^{1}and Mark S. Gordon

^{1,a)}

### Abstract

Improvements in the manner in which the potential energy surface (PES) is generated in the vibrational self-consistent field (VSCF) method have been implemented. The PES can now be computed over a flexible range of displacements and following normal mode displacement vectors expressed in internal rather than Cartesian coordinates, leading to higher accuracy of the calculated vibrational frequencies. The coarse-grained parallelization of the PES calculations, which is computationally by far the most expensive part of the VSCF method, enables the usage of higher levels of theory and larger basis sets. The new VSCF procedure is discussed and applied to three examples, , , and , to illustrate its accuracy and applicability.

This work was supported by a grant from the Air Force Office of Scientific Research. The authors are grateful for many helpful discussions with Dr. Jerry Boatz, Dr. Ryan Olson, Dr. Michael Schmidt, and Professors Benny Gerber and Lyudmila Slipchenko. The coarse-grained parallel calculations were carried out on the Scalable Computing Laboratory IBM Power4 cluster, and on the Iowa State University BlueGene/L computer, made available via grants from the National Science Foundation (Professor S. Aluru, PI), the Iowa State University Plant Sciences Institute, and Iowa State University.

I. INTRODUCTION

II. THEORETICAL APPROACH

A. Computational details

B. Adaptive grid of the PES points

C. Internal versus Cartesian normal mode displacement vectors

D. Recommended procedure

III. RESULTS AND DISCUSSION

A.

B.

C.

D. Summary

E. Coarse grained parallelization of the PES calculation

IV. CONCLUSIONS

### Key Topics

- Normal modes
- 41.0
- Set theory
- 12.0
- Ozone
- 8.0
- Basis sets
- 7.0
- Potential energy surfaces
- 7.0

## Figures

Optimized geometries of (a) at CCSD level of theory with the cc-pVTZ basis set in symmetry; (b) at CCSD(T) level of theory with the cc-pVTZ basis set in symmetry; (c) at CCSD(T) level of theory with the cc-pVDZ basis set in symmetry. Geometry parameters are listed in the set of internal coordinates that were used as the basis for the decomposition of the normal mode displacement vectors. Experimental values (Ref. 18) are given in parentheses.

Optimized geometries of (a) at CCSD level of theory with the cc-pVTZ basis set in symmetry; (b) at CCSD(T) level of theory with the cc-pVTZ basis set in symmetry; (c) at CCSD(T) level of theory with the cc-pVDZ basis set in symmetry. Geometry parameters are listed in the set of internal coordinates that were used as the basis for the decomposition of the normal mode displacement vectors. Experimental values (Ref. 18) are given in parentheses.

Flow chart of the procedure for calculating vibrational frequencies using the VSCF method in internal coordinates. VSCF-diagonal stands for anharmonic frequencies calculated without coupling, while VSCF-PT2 stands for anharmonic coupled vibrational frequencies calculated using a second order perturbation theory correction.

Flow chart of the procedure for calculating vibrational frequencies using the VSCF method in internal coordinates. VSCF-diagonal stands for anharmonic frequencies calculated without coupling, while VSCF-PT2 stands for anharmonic coupled vibrational frequencies calculated using a second order perturbation theory correction.

CCSD(T)/cc-pVTZ in Cartesian coordinates (dotted line) and internal coordinates (solid line). (a) H1–O2 bond distance is plotted against O4–N3–O2–H1 torsion angle at the VSCF points along the torsion mode ; (b) H1–O2 bond distance is plotted against the N3–O2–H1 angle at the VSCF points along the bending mode .

CCSD(T)/cc-pVTZ in Cartesian coordinates (dotted line) and internal coordinates (solid line). (a) H1–O2 bond distance is plotted against O4–N3–O2–H1 torsion angle at the VSCF points along the torsion mode ; (b) H1–O2 bond distance is plotted against the N3–O2–H1 angle at the VSCF points along the bending mode .

CCSD(T)/cc-pVDZ in Cartesian coordinates (dotted line) and internal coordinates (solid line). (a) the H5–O2 bond distance is plotted against the H5–O2–N1–O3 torsion angle at the VSCF points along torsion mode . (b) the H5–O2 bond distance is plotted against the H5–O2–N1 bending angle on the VSCF points along bending mode .

CCSD(T)/cc-pVDZ in Cartesian coordinates (dotted line) and internal coordinates (solid line). (a) the H5–O2 bond distance is plotted against the H5–O2–N1–O3 torsion angle at the VSCF points along torsion mode . (b) the H5–O2 bond distance is plotted against the H5–O2–N1 bending angle on the VSCF points along bending mode .

Total errors obtained as a difference between calculated and experimental vibrational frequencies.

Total errors obtained as a difference between calculated and experimental vibrational frequencies.

Coarse-grained parallelization of PES calculations in the VSCF method.

Coarse-grained parallelization of PES calculations in the VSCF method.

Speed up curve for parallelized VSCF at the MP2 level of theory with the cc-pVDZ basis set carried out on molecule. The dashed line with data point represented as triangles is the theoretical curve that follows linear speed up. The actual speed up is represented by the solid line with circles as data points. Beneath each point in parentheses are given the number of points per each group that needed to be calculated, followed up by the timings in minutes.

Speed up curve for parallelized VSCF at the MP2 level of theory with the cc-pVDZ basis set carried out on molecule. The dashed line with data point represented as triangles is the theoretical curve that follows linear speed up. The actual speed up is represented by the solid line with circles as data points. Beneath each point in parentheses are given the number of points per each group that needed to be calculated, followed up by the timings in minutes.

