^{1,a)}, Murat Keçeli

^{1}and So Hirata

^{1,b)}

### Abstract

A procedure to determine optimal vibrational coordinates is developed on the basis of an earlier idea of Thompson and Truhlar [J. Chem. Phys.77, 3031 (1982)]10.1063/1.444226. For a given molecule, these coordinates are defined as the unitary transform of the normal coordinates that minimizes the energy of the vibrational self-consistent-field (VSCF) method for the ground state. They are justified by the fact that VSCF in these coordinates becomes exact in two limiting cases: harmonic oscillators, where the optimized coordinates are normal, and noninteracting anharmonic oscillators, in which the optimized coordinates are localized on individual oscillators. A robust and general optimization algorithm is developed, which decomposes the transformation matrix into a product of Jacobi matrices, determines the rotation angle of each Jacobi matrix that minimizes the energy, and iterates the process until a minimum in the whole high dimension is reached. It is shown that the optimized coordinates are neither entirely localized nor entirely delocalized (or normal) in any of the molecules (the water, water dimer, and ethylene molecules) examined (apart from the aforementioned limiting cases). Rather, high-frequency stretching modes tend to be localized, whereas low-frequency skeletal vibrations remain normal. On the basis of these coordinates, we introduce two new vibrational structure methods: optimized-coordinate VSCF (oc-VSCF) and optimized-coordinate vibrational configuration interaction (oc-VCI). For the modes that become localized, oc-VSCF is found to outperform VSCF, whereas, for both classes of modes, oc-VCI exhibits much more rapid convergence than VCI with respect to the rank of excitations. We propose a rational configuration selection for oc-VCI when the optimized coordinates are localized. The use of the optimized coordinates in VCI with this configuration selection scheme reduces the mean absolute errors in the frequencies of the fundamentals and the first overtones/combination tones from 104.7 (VCI) to 10.7 (oc-VCI) and from 132.4 (VCI) to 8.2 (oc-VCI) cm^{−1} for the water molecule and the water dimer, respectively. It is also shown that the degree of coupling in the potential for ethylene is reduced effectively from four modes to three modes by the transformation from the normal to optimized coordinates, which enhances the accuracy of oc-VCI with low-rank excitations.

We thank Matthew Hermes for an insightful discussion. This work has been supported by the Department of Energy, Office of Science, Office of Basic Energy Sciences (Grant No. DE-FG02-11ER16211). S.H. is a Camille Dreyfus Teacher-Scholar, a Scialog Fellow of the Research Corporation for Science Advancement, and an Alumni Research Scholar of the University of Illinois. M.K. was a University Block Grant Fellow of the University of Illinois.

I. INTRODUCTION

II. THEORY

A. Optimized-coordinate VSCF

B. Optimized-coordinate VCI

III. OPTIMIZATION ALGORITHM

IV. APPLICATIONS

A. The hydrogen fluoride tetramer

B. The water molecule

C. The water dimer

D. Ethylene

V. CONCLUSION

### Key Topics

- Oscillators
- 15.0
- Wave functions
- 11.0
- Ground states
- 9.0
- Optimization
- 6.0
- Excited states
- 4.0

## Figures

The algorithm of the coordinate optimization.

The algorithm of the coordinate optimization.

(a) The normal and (b) the optimized coordinates of the hydrogen fluoride tetramer.

(a) The normal and (b) the optimized coordinates of the hydrogen fluoride tetramer.

The VSCF zero-point energy of the water molecule as a function of the rotation angle (θ_{ ij }) of the unitary transformation of two coordinates, , while the third remains to be the normal coordinate. At θ_{ ij } = 0, *Q* _{1}, *Q* _{2}, and *Q* _{3} are the bending, symmetric stretching, and anti-symmetric stretching normal coordinates, respectively.

The VSCF zero-point energy of the water molecule as a function of the rotation angle (θ_{ ij }) of the unitary transformation of two coordinates, , while the third remains to be the normal coordinate. At θ_{ ij } = 0, *Q* _{1}, *Q* _{2}, and *Q* _{3} are the bending, symmetric stretching, and anti-symmetric stretching normal coordinates, respectively.

The normal and optimized coordinates of the O–H stretching modes of the water molecule superposed on the two-dimensional cross-section of the PES. The contours are drawn with an interval of 0.01 hartree.

The normal and optimized coordinates of the O–H stretching modes of the water molecule superposed on the two-dimensional cross-section of the PES. The contours are drawn with an interval of 0.01 hartree.

The normal and the optimized coordinates of the water dimer and the associated harmonic frequencies (the intramolecular modes only). (a) The normal and the optimized coordinates for the bending modes, which coincide with each other. (b) The normal and (c) the optimized coordinates for the O–H stretching modes.

The normal and the optimized coordinates of the water dimer and the associated harmonic frequencies (the intramolecular modes only). (a) The normal and the optimized coordinates for the bending modes, which coincide with each other. (b) The normal and (c) the optimized coordinates for the O–H stretching modes.

The normal and the optimized coordinates of ethylene with the associated harmonic frequencies. (a) The normal and the optimized coordinates (which coincide with each other) for the C=C stretching, bending, and out-of-plane modes. (b) The normal and (c) the optimized coordinates for the CH stretching modes.

The normal and the optimized coordinates of ethylene with the associated harmonic frequencies. (a) The normal and the optimized coordinates (which coincide with each other) for the C=C stretching, bending, and out-of-plane modes. (b) The normal and (c) the optimized coordinates for the CH stretching modes.

## Tables

The zero-point energy (ZPE) and the fundamental frequencies (in cm^{−1}) of the hydrogen fluoride tetramer obtained by VSCF, oc-VSCF, and VCI[4]-(8). The deviations and mean absolute deviations (MAD) from VCI[4]-(8) are given in the parentheses and in the last row, respectively.

