^{1,a)}, Sudip Chattopadhyay

^{2,b)}, Uttam Sinha Mahapatra

^{3,c)}and Karl F. Freed

^{4,d)}

### Abstract

The improved virtual orbital-complete active space configuration interaction (IVO-CASCI) method is extended to determine the geometry and vibrational frequencies for ground and *excited* electronic states using an analytical total energy gradient scheme involving both first and second order analytical derivatives. Illustrative applications consider the ground state geometries of the benzene , biphenyl , and alanine dipeptide molecules. In addition, the IVO-CASCI geometry optimization has been performed for the first excited singlet and triplet states of benzene to assess its applicability for excited and open-shell systems. The symmetry benzene triplet optimization produces a saddle point, and a descent along the unstable mode produces the stable minimum. Comparisons with Hartree–Fock, second order Möller–Plesset perturbation theory, complete active space self-consistent field (CASSCF), and density functional theory demonstrate that the IVO-CASCI approach generally fares comparable to or better for all systems studied. The vibrational frequencies of the benzene and biphenyl molecules computed with the analytical gradient based IVO-CASCI method agree with the experiment and with other accurate theoretical estimates. Satisfactory agreement between our results, other benchmark calculations, and available experiment demonstrates the efficacy and potential of the method. The close similarity between CASSCF and IVO-CASCI optimized geometries and the greater computational efficiency of the IVO-CASCI method suggests the replacement of CASSCF treatments by the IVO-CASCI approach, which is free from the convergence problems that often plague CASSCF treatments.

This research is supported in part by the Department of Science and Technology (DST), India (Grant No. SR/S1/PC-32/2005) and the NSF (Grant No. CHE-0749788). The authors also thank Dr. Jeff Hammond for helpful discussions.

I. INTRODUCTION

II. THEORY

A. Generation of improved virtual orbitals

B. Formal structure of the first order energy gradients

III. NUMERICAL APPLICATIONS

A. Benzene

B. Biphenyl

1. Twisted geometry

2. Coplanar geometry

3. Perpendicular geometry

C. Alanine dipeptide

IV. CONCLUSIONS

### Key Topics

- Density functional theory
- 26.0
- Excited states
- 19.0
- Ground states
- 14.0
- Wave functions
- 11.0
- Peptides
- 10.0

## Figures

Labeling of molecular structure for the triplet states of benzene.

Labeling of molecular structure for the triplet states of benzene.

Energy as a function of the distance along the reaction path in mass weighted Cartesian space. The path is along the direction of the largest magnitude of the downhill component of the imaginary normal mode.

Energy as a function of the distance along the reaction path in mass weighted Cartesian space. The path is along the direction of the largest magnitude of the downhill component of the imaginary normal mode.

Structure of coplanar biphenyl molecule.

Structure of coplanar biphenyl molecule.

The SCF , MP2 (×), DFT , and IVO-CASCI (◻) ground state energies of biphenyl molecules as a function of torsional angle.

The SCF , MP2 (×), DFT , and IVO-CASCI (◻) ground state energies of biphenyl molecules as a function of torsional angle.

Geometrical structure of alanine dipeptide (structure I).

Geometrical structure of alanine dipeptide (structure I).

Geometrical structure of alanine dipeptide (structure II).

Geometrical structure of alanine dipeptide (structure II).

## Tables

Optimized and experimental geometries for the ground , first excited singlet , and triplet states of benzene from theories using the cc-pVDZ basis set. The bond angles and bond distances are given in degrees and angstrom, respectively.

Optimized and experimental geometries for the ground , first excited singlet , and triplet states of benzene from theories using the cc-pVDZ basis set. The bond angles and bond distances are given in degrees and angstrom, respectively.

Optimized triplet state geometries of benzene. Bond distances, bond angle, energy with respect to ground state, and CPU time are in angstrom, degrees, kcal/mol, and seconds.

Optimized triplet state geometries of benzene. Bond distances, bond angle, energy with respect to ground state, and CPU time are in angstrom, degrees, kcal/mol, and seconds.