## Tables

MP2 and CCSD diagonal frequencies with the cc-pVDZ (ccd), cc-pVTZ (cct), and cc-pVQZ (ccq) basis sets. The amplitude maps onto the energy level , see text. Increasing the value of amp corresponds to a larger spacing of the PES points along the normal mode.

MP2 and CCSD diagonal frequencies with the cc-pVDZ (ccd), cc-pVTZ (cct), and cc-pVQZ (ccq) basis sets. The amplitude maps onto the energy level , see text. Increasing the value of amp corresponds to a larger spacing of the PES points along the normal mode.

, MP2 and CCSD calculations with the cc-pVDZ (ccd), cc-pVTZ (cct), and cc-pVQZ (ccq) basis sets. The actual values of harmonic (harm), scaled harmonic [harm(scal)], anharmonic in Cartesian coordinates [anhr(cart)], anharmonic in internals using two bonds and the angle [anhr(int)2ba], anharmonic in internals using three bonds [anhr(int)3b], and experimental (Refs. 27 and 28) (exp) values are given.

, MP2 and CCSD calculations with the cc-pVDZ (ccd), cc-pVTZ (cct), and cc-pVQZ (ccq) basis sets. The actual values of harmonic (harm), scaled harmonic [harm(scal)], anharmonic in Cartesian coordinates [anhr(cart)], anharmonic in internals using two bonds and the angle [anhr(int)2ba], anharmonic in internals using three bonds [anhr(int)3b], and experimental (Refs. 27 and 28) (exp) values are given.

Experimental frequencies (Refs. 27 and 28) and the errors between calculated and experimental frequencies for . Calculations were performed at MP2 and CCSD levels of theory with the cc-pVDZ (ccd), cc-pVTZ (cct), and cc-pVQZ (ccq) basis sets. The following frequencies were calculated: harmonic (harm), scaled harmonic [harm(scal)], anharmonic in Cartesian coordinates [anhr(cart)], and anharmonic in internal coordinates [anhr(int)].

Experimental frequencies (Refs. 27 and 28) and the errors between calculated and experimental frequencies for . Calculations were performed at MP2 and CCSD levels of theory with the cc-pVDZ (ccd), cc-pVTZ (cct), and cc-pVQZ (ccq) basis sets. The following frequencies were calculated: harmonic (harm), scaled harmonic [harm(scal)], anharmonic in Cartesian coordinates [anhr(cart)], and anharmonic in internal coordinates [anhr(int)].

Experimental frequencies (Ref. 19) and the errors between calculated and experimental frequencies for . Calculations were performed with PM2 and CCSD(T) using the cc-pVDZ (ccd), and cc-pVTZ (cct) basis sets. The following frequencies were calculated: harmonic (harm), scaled harmonic [harm(scal)], anharmonic in Cartesian coordinates [anhr(cart)], and anharmonic in internal coordinates [anhr(int)].

Experimental frequencies (Ref. 19) and the errors between calculated and experimental frequencies for . Calculations were performed with PM2 and CCSD(T) using the cc-pVDZ (ccd), and cc-pVTZ (cct) basis sets. The following frequencies were calculated: harmonic (harm), scaled harmonic [harm(scal)], anharmonic in Cartesian coordinates [anhr(cart)], and anharmonic in internal coordinates [anhr(int)].

Calculated anharmonic vibrational frequencies for and at CCSD(T) with the cc-pVDZ basis set. Frequencies were calculated without coupling (diag) or with coupling but only using VSCF level without a PT2 correction (vscf). Depending on the type of coordinates in which the PES was created, diag(cart) and vscf(cart) correspond to Cartesian coordinates, while diag(int) and vscf(int) correspond to internal coordinates. Difference between diag(cart) and vscf(cart) is labeled as diff(cart), while difference in diag(int) and vscf(int) is labeled as diff(int). Difference between diag(cart) and diag(int) is given as diff(diag).

Calculated anharmonic vibrational frequencies for and at CCSD(T) with the cc-pVDZ basis set. Frequencies were calculated without coupling (diag) or with coupling but only using VSCF level without a PT2 correction (vscf). Depending on the type of coordinates in which the PES was created, diag(cart) and vscf(cart) correspond to Cartesian coordinates, while diag(int) and vscf(int) correspond to internal coordinates. Difference between diag(cart) and vscf(cart) is labeled as diff(cart), while difference in diag(int) and vscf(int) is labeled as diff(int). Difference between diag(cart) and diag(int) is given as diff(diag).

Experimental frequencies (Ref. 19) and the errors between calculated and experimental frequencies for . Calculations were performed with MP2 and CCSD(T) and the cc-pVDZ (ccd), and cc-pVTZ (cct) basis sets. The following frequencies were calculated: harmonic (harm), scaled harmonic (harm(scal)), anharmonic in Cartesian coordinates [anhr(cart)], and anharmonic in internal coordinates [anhr(int)].

Experimental frequencies (Ref. 19) and the errors between calculated and experimental frequencies for . Calculations were performed with MP2 and CCSD(T) and the cc-pVDZ (ccd), and cc-pVTZ (cct) basis sets. The following frequencies were calculated: harmonic (harm), scaled harmonic (harm(scal)), anharmonic in Cartesian coordinates [anhr(cart)], and anharmonic in internal coordinates [anhr(int)].

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