref. 54.

Spectral areas, peak positions and their contributions for the normal vibrations of oxazole. See Fig. 5 for a visual representation of the regions. The experimental data are taken from Refs. 60 and 61, and are here fitted such that the data fit with the dominant mode in our analysis.

Spectral areas, peak positions and their contributions for the normal vibrations of oxazole. See Fig. 5 for a visual representation of the regions. The experimental data are taken from Refs. 60 and 61, and are here fitted such that the data fit with the dominant mode in our analysis.

Spectral regions, peak positions, and their contributions for the normal vibrations of cyclopropene. See Fig. 8 for a visual representation of the regions. The experimental data are taken from Refs. 62 and 63, and are here fitted such that the reported peak positions match with ours.

Spectral regions, peak positions, and their contributions for the normal vibrations of cyclopropene. See Fig. 8 for a visual representation of the regions. The experimental data are taken from Refs. 62 and 63, and are here fitted such that the reported peak positions match with ours.

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