Electronic Hyperpolarizability and Chemical Structure - American

Chapter 5

Electronic Hyperpolarizability and Chemical Structure David N. Beratan

Downloaded by MONASH UNIV on May 18, 2013 | http://pubs.acs.org Publication Date: March 11, 1991 | doi: 10.1021/bk-1991-0455.ch005

Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109

The chemical structure dependence of electronic hyperpolarizability is discussed. Strategies for developing structure-function relationships for nonlinear optical chromo phores are presented. Some of the important parameters in these relationships, including the relative ionization potential of reduced donor and acceptor and the chain length, are discussed. The correspondence between molecular orbital and classical anharmonic oscillator models for nonlinear polarizability is described. Goals The goal of this tutorial chapter is to give chemists a starting point to build an under standing of the chemical basis for electronic hyperpolarizability [1-4]. The discussion is intentionally oversimplified. Many important effects are suppressed to emphasize generic molecular structure - hyperpolarizability relationships. We will focus on single molecule properties. Complicating effects, described elsewhere in this volume, are essential for modeling bulk material properties, a process that just starts with an understanding of the molecular hyperpolarizability. The intent here is to present sufficient background to stimulate structure-function studies of this property. The emphasis of the theoretical discussion is (1) derivation and interpretation of the sum on states perturbation theory for charge polarization; (2) development of physical models for the hyperpolarizability to assist molecular design (e.g., reduction of molec ular orbital representations to the corresponding anharmonic oscillator description for hyperpolarizability). What is electronic (hyper)polarizability? Applied electric fields, whether static or oscillating, distort (polarize) the electron distribution and nuclear positions in molecules. Much of this volume describes effects that arise from the electronic polarization. Nuclear contributions to the overall polariza tion can be quite large, but occur on a slower time-scale than the electronic polarization. Electronic motion can be sufficiently rapid to follow the typical electric fields associated with incident U V to near IR radiation. This is the case if the field is sufficiently off resonance relative to electronic transitions and the nuclei are fixed (see ref 5 for contri butions arising from nuclear motion). Relaxation between states need not be rapid, so

0097-6156/91/0455-0089$06.00/0 © 1991 American Chemical Society In Materials for Nonlinear Optics; Marder, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1991.

90

MATERIALS FOR NONLINEAR OPTICS: CHEMICAL PERSPECTIVES

instantaneous response models are inadequate if the frequencies approach single or multiphoton electronic resonances with sufficient oscillator strength [6-8]. The term "nonlinear optics" suggests the physical response of interest: optical effects arising from nonlinear polarization of the molecule in an applied field. Since the electronic polarization is ef fectively instantaneous, the polarization at time t can be thought of as the response to a static field, E = So cos(a;*), that is nonlinear in S(t).


Polarization of springs and molecules The calculation of linear and nonlinear polarizabilities for classical oscillators is a familiar textbook problem [9,10]. Analogies are, therefore, often drawn between the polarizability of molecules and the polarizability of harmonic and anharmonic classical springs. Harmonic springs display only linear polarizability. Nonlinear polarizabilities of classical springs are associated with the anharmonic spring constants [10]. One might not immediately expect molecular polarizabilities to be analogous to classical spring polarizabilities because of the anharmonic nature of screened coulombic interactions. Why then are these analogies so ubiquitous, and what chemical structures correspond to anharmonic springs? The potential for a particle bound to a massless one-dimensional harmonic spring is 2

V(x) = ^m^(x-xo)

(la)

For simplicity, the discussion is limited to polarization in one-dimension. V(x) is the energy cost of compressing or extending the spring beyond its equilibrium position XQ. The force exerted by the spring on the particle at position x is (16)

- ^V(x)

A charge Z bound to the spring produces a polarization V

(lc)

= Z(x-x )

x

0

The force due to an electric field S acting on the particle is ZS. The polarization of the particle in an oscillating electric field is calculated by solving the classical equation of motion for x(t), neglecting damping effects (important near resonance) (2a)

mx = F mx = - -^-V(x) + ZS coscjt ax

(26)

mx = — mw\x + ZSQ cosut

(2c)

0

Z 2

ra(u>g — u Y V (t) = Zx(t) x

(2c)

The polarization of the harmonic spring is linear in the applied electric field. Anharmonicities in V (t) are introduced by adding cubic or higher order terms in (x — x ) to V(x). In general, x

0

N n

V

X

= Y^ anCos ut

In Materials for Nonlinear Optics; Marder, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1991.