Comparison of calculated and experimental vibrational frequencies (in ) of ground and first excited singlet state of benzene. [Entries are the differences with respect to experiment (theory experiment).]

Comparison of calculated and experimental vibrational frequencies (in ) of ground and first excited singlet state of benzene. [Entries are the differences with respect to experiment (theory experiment).]

Calculated harmonic vibrational frequencies (in ) for the first excited and states of benzene.

Calculated harmonic vibrational frequencies (in ) for the first excited and states of benzene.

Calculated vibrational frequencies (in ) for the first excited triplet state of benzene at the saddle point and at the minima along the internal reaction coordinate (Fig. 2).

Calculated vibrational frequencies (in ) for the first excited triplet state of benzene at the saddle point and at the minima along the internal reaction coordinate (Fig. 2).

Computed equilibrium geometry of biphenyl molecule from basis set calculations. The bond angles, bond distances, and energies are given in degrees, angstrom, and a.u., respectively.

Computed equilibrium geometry of biphenyl molecule from basis set calculations. The bond angles, bond distances, and energies are given in degrees, angstrom, and a.u., respectively.

Calculated equilibrium geometry for the twisted biphenyl molecule from the pc-1 basis set (except for the CASSCF) calculation. The bond angles, bond distances, and energy are given in degrees, angstrom, and a.u., respectively.

Calculated equilibrium geometry for the twisted biphenyl molecule from the pc-1 basis set (except for the CASSCF) calculation. The bond angles, bond distances, and energy are given in degrees, angstrom, and a.u., respectively.

Computed equilibrium geometry for the coplanar biphenyl molecule from basis set calculations. The bond angles, bond distances, and relative energy with respect to the twisted geometry are given in degrees, angstrom, and kJ/mol, respectively. The bond lengths and bond angles given here are the differences with respect to experiment.

Computed equilibrium geometry for the coplanar biphenyl molecule from basis set calculations. The bond angles, bond distances, and relative energy with respect to the twisted geometry are given in degrees, angstrom, and kJ/mol, respectively. The bond lengths and bond angles given here are the differences with respect to experiment.

Computed equilibrium geometry for the coplanar biphenyl molecule from the pc-1 basis set (except for the CASSCF) calculation. Entries (for bond angles and bond lengths) are the differences with respect to experiment. The bond angles, bond distances, and energy are given in degrees, angstrom, and kJ/mol, respectively.

Computed equilibrium geometry for the coplanar biphenyl molecule from the pc-1 basis set (except for the CASSCF) calculation. Entries (for bond angles and bond lengths) are the differences with respect to experiment. The bond angles, bond distances, and energy are given in degrees, angstrom, and kJ/mol, respectively.

Comparison of calculated and experimental harmonic vibrational frequencies (in ) for the first few modes of twisted biphenyl molecule.

Comparison of calculated and experimental harmonic vibrational frequencies (in ) for the first few modes of twisted biphenyl molecule.

Computed equilibrium geometry for the perpendicular biphenyl molecule from basis set calculations. The bond angles, bond distances, and (stabilization energy) energies are given in degrees, angstrom, and kJ/mol, respectively.

Computed equilibrium geometry for the perpendicular biphenyl molecule from basis set calculations. The bond angles, bond distances, and (stabilization energy) energies are given in degrees, angstrom, and kJ/mol, respectively.

Computed equilibrium geometry for the perpendicular biphenyl molecule from pc-1 basis set (except for CASSCF) calculations. The bond angles, bond distances, and energies are given in degrees, angstrom, and kJ/mol, respectively.

Computed equilibrium geometry for the perpendicular biphenyl molecule from pc-1 basis set (except for CASSCF) calculations. The bond angles, bond distances, and energies are given in degrees, angstrom, and kJ/mol, respectively.

Some representative structural parameters for the ground state of alanine dipeptide determined with various theories using the DZP basis set. Bond lengths, bond angles, and relative stability are expressed in angstrom, degrees, and Kcal/mol, respectively.

Some representative structural parameters for the ground state of alanine dipeptide determined with various theories using the DZP basis set. Bond lengths, bond angles, and relative stability are expressed in angstrom, degrees, and Kcal/mol, respectively.

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