Young - Computational chemistry

Computational Chemistry: A Practical Guide for Applying Techniques to Real-World Problems. David C. Young Copyright ( 2001 John Wiley & Sons, Inc.

ISBNs: 0-471-33368-9 (Hardback); 0-471-22065-5 (Electronic)

32 Nonlinear Optical Properties

Nonlinear optical properties are of interest due to their potential usefulness for unique optical devices. Some of these applications are frequency-doubling devices, optical signal processing, and optical computers.

Most of the envisioned practical applications for nonlinear optical materials would require solid materials. Unfortunately, only gas-phase calculations have been developed to a reliable level. Most often, the relationship between gasphase and condensed-phase behavior for a particular class of compounds is determined experimentally. Theoretical calculations for the gas phase are then scaled accordingly.

32.1NONLINEAR OPTICAL PROPERTIES

When light is incident on a material, the optical electric ®eld E results in a polarization P of the material. The polarization can be expressed as the sum of the linear polarization PL and a nonlinear polarization PNL:

P ˆ PL ‡ PNL	…32:1†
PL ˆ w…1† E	…32:2†
PNL ˆ w…2† EE ‡ w…3† EEE ‡	…32:3†

The susceptibility tensors w…n† give the correct relationship for the macroscopic material. For individual molecules, the polarizability a, hyperpolarizability b, and second hyperpolarizability g, can be de®ned; they are also tensor quantities. The susceptibility tensors are weighted averages of the molecular values, where the weight accounts for molecular orientation. The obvious correspondence is correct, meaning that w…1† is a linear combination of a values, w…2† is a linear combination of b values, and so on.

The molecular quantities can be best understood as a Taylor series expansion. For example, the energy of the molecule E would be the sum of the energy without an electric ®eld present, E0, and corrections for the dipole, polarizability, hyperpolarizability, and the like:

E ˆ E0	ÿ m E ÿ	2! a E2		ÿ	3! b E3		ÿ	4! g E4		ÿ	…32:4†
		1			1			1

256

32.2 COMPUTATIONAL ALGORITHMS

257

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:

	qm
aij ˆ	i	E!0				…32:5†
	qEj
	q2m			!
bijk ˆ	i					…32:6†
	qEjqE		k
				E!0
	q3m				!
gijkl ˆ	i					…32:7†
	qEjqEkqEl
					E!0

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

258	32	NONLINEAR OPTICAL PROPERTIES
TABLE 32.1 Nonlinear Optical Properties
ÿns;n1,n2,n3		Abbreviation	Name
		Polarizability a
0;0			Static polarizability
ÿn;n			Frequency-dependent polarizability

Hyperpolarizability b

0;0,0 ÿn;n,0 ÿ2n;n,n 0;n,ÿn

ÿ…n1‡n2†;n1,n2

0;0,0,0 ÿ3n;n,n,n

ÿn;n,n,ÿn

ÿn1;n1,n2,ÿn2

0;0,n,ÿn ÿ2n;0,n,n

ÿn;n,0,0

ÿns;n1,n1,n2

ÿ…n1‡n2†;0,n1,n2

ÿ2n1‡n2;n1,n1,ÿn2

	Static hyperpolarizability
EOPE	Electro-optics Pockels e¨ect
SHG	Second harmonic generation
OR	Optical recti®cation
	Two-wave mixing
Second Hyperpolarizability g
	Static second hyperpolarizability
THG	Third harmonic generation
IDRI or DFWM	Intensity-dependent refractive index or
	degenerate four-wave mixing
OKE	Optical Kerr e¨ect or AC Kerr e¨ect
DCOR	DC-induced optical recti®cation
DC-SHG or EFISH	DC-induced second harmonic generation
	or electric-®eld-induced second
	harmonic
EOKE	Electro-optic Kerr e¨ect
	Three-wave mixing
	DC-induced two-wave mixing
CARS	Coherent anti-Stokes Raman scattering

the derivatives de®ned above either analytically or numerically. Analytic derivatives have been formulated for a few methods. This is sometimes called the derivative Hartree±Fock method or DHF (note that the acronym DHF is also used for the Dirac±Hartree±Fock method). Numerical derivatives can be used with any method but require a large amount of CPU time. The researcher should pay close attention to numerical precision when using numerical derivatives.

A second method is to use a perturbation theory expansion. This is formulated as a sum-over-states algorithm (SOS). This can be done for correlated wave functions and has only a modest CPU time requirement. The randomphase approximation is a time-dependent extension of this method.

The electric ®eld can be incorporated in the Hamiltonian via a ®nite ®eld term or approximated by a set of point charges. This allows the computation of corrections to the dipole only, which is generally the most signi®cant contribution.

Time-dependent calculations have been completed with a number of di¨erent methods. There are three formulations giving equivalent results; TDHF,

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).

Computational Chemistry: A Practical Guide for Applying Techniques to Real-World Problems. David C. Young Copyright ( 2001 John Wiley & Sons, Inc.