(3)

5. BERATAN

Electronic Hyperpolarizability and Chemical Structure

91

The nonlinear polarizabilities in the classical spring problem arise from anharmonic contributions to the spring constant. Resolution of eq. 3 into harmonics of frequency nu> using trigonometric identities provides an understanding of how specific orders of anharmonicity in V(x) lead to anharmonic polarizations at frequencies different from that of the applied field £(t). In the classical problem, the coefficients a are determined by the anharmonicity constants in V(x) [10].


n

Classical anharmonic spring models with or without damping [9], and the corre sponding quantum oscillator models seem well removed from the molecular problems of interest here. The quantum systems are frequendy described in terms of coulombic or muffin tin potentials that are intrinsically anharmonic. We will demonstrate their correspondence after first discussing the quantum approach to the nonlinear polarizability problem. Since we are calculating the polarization of electrons in molecules in the presence of an external electric field, we will determine the polarized molecular wave functions expanded in the basis set of unperturbed molecular orbitals and, from them, the nonlinear polarizability. At the heart of this strategy is the assumption that perturbation theory is appropriate for treating these small effects (see below). This is appropriate if the polarized states differ in minor ways from the unpolarized states. The electric dipole operator defines the interaction between the electric field and the molecule. Because the polarization operator (eq lc) is proportional to the dipole operator, there is a direct link between perturbation theory corrections (stark effects) and electronic polar izability [6,11,12]. Electric field dependent electronic polarization in a molecule yields a field dependent dipole moment: 2

3

(e) = i(e = o) + ae + pe

fl

+ fS +

f

•••

(4)

The molecular hyperpolarizabilities are /?, 7, • • • and a is the molecular polarizability. Typical values of are ~ 10~ esu (esu units mean that the dimensions are in CGS units and the charge is in electrostatic units, thus "/? in esu" meansftin units of cm esu /erg ) [1-4]. For an electric field typical of Q-switched laser light, ~ 10 statvolts/cm, the contribution to \x — /i(0) from f3£ is 10~ debye. These polarizations are infinitesimal on the scale of our usual chemical thinking. Yet, these small polarizations are responsible for the exotic effects described throughout this volume. The perturbation theory approach used to describe these properties is justified by the fact that so little charge actually moves. 30

3

3

2

4

2

4

Polarization, polarizability, perturbation theory, and stark effects 2

The square of the molecular wave function |#| , defines the molecular charge distri bution. The wave function of state i, #;(£)» can be calculated in the presence or absence of an electric field. Details of the zero field occupied and unoccupied states determine the size of the hyperpolarizability. Summing the expectation value of the electronic position over occupied states (1 to M) gives the polarization [11] M

V

x

= -eJ2 1=1

(5)

(We limit our discussion to polarization and polarizability in a single dimension for pedagogical reasons). The zero field wave functions substituted in the eq 5 yield the ground state dipole moment of the molecule. When the effect of the field S on molecular eigenstates is sufficiently weak, (6a)


92


This simple polynomial expansion for V is appropriate because the \£j's themselves can be written = * + ] T c S ^ (66) x

n

n)

n

n>0

where c i , c , ••• come from time-independent perturbation theory. Thus, the simple taylor's series in S form is carried through after the integration over the a:-coordinate in eq 5. Since the wavelength of the radiation is long compared to the molecular size, the dipole approximation for the interaction between the field and the molecule is appropriate so 2

H = molecular hamiltonian,

(7a)

(76)