The zero-point energy (ZPE) and the fundamental frequencies (in cm^{−1}) of the hydrogen fluoride tetramer obtained by VSCF, oc-VSCF, and VCI[4]-(8). The deviations and mean absolute deviations (MAD) from VCI[4]-(8) are given in the parentheses and in the last row, respectively.

The scaled force constants (in cm^{−1}) of the O–H stretching modes of the water molecule in the normal and optimized coordinates.

The scaled force constants (in cm^{−1}) of the O–H stretching modes of the water molecule in the normal and optimized coordinates.

The zero-point energy (ZPE) and the frequencies of the fundamentals, the first overtones, and combination tones (in cm^{−1}) of the water molecule calculated by VSCF, oc-VSCF, VCI-(2), oc-VCI-(2), VCI[*m*]-(8) (*m* = 1 − 3), and oc-VCI[*m*]-(8) (*m* = 1 − 3). The deviations and mean absolute deviations (MAD) from the VCI[3]-(8) results are given in the parentheses and in the rows labeled by MAD, respectively.

The zero-point energy (ZPE) and the frequencies of the fundamentals, the first overtones, and combination tones (in cm^{−1}) of the water molecule calculated by VSCF, oc-VSCF, VCI-(2), oc-VCI-(2), VCI[*m*]-(8) (*m* = 1 − 3), and oc-VCI[*m*]-(8) (*m* = 1 − 3). The deviations and mean absolute deviations (MAD) from the VCI[3]-(8) results are given in the parentheses and in the rows labeled by MAD, respectively.

The frequencies of the fundamentals, the first overtones, and combination tones (in cm^{−1}) of the water molecule calculated by oc-VCI-(2) with a QFF, a hybrid PES, and a grid PES and by oc-VCI[3]-(8) with the grid PES. The deviations and mean absolute deviations (MAD) from the experimental values are given in the parentheses and in the rows labeled by MAD, respectively.

The frequencies of the fundamentals, the first overtones, and combination tones (in cm^{−1}) of the water molecule calculated by oc-VCI-(2) with a QFF, a hybrid PES, and a grid PES and by oc-VCI[3]-(8) with the grid PES. The deviations and mean absolute deviations (MAD) from the experimental values are given in the parentheses and in the rows labeled by MAD, respectively.

The zero-point energy (ZPE) and fundamental frequencies (in cm^{−1}) of the water dimer (intramolecular modes only) calculated by VSCF, oc-VSCF, VCI[*m*]-(8) (*m* = 1 − 4), oc-VCI[*m*]-(8) (*m* = 1 − 4), VCI-(8), and oc-VCI-(8). The deviations and mean absolute deviations (MAD) from the VCI-(8) results are given in the parentheses and in the rows labeled by MAD, respectively.

The zero-point energy (ZPE) and fundamental frequencies (in cm^{−1}) of the water dimer (intramolecular modes only) calculated by VSCF, oc-VSCF, VCI[*m*]-(8) (*m* = 1 − 4), oc-VCI[*m*]-(8) (*m* = 1 − 4), VCI-(8), and oc-VCI-(8). The deviations and mean absolute deviations (MAD) from the VCI-(8) results are given in the parentheses and in the rows labeled by MAD, respectively.

The frequencies of the fundamentals, overtones, and combination tones (in cm^{−1}) of the O–H stretching modes of the water dimer calculated by VCI-(1), oc-VCI-(1), VCI-(2), and oc-VCI-(2) as well as oc-VCI-(8). The deviations and mean absolute deviations (MAD) from the oc-VCI-(8) results are given in the parentheses and in the rows labeled by MAD, respectively. The assignments for the overtones and combination tones are based on the oc-VCI-(8) results.

The frequencies of the fundamentals, overtones, and combination tones (in cm^{−1}) of the O–H stretching modes of the water dimer calculated by VCI-(1), oc-VCI-(1), VCI-(2), and oc-VCI-(2) as well as oc-VCI-(8). The deviations and mean absolute deviations (MAD) from the oc-VCI-(8) results are given in the parentheses and in the rows labeled by MAD, respectively. The assignments for the overtones and combination tones are based on the oc-VCI-(8) results.

The zero-point energy (ZPE) and fundamental frequencies (in cm^{−1}) of ethylene calculated by VSCF, oc-VSCF, VCI[*m*]-(6) (*m* = 1 − 5), oc-VCI[*m*]-(6) (*m* = 1 − 5), VCI-(6), and oc-VCI-(6). The deviations and mean absolute deviations (MAD) from the VCI-(6) results are given in the parentheses and in the rows labeled by MAD, respectively.

The zero-point energy (ZPE) and fundamental frequencies (in cm^{−1}) of ethylene calculated by VSCF, oc-VSCF, VCI[*m*]-(6) (*m* = 1 − 5), oc-VCI[*m*]-(6) (*m* = 1 − 5), VCI-(6), and oc-VCI-(6). The deviations and mean absolute deviations (MAD) from the VCI-(6) results are given in the parentheses and in the rows labeled by MAD, respectively.

The deviations and mean absolute deviation (MAD) in the zero-point energy (ZPE) and fundamental frequencies (in cm^{−1}) of ethylene obtained with *n*MR-QFF (*n* = 1 − 3) from those of 4MR-QFF.

The deviations and mean absolute deviation (MAD) in the zero-point energy (ZPE) and fundamental frequencies (in cm^{−1}) of ethylene obtained with *n*MR-QFF (*n* = 1 − 3) from those of 4MR-QFF.

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