ISBNs: 0-471-33368-9 (Hardback); 0-471-22065-5 (Electronic)

33 Relativistic E¨ects

The SchroÈdinger equation is a nonrelativistic description of atoms and molecules. Strictly speaking, relativistic e¨ects must be included in order to obtain completely accurate results for any ab initio calculation. In practice, relativistic e¨ects are negligible for many systems, particularly those with light elements. It is necessary to include relativistic e¨ects to correctly describe the behavior of very heavy elements. With increases in computer capability and algorithm e½ciency, it will become easier to perform heavy atom calculations and thus an understanding of relativistic corrections is necessary.

This chapter provides only a brief discussion of relativistic calculations. Currently, there is a small body of references on these calculations in the computational chemistry literature, with relativistic core potentials comprising the largest percentage of that work. However, the topic is important both because it is essential for very heavy elements and such calculations can be expected to become more prevalent if the trend of increasing accuracy continues.

33.1RELATIVISTIC TERMS IN QUANTUM MECHANICS

The fact that an electron has an intrinsic spin comes out of a relativistic formulation of quantum mechanics. Even though the SchroÈdinger equation does not predict it, wave functions that are antisymmetric and have two electrons per orbital are used for nonrelativistic calculations. This is necessary in order to obtain results that are in any way reasonable.

Mass defect is the phenomenon of the electrons increasing in mass as they approach a signi®cant percentage of the speed of light. This is particularly signi®cant for s orbitals near the nucleus of heavy atoms. Mass defect must only be included in calculations on the heaviest atoms, typically atomic number 55 and up. The e¨ect of mass defect is to contract the s and p orbitals closer to the nucleus. This creates an additional shielding of the nucleus, causing the d and f orbitals to expand, making bond lengths longer. This e¨ect is most pronounced for the group 11 elements: gold, silver, and copper.

There are many moving charges within an atom. These motions are the intrinsic electron spin, electron orbital motion, and nuclear spin. Every one of these moving charges creates a magnetic ®eld. Spin couplings are magnetic interactions due to the interaction of these magnetic ®elds. Spin±orbit coupling tends to be most signi®cant for the lightest transition metals and spin±spin

261

262 33 RELATIVISTIC EFFECTS

couplings tend to be important for the heaviest actinides. For elements between these extremes, spin±orbit coupling is often included and other spin-coupling terms are sometimes included. The size of p orbitals is often relatively unchanged by relativistic e¨ects due to the mass defect and spin±orbit e¨ects canceling out.

Also arising from relativistic quantum mechanics is the fact that there should be both negative and positive energy states. One of these corresponds to electron energies and the other corresponds to the electron antiparticle, the positron.

33.2 EXTENSION OF NONRELATIVISTIC COMPUTATIONAL TECHNIQUES

The relativistic SchroÈdinger equation is very di½cult to solve because it requires that electrons be described by four component vectors, called spinnors. When this equation is used, numerical solution methods must be chosen.

The most common description of relativistic quantum mechanics for Fermion systems, such as molecules, is the Dirac equation. The Dirac equation is a oneelectron equation. In formulating this equation, the terms that arise are intrinsic electron spin, mass defect, spin couplings, and the Darwin term. The Darwin term can be viewed as the e¨ect of an electron making a high-frequency oscillation around its mean position.

The Dirac equation can be readily adapted to the description of one electron in the ®eld of the other electrons (Hartree±Fock theory). This is called a Dirac± Fock or Dirac±Hartree±Fock (DHF) calculation.

33.3CORE POTENTIALS

The most common way of including relativistic e¨ects in a calculation is by using relativisticly parameterized e¨ective core potentials (RECP). These core potentials are included in the calculation as an additional term in the Hamiltonian. Core potentials must be used with the valence basis set that was created for use with that particular core potential. Core potentials are created by ®tting a potential function to the electron density distribution from an accurate relativistic calculation for the atom. A calculation using core potentials does not have any relativistic terms, but the e¨ect of relativity on the core electrons is included.

The use of RECP's is often the method of choice for computations on heavy atoms. There are several reasons for this: The core potential replaces a large number of electrons, thus making the calculation run faster. It is the least computation-intensive way to include relativistic e¨ects in ab initio calculations. Furthermore, there are few semiempirical or molecular mechanics methods that are reliable for heavy atoms. Core potentials were discussed further in Chapter 10.

33.5 EFFECTS ON CHEMISTRY

263

33.4EXPLICIT RELATIVISTIC CALCULATIONS

There are also ways to perform relativistic calculations explicitly. Many of these methods are plagued by numerical inconsistencies, which make them applicable only to a select set of chemical systems. At the expense of time-consuming numerical integrations, it is possible to do four component calculations. These calculations take about 100 times as much CPU time as nonrelativistic Hartree±Fock calculations. Such calculations are fairly rare in the literature.

Many researchers have performed calculations that include the two largemagnitude components of the spinnors. This provides a balance between high accuracy and making the calculation tractable. Such calculations are often done on atoms in order to obtain the wave function description used to create relativistic core potentials.

There are several ways to include relativity in ab initio calculations more e½ciently at the expense of a bit of accuracy. One popular technique is the Dirac±Hartree±Fock technique, which includes the one-electron relativistic terms. Another option is computing energy corrections to the nonrelativistic wave function without changing that wave function.