1

H = electric dipole operator = —eSx


and the total electronic hamiltonian is H + H'. As an example of the connection between perturbation theory wave function cor rections and polarizability, we now calculate the linear polarizability, a . The states are corrected to first order in H'. Since the polarization operator (Zx) is field independent, polarization terms linear in the electric field arise from products of the unperturbed states and their first-order corrections from the dipole operator. The corrected states are [12] x

]dx t

t

i=i J

(106) We have added an ellipse in the £ and £ terms to emphasize that further corrections to terms of these orders arise from wave function corrections of higher-order than explicitly written in eq 10a. To generate /? more specifically, one writes the wave functions to second-order in the field 3

4

E „

+

0)

!—I—-tfV

m

(11a)

jp(0) T7.V

r (0) F

>
\ H ' \ * ] ?

0)

0)

}

>

(0)

(116)

0)

(^ -^ )(^ -4 ) 2

and calculates < \£ |x|\I>; >, grouping terms of order S as described in eqs 9-10. The result is t

M

(12a) In the low frequency limit this is equivalent to the time-dependent perturbation theory expression [1-4]: M

P^Y^HYl

i=l

XinXnmXmi

{F (hu ,U ,U ) X

in

im

+ F (foj, 2

2tlU ,U U )] ini

im

(126)

nî m^t

where fiuj is the energy associated with an incident photon and + F ] is an energy dispersion term. In the limit that UJ < u;; ,u; , this expression reduces to eq 12a. 2

n

tm


94


In general, grouping the terms in e / x\9

m

+9

w

+9

w

2

+ ---\ dx

(13)

of appropriate order in S and using the identity s

r

(14) g=0 p=0

to make the perturbation energy-polarizability connection gives the polarizability in terms of stark energies (where e is the jth order energy correction due to the field). Expres sions for the energy perturbation terms are well known. If the stark energy corrections are not small, this approach is limited. In such cases, perturbation theory may be useful, but the zero-order hamiltonian and perturbation may be different. Also, time-dependent treatments may be needed if the dependence of the polarizability on the light frequency is of interest [8].


(j)

Evaluation of molecular hyperpolarizabilities Most numerical methods for calculating molecular hyperpolarizability use sum over states expressions in either a time-dependent (explicitly including field dependent disper sion terms) or time-independent perturbation theory framework [13,14]. Sum over states methods require an ability to determine the excited states of the system reliably. This can become computationally demanding, especially for high order hyperpolarizabilities [15]. An alternative strategy adds a finite electric field term to the hamiltonian and computes the hyperpolarizability from the derivatives of the field dependent molecular dipole mo ment. Finite-field calculations use the ground state wave function only and include the influence of the field in a self-consistent manner [16]. As with the solution of other many-body electronic structure problems, determination of the unperturbed eigenvalues is numerically challenging and involves compromises in the following areas: (1) approximations to the hamiltonian to simplify the problem (e.g., use of semi-empirical molecular orbital methods) (2) use of incomplete basis sets (3) neglect of highly excited states (4) neglect of screening effects due to other molecules in the condensed phase. Methods that are known to calculate transition matrix elements reliably for the sys tems of interest (e.g., TT-electron systems) have been used extensively [13,17]. Especially for /3 calculations, where relatively few electronic states often dominate the hyperpolar izability, numerical methods are reliable. However, 7 calculations are more complicated because of the larger number of contributing terms and the possibility of subtle cancel lations that can occur only when the full series is summed. General aspects of /9 and 7 calculations are discussed in the next section. Structure-function properties for (3 A sum rule exists for electronic transitions (15) ex


5. BERATAN


95

where f is the oscillator strength of each distinct one-electron transition, N is the number of electrons in the molecule, and each electronic state is doubly occupied [18-20]. Oscillator strength is defined in the usual way 9yex

2

8ir mc

(16)

where v ^ is the energy of the transition in wavenumbers. For strong transitions (e ~ g

