Chapter 7
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Comparison of Local, Nonlocal, and Hybrid Density Functionals Using Vibrational Absorption and Circular Dichroism Spectroscopy P. J . Stephens , F. J. Devlin , C. S. Ashvar , K. L. Bak , P. R. Taylor , and M. J. Frisch 1
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Department of Chemistry, University of Southern California, Los Angeles, CA 90089-0482 UNI-C, Olof Palmes Allé 38, DK-8200 Aarhus N, Denmark San Diego Supercomputer Center, P.O. Box 85608, San Diego, CA 92186-9784 Lorentzian, Inc., 140 Washington Avenue, North Haven, CT 06473 1
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A b i n i t i o calculations of vibrational unpolarized absorption and circular dichroism spectra of 6,8dioxabicyclo[3.2.1] octane are reported. The harmonic force field is calculated via Density Functional Theory using three density functionals: L S D A , B L Y P , B 3 L Y P . The basis set is 6-31G*. Spectra calculated using the hybrid B 3 L Y P functional give the best agreement with experimental spectra, demonstrating that this functional is superior in accuracy to the BLYP and LSDA functionals. Density Functional Theory (DFT) (1) is increasingly the methodology of choice i n ab initio calculations. A t the same time, the number, variety, and sophistication of density functionals is also increasing. The choice of D F T is thus accompanied by the problem of selecting the optimum functional. In this paper, we demonstrate the utility of vibrational unpolarized absorption spectra and vibrational circular dichroism spectra in assessing the accuracies of density functionals. Specifically, we report calculations of the vibrational absorption and circular dichroism spectra of 6,8-dioxabicyclo[3.2.1]octane, 1, using three density functionals and evaluate the relative accuracies of these functionals by comparison of the predicted spectra to experiment. Broadly speaking, density functionals in active use today can be classed as: (i) local; (ii) non-local; or (iii) hybrid. Local functionals 0097-6156/96/0629-0105$15.00/0 © 1996 American Chemical Society In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
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106
C H E M I C A L APPLICATIONS O F DENSITY-FUNCTIONAL T H E O R Y
are the simplest and were the first to be used. Non-local (or "gradient") corrections were then added, creating non-local functionals. Very recently, even more sophisticated functionals have been introduced, based on the Adiabatic Connection Method of Becke (2), which are referred to as hybrid functionals. In this work we utilize one functional from each class: (i) the L o c a l Spin Density Approximation ( L S D A ) functional; (ii) the non-local Becke-LeeYang-Parr ( B L Y P ) functional; and (iii) the Becke 3-Lee-Yang-Parr ( B 3 L Y P ) hybrid functional. Since the development and implementation of analytical derivative methods for D F T energy gradients, vibrational frequencies have been used extensively in evaluating the accuracies of density functionals (3). The recent development and implementation of analytical derivative methods for D F T energy second derivatives (Hessians) (4) has greatly increased the efficiency of calculations of D F T harmonic frequencies. In contrast, vibrational intensities have not been substantially utilized. In this work we demonstrate the advantages of incorporating vibrational intensities in the evaluation of density functionals. W e further demonstrate the additional advantages of the use of both unpolarized absorption and circular d i c h r o i s m intensities. U n p o l a r i z e d vibrational absorption spectroscopy is widely utilized and well-understood. Vibrational Circular Dichroism ( V C D ) spectroscopy (5) is not yet as widely utilized. Our work will illustrate its substantial potential. Methods D F T harmonic force fields and atomic polar tensors (APTs) were calculated using G A U S S I A N 9 2 / D F T and the three density functionals: 1) L S D A (local spin density approximation): this uses the standard local exchange functional (6) and the local correlation functional of Vosko, Wilk, and Nusair ( V W N ) (7). 2) B L Y P : this combines the standard local exchange functional with the gradient correction of Becke (6) and uses the Lee-Yang-Parr correlation functional (8) (which also includes density gradient terms). 3) B e c k e 3 L Y P : this functional is a hybrid of exact (Hartree-Fock) exchange with local and gradient-corrected exchange and correlation terms, as first suggested by Becke (2). The exchange-correlation functional proposed and tested by Becke was E
x c
= (1 - a ) E ^ 0
Here ΔΕ
Χ
88
and ΔΕ™
91
D A
+ a E f + a AE 0
x
x
B88
+E*
D A
+ a AE™ c
(1)
91
is Becke's gradient correction to the exchange functional, is the Perdew-Wang gradient correction to the correlation
functional (9). Becke suggested coefficients a = 0.2, a = 0 . 7 2 , and 0
x
In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
7. STEPHENS ET AL.
107
Comparison of Density Functioruûs
a =0.81 based on fitting to heats of formation of small molecules. Only single-point energies were involved in the fit; no molecular geometries or frequencies were used. The B e c k e 3 L Y P functional in Gaussian 9 2 / D F T uses the values of a , a , and a suggested by Becke but uses L Y P for the correlation functional. Since L Y P does not have an easily separable local component, the V W N local correlation expression has been used to provide the different coefficients of local and gradient corrected correlation functionals: c
0
Downloaded by STANFORD UNIV GREEN LIBR on October 10, 2012 | http://pubs.acs.org Publication Date: May 5, 1996 | doi: 10.1021/bk-1996-0629.ch007
E°
3LYP
= (1 - a ) E ^ 0
D A
x
c
+ a E f + a AE° + a E ^ + (1 - a ) E 0
88
x
c
c
c
VWN
(2)
The standard fine grid in Gaussian 92/DFT (70) was used in all D F T calculations. This grid was produced from a basic grid having 75 radial shells and 302 angular points per radial shell for each atom and by reducing the number of angular points for different ranges of radial shells, leaving about 7000 points per atom while retaining similar accuracy to the original (75,302) grid. Becke's numerical integration techniques (11) were employed. Atomic axial tensors ( A A T s ) (12) were calculated using the distributed origin gauge (12,13), in which the A A T of nucleus λ , (M„ )°, with respect to origin Ο is given by p
(Μ*)
0
= (Ο
Χ
+- ^ Σ ^
