Young D.C. - Computational chemistry (2001)(en)

31.3 EMPIRICAL METHODS 253

local origin (LORG) methods are similar. Both are based on identities and closure relations that are rigorously correct for complete basis sets. These are reasonable approximations for ®nite basis sets. The two methods are equivalent in the limit of a complete basis set.

The individual gauges for atoms in molecules (IGAIM) method is based on Bader's atoms in molecules analysis scheme. This method yields results of comparable accuracy to those of the other methods. However, this technique is seldom used due to large CPU time demands.

There have also been methods designed for use with perturbation theory and MCSCF calculations. Correlation e¨ects are necessary for certain technically di½cult molecules, such as CO, N2, HCN, F2, and N2O.

Density functional theory calculations have shown promise in recent studies. Gradient-corrected or hybrid functionals must be used. Usually, it is necessary to employ a moderately large basis set with polarization and di¨use functions along with these functionals.

The methods listed thus far can be used for the reliable prediction of NMR chemical shifts for small organic compounds in the gas phase, which are often reasonably close to the liquid-phase results. Heavy elements, such as transition metals and lanthanides, present a much more di½cult problem. Mass defect and spin-coupling terms have been found to be signi®cant for the description of the NMR shielding tensors for these elements. Since NMR is a nuclear e¨ect, core potentials should not be used.

31.2SEMIEMPIRICAL METHODS

There is one semiempirical program, called HyperNMR, that computes NMR chemical shifts. This program goes one step further than other semiempiricals by de®ning di¨erent parameters for the various hybridizations, such as sp2 carbon vs. sp3 carbon. This method is called the typed neglect of di¨erential overlap method (TNDO/1 and TNDO/2). As with any semiempirical method, the results are better for species with functional groups similar to those in the set of molecules used to parameterize the method.

Another semiempirical method, incorporated in the VAMP program, combines a semiempirical calculation with a neural network for predicting the chemical shifts. Semiempirical calculations are useful for large molecules, but are not generally as accurate as ab initio calculations.

31.3EMPIRICAL METHODS

The simplest empirical calculations use a group additivity method. These calculations can be performed very quickly on small desktop computers. They are most accurate for a small organic molecule with common functional groups. The prediction is only as good as the aspects of molecular structure being par-

254 31 COMPUTING NMR CHEMICAL SHIFTS

ameterized. For example, they often do not distinguish between cis and trans isomers. Due to the limited accuracy, this method is more often used as a tool to check for reasonable results, but not as a rigorous prediction method.

Another technique employs a database search. The calculation starts with a molecular structure and searches a database of known spectra to ®nd those with the most similar molecular structure. The known spectra are then used to derive parameters for inclusion in a group additivity calculation. This can be a fairly sophisticated technique incorporating weight factors to account for how closely the known molecule conforms to typical values for the component functional groups. The use of a large database of compounds can make this a very accurate technique. It also ensures that liquid, rather than gas-phase, spectra are being predicted.

31.4RECOMMENDATIONS

In general, the computation of absolute chemical shifts is a very di½cult task. Computing shifts relative to a standard, such as TMS, can be done more accurately. With some of the more approximate methods, it is sometimes more reliable to compare the shifts relative to the other shifts in the compound, rather than relative to a standard compound. It is always advisable to verify at least one representative compound against the experimental spectra when choosing a method. The following rules of thumb can be drawn from a review of the literature:

1.Database techniques are very fast and very accurate for organic molecules with common functional groups.

2.Ab initio methods are accurate and can be reliably applied to unusual structures and inorganic compounds. In most cases, HF calculations are fairly good for organic molecules. Large basis sets should be used.

3.For large molecules, the choice between semiempirical calculations and empirical calculations should be based on a test case.

4.Correlated and relativistic quantum mechanical calculations give the highest possible accuracy and are necessary for heavy atoms or correla- tion-sensitive systems.

BIBLIOGRAPHY

Introductory descriptions are in

M. F. Schlecht, Molecular Modeling on the PC Wiley-VCH, New York (1998). E. K. Wilson, Chem. & Eng. News Sept. 28 (1998).

P.W. Atkins, R. S. Friedman, Molecular Quantum Mechanics Third Edition Oxford, Oxford (1997).

BIBLIOGRAPHY 255

M. Karplus, R. N. Porter, Atoms & Molecules: An Introduction For Students of Physical Chemistry W. A. Benjamin, Inc., Menlo Park (1970).