Relativistic density functional theory can be used for all electron calculations. Relativistic DFT can be formulated using the Pauli formula or the zero-order regular approximation (ZORA). ZORA calculations include only the zero-order term in a power series expansion of the Dirac equation. ZORA is generally regarded as the superior method. The Pauli method is known to be unreliable for very heavy elements, such as actinides.

Molecular mechanics and semiempirical calculations are all relativistic to the extent that they are parameterized from experimental data, which of course include relativistic e¨ects. There have been some relativistic versions of PM3, CNDO, INDO, and extended Huckel theory. These relativistic semiempirical calculations are usually parameterized from relativistic ab initio results.

33.5EFFECTS ON CHEMISTRY

As described above, relativistic e¨ects are responsible for shifts in the bond lengths of compounds, particularly those involving group 11 elements. This is called the gold maximum. For example, the Ag2 bond length predicted by nonrelativistic calculations will be in error by ÿ0.1 AÊ . The AuH deviation of ÿ0.2 AÊ and Au2 deviation of ÿ0.4 AÊ along with the Hg2 2‡ deviation of ÿ0.3 AÊ are among the largest known.

Relativistic e¨ects are cited for changes in energy levels, resulting in the yellow color of gold and the fact that mercury is a liquid. Relativistic e¨ects are also cited as being responsible for about 10% of lanthanide contraction. Many more speci®c examples of relativistic e¨ects are reviewed by PyykkoÈ (1988).

264 33 RELATIVISTIC EFFECTS

33.6RECOMMENDATIONS

The most di½cult part of relativistic calculations is that a large amount of CPU time is necessary. This makes the problem more di½cult because even nonrelativistic calculations on elements with many electrons are CPU-intensive. The following lists relativistic calculations in order of increasing reliability and thus increasing CPU time requirements:

1.Relativistic semiempirical calculations

2.Relativistic e¨ective core potentials

3.Dirac±Hartree±Fock

4.Relativistic density functional theory

5.Relativistic correlated calculations using the DHF Hamiltonian

6.Two-component calculations

7.Four-component calculations

BIBLIOGRAPHY

Introductory descriptions are

F.Jensen, Introduction to Computational Chemistry John Wiley & Sons, New York (1999).

M. Jacoby, Chem. & Eng. News March 23, 48 (1998).

C.B. Kellogg, An Introduction to Relativistic Electronic Structure Theory in Quantum Chemistry http://zopyros.ccqc.uga.edu/@kellogg/docs/rltvt/rltvt.html (1996).

I. N. Levine, Quantum Chemistry Fourth Edition Prentice Hall, Englewood Cli¨s (1991). W. H. E. Schwartz, Theoretical Models of Chemical Bonding Z. B. MaksicÏ, Ed.,

Springer-Verlag, Berlin (1990).

K. S. Pitzer, Acct. Chem. Res. 12, 271 (1979).

P. PyykkoÈ, J.-P. Desclaux, Acct. Chem. Res. 12, 276 (1979).

Review articles are

C. van Wullen, J. Comp. Chem. 20, 51 (1999).

K. Balasubramanian, Encycl. Comput. Chem. 4, 2471 (1998).

P. Schwerdtfeger, M. Seth, Encycl. Comput. Chem. 4, 2480 (1998). B. A. Hess, Encycl. Comput. Chem. 4, 2499 (1998).

J. Almlof, O. Gropen, Rev. Comput. Chem. 8, 203 (1996).

E.Engel, R. M. Dreizler, Density Functional Theory II Springer, Berlin (1996).

Y. Ishikawa, U. Kaldor, Computational Chemistry ± Reviews of Current Trends

J. Leszczynski, Ed., 1, World Scienti®c, Singapore (1996).

B.A. Hess, C. M. Marian, S. D. Peyerimho¨, Modern Electronic Structure Theory D. R. Yarkony, Ed., 152, World Scienti®c, Singapore (1995).

BIBLIOGRAPHY 265

K. Balasubramanian, Handbook on the Physics and Chemistry of Rare Earths K. A. Gschneidner, Jr., L. Eyring, Eds., 18, 29, Elsevier, Amsterdam (1994).

P. PyykkoÈ, Chem. Rev. 88, 563 (1988).

S. Wilson, Methods of Computational Chemistry Volume II Plenum, New York (1988). P. PyykkoÈ, Adv. Quantum Chem. 11, 353 (1978).

S.R. Langho¨, C. W. Kern, Modern Theoretical Chemistry 381, Plenum, New York (1977).

M. Bar®eld, R. J. Spear, S. Sternkell, Chem. Rev. 76, 593 (1976).

M. Bar®eld, B. Chakrabarti, Chem. Rev. 69, 757 (1969).

Books on relativistic quantum theory are

K. Balasubramanian, Relativistic E¨ects in Chemistry John Wiley & Sons, New York (1997).

R. Landau, Quantum Mechanics II John Wiley & Sons, New York, (1996). W. Greiner, Relativistic Quantum Mechanics Springer-Verlag, Berlin (1990).

A database of relativistic quantum mechanics references is at

http://www.csc.®/lul/rtam/