5


10 M

_ 1

x

1

cm- ) f

~ 1 and X ^

9itx

9

x

~ 2 °A [18]. A l l f 's 9iex

are positive and J2fg,**

has an upper bound defined by eq 15 (for example, the number of 7r-electrons in the calculation of a typical organic chromophore). Molecules with large values of often have intense charge transfer transitions with oscillator strengths ~ 1. We observe that: (1) eq 12 introduces numerators cubic in dipole matrix elements, with denominators quadratic in energy, (2) the transition matrix element between frontier orbitals in donor-acceptor 7r-electron systems is about as large as it can ever be in this family of compounds, and (3) the sum rule shows us that the frontier orbitals contain much of the total oscillator strength of the molecule. For these reasons, it is not surprising that eq 12 is often dominated by the frontier charge transfer states, i.e., those with the largest numerators and smallest denominators in the summation. Often, much of the quadratic (/?) nonlinearity in large hyperpolarizable systems is dominated by contributions from the first few excited states and small corrections (of about a factor of two) occur on addition of the next 50 or so states, followed by apparent convergence of the calculations [21,22]. Eq 12a generates the two-state approximation when a single transition involving both a large oscillator strength and a significant dipole moment change exists. In this case, i — g and n = m — ex so the summations introduce a single dominant term. Such transitions, because of the dipole moment change, are often called charge transfer transitions. In this case eq 12a reduces to

0(2 — state approx.) = 6e

3

Xg ex X ,ex ex

X x,t e

X

A a

g,ez

(17a)

vex,gwex,g

Regrouping terms the two-state approximation becomes

f3(2 — state approx.) — 6e

3

•^g,er(êz,ez ~ ^g,g) (E - E f g

(176)

ex

where X and X are the excited and ground state dipole moments. The molecular hyperpolarizability in the two-state limit is proportional to the oscillator strength of the charge transfer transition, the amount of charge moved, and the square of the transition wavelength (as the wavelength becomes long, the transparency of the material decreases and the nonresonant model may not be adequate). ^ and 4? in the two-state model should be understood to include contributions from the donor, acceptor, and bridge; mixing between the sites is considerable. eXy€X

9y9

g

ex

A simple analysis of eq 17 shows the general aspects of the structure-function rela tions expected to control /3. This demonstration is simplistic in its lack of explicit bridge orbital structure, but it demonstrates the compromises needed to optimize /?. Consider


96


two interacting orbitals, D and A coupled by the matrix element t =< < D\H\D >= A , and < A\H\A >= —A. 2A is the relative ionization potential of the accessible donor and acceptor orbitals. Writing the donor and acceptor localized states as D

^

)

g

i 3)

= c^ + c A D

êx=C

(

C X D

(18a)

A

VD + C ^

X )

^

(186)

respectively, the Schrodinger equation in matrix form is


( V

-*'.,)(£)-

The donor and acceptor orbitals are centered at ±a, so < D\x\D >= — < A\x\A = -fa, and we make the usual assumption < (J>D\X\A >= 0. Using these relationships, the constants in eq 18 can be calculated —(±)

2

= ±y/l

+ (A/t)

- (A/*)

(lSd)

CD

E±/t

= ±y/l

2

+ (A/t)

(18c)

where the + and — terms correspond to V (+) and and l/(E - E ) as a function of 2A/*. Note that the transition matrix element and the energy terms peak for the totally symmetric system and decreases as the system becomes asymmetric. On the other hand, the dipole moment change peaks for asymmetric systems and vanishes for symmetric systems. The characteristic parameter, Aft, can be varied by changing relative ionization potentials of donor and acceptor. An important question to address is whether known systems are on the rising or falling side of the 0 plot. Calculations that include structural details of the bridge are needed, but this plot shows the interplay between charge localization and delocalization needed to maximize /?. g

ex

2

2

t x

g

ex

g

ex

The anharmonic oscillator - molecular orbital theory connection We have shown the molecular orbital theory origin of structure - function relation ships for electronic hyperpolarizability. Yet, much of the common language of nonlinear optics is phrased in terms of anharmonic oscillators. How are the molecular orbital and oscillator models reconciled with one another? The potential energy function of a spring maps the distortion energy as a function of its displacement. A connection can indeed be drawn between the molecular orbitals of a molecule and its corresponding "effective oscillator". As an example, we return to the two orbital calculation of