Books about NMR modeling are

B.Born, H. W. Spiess, Ab Initio Calculations of Conformational E¨ects on 13C NMR Spectra of Amorphous Polymers Springer-Verlag, New York (1997).

I. Ando, G. A. Webb, Theory of NMR Parameters Academic Press, London (1983).

The following book gives a tutorial and examples for using ab initio methods. Some printings have an error in the listed TMS values. An eratta is available from Gaussian, Inc.

J.B. Foresman, á. Frisch, Exploring Chemistry with Electronic Structure Methods Second Edition Gaussian, Pittsburgh (1996).

Review articles are

U. Fleischer, C. van WuÈllen, w. Kutzelnigg, Encycl. Comput. Chem. 3, 1827 (1998). M. BuÈhl, Encycl. Comput. Chem. 3, 1835 (1998).

M. Kaupp, V. G. Malkin, O. L. Malkina, Encycl. Comput. Chem. 3, 1857 (1998).

C.J. Jameson, Annu. Rev. Phys. Chem. 47, 135 (1996).

D.B. Chesnut, Rev. Comput. Chem. 8, 245 (1996).

J.R. Cheeseman, G. W. Trucks, T. A. Keith, M. J. Frisch, J. Chem. Phys. 104, 5497 (1996).

D. B. Chesnut, Annual Reports on NMR Spectroscopy 29, 71 (1994).

C.J. Jameson, Chem. Rev. 91, 1375 (1991).

D.B. Chesnut, Annual Reports on NMR Spectroscopy 21, 51 (1989).

C. Giessner-Prettre, B. Pullman, Quarterly Reviews of Biophysics 20, 113 (1987).

C.J. Jameson, H. J. Osten, Annual Reports on NMR Spectroscopy 17, 1 (1986).

The group additivity technique is presented in

E.Pretsch, J. Seibl, W. Simon, T. Clerc, Tabellen zur StrukturaufklaÈrung Organischer Verbindungen mit Spektroskopischen Methoden Springer-Verlag, Berlin (1981).

As implied by this, the polarizabilities can be formulated as derivatives of the dipole moment with respect to the incident electric ®eld. Below these derivatives are given, with subscripts added to indicate their tensor nature:

These expressions are only correct for wave functions that obey the Hellmann± Feynman theorem. However, these expressions have been used for other methods, where they serve as a reasonable approximation. Methods that rigorously obey the Hellmann±Feynman theorem are SCF, MCSCF, and Full CI. The change in energy from nonlinear e¨ects is due to a change in the electron density, which creates an induced dipole moment and, to a lesser extent, induced higher-order multipoles.

After examining these de®nitions, several conclusions can be drawn, which have been veri®ed theoretically and experimentally. One is that a molecule with a center of inversion will have no hyperpolarizability …b ˆ 0†. Molecules with a large dipole moment and a means for electron density to shift will have large hyperpolarizabilities. For example, organic systems with electron-donating groups and electron-withdrawing groups at opposite ends of a conjugated system generally have large hyperpolarizabilities.

The de®nitions given above re¯ect static polarizabilities that are due to the presence of a static electric ®eld. Nonlinear optical properties are the result of the oscillating electric ®eld component of the incident light. Static hyperpolarizabilities are often computed and then employed to predict nonlinear optical properties by using an experimentally determined correction factor. Alternatively, time-dependent calculations can be used to predict experimental results directly. There are several di¨erent nonlinear optical properties due to several incoming photons of light …n1; n2; n3† and result in an exiting photon of the same or a di¨erent frequency …ns†. The list of outgoing and incoming photons is typically denoted with the notation ÿns; n1, n2, n3. The nonlinear optical properties are summarized in Table 32.1. Each of these can be computed from the appropriate frequency-dependent terms.

32.2COMPUTATIONAL ALGORITHMS

There are several ways in which to compute polarizabilities and hyperpolarizabilities from semiempirical or ab initio wave functions. One option is to take

32.4 RECOMMENDATIONS 259

RPA, and CPHF. Time-dependent Hartree±Fock (TDHF) is the Hartree±Fock approximation for the time-dependent SchroÈdinger equation. CPHF stands for coupled perturbed Hartree±Fock. The random-phase approximation (RPA) is also an equivalent formulation. There have also been time-dependent MCSCF formulations using the time-dependent gauge invariant approach (TDGI) that is equivalent to multicon®guration RPA. All of the time-dependent methods go to the static calculation results in the n ˆ 0 limit.

32.3LEVEL OF THEORY

Polarizabilities and hyperpolarizabilities have been calculated with semiempirical, ab initio, and DFT methods. The general conclusion from these studies is that a high level of theory is necessary to correctly predict nonlinear optical properties.

Semiempirical calculations tend to be qualitative. In some cases, the correct trends have been predicted. In other cases, semiempirical methods give incorrect signs as well as unreasonable magnitudes.

Ab initio methods can yield reliable, quantitatively correct results. It is important to use basis sets with di¨use functions and high-angular-momentum polarization functions. Hyperpolarizabilities seem to be relatively insensitive to the core electron description. Good agreement has been obtained between ECP basis sets and all electron basis sets. DFT methods have not yet been used widely enough to make generalizations about their accuracy.

Explicitly correlated wave functions have been shown to give very accurate results. Unfortunately, these calculations are only tractable for very small molecules.

There have been some attempts to compute nonlinear optical properties in solution. These studies have shown that very small variations in the solvent cavity can give very large deviations in the computed hyperpolarizability. The valence bond charge transfer (VB-CT) method created by Goddard and coworkers has had some success in reproducing solvent e¨ect trends and polymer results (the VB-CT-S and VB-CTE forms, respectively).

32.4RECOMMENDATIONS

Unfortunately, it is necessary to use very computationally intensive methods for computing accurate nonlinear optical properties. The following list of alternatives is ordered, starting with the most accurate and likewise most computationintensive techniques:

1.Time-dependent calculations with highly correlated methods

2.Explicitly correlated methods

3.CCSD(T)

26032 NONLINEAR OPTICAL PROPERTIES

4.CISD, CCSD, or MP4

5.TDHF, RPA, or CPHF

6.MP2 or MP3

7.SCF or DFT

8.Semiempirical methods where they have been shown to reproduce the correct trends

BIBLIOGRAPHY

Introductory descriptions are in

S.P. Karna, A. T. Yeates, Nonlinear Opticial Materials: Theory and Modeling S. P. Karna, A. T. Yeates, Eds., 1, American Chemical Society, Washington (1996).

R. W. Boyd, Nonlinear Optics Academic Press, San Diego (1992).

Review articles are

H. A. Kurtz, D. S. Dudis, Rev. Comput. Chem. 12, 241 (1998).

R. J. Bartlett, H. Sekino, Nonlinear Opticial Materials: Theory and Modeling S. P. Karna, A. T. Yeates, Eds., 23, American Chemical Society, Washington (1996).

D. M. Bishop, Adv. Quantum Chem. 25, 1 (1994).

D. P. Shelton, J. E. Rice, Chem. Rev. 94, 3 (1994).

J. L. BreÂdas, C. Adant, P. Tackx, A. Persoons, Chem. Rev. 94, 243 (1994). D. R. Kanis, M. A. Ratner, T. J. Marks, Chem. Rev. 94, 195 (1994).

A. A. Hasanein, Adv. Chem. Phys. 85, 415 (1994).

W. T. Co¨ey, Y. D. Kalmykov, E. S. Massawe, Adv. Chem. Phys. 85, 667 (1994). D. M. Bishop, Rev. Mod. Phys. 62, 343 (1990).

Mathematical treatments are in

W. Alexiewicz, B. Kasprowicz-Kielich, Adv. Chem. Phys. 85, 1 (1994). D. L. Andrews, Adv. Chem. Phys. 85, 545 (1994).

G.C. Schatz, M. A. Ratner, Quantum Mechanics in Chemistry Prentice Hall, Englewood Cli¨s (1993).

C.E. Dykstra, J. D. Augspurser, B. Kirtman, D. J. Malik, Rev. Comput. Chem. 1, 83 (1990).

Other pertinent articles are

P.Korambath, H. A. Kurtz, Nonlinear Opticial Materials: Theory and Modeling S. P. Karna, A. T. Yeates, Eds., 133, American Chemical Society, Washington (1996).

W. A. Goddard, III, D. Lu, G. Chen, J. W. Perry, Computer-Aided Molecular Design 341 C. H. Reynolds, M. K. Holloway, H. K. Cox, Ed., American Chemical Society, Washington (1995).

W. A. Parkinson, J. Oddershede, J. Chem. Phys. 94, 7251 (1991).

H. A. Kurtz, J. J. P. Stewart, K. M. Dieter, J. Comput. Chem. 11, 82 (1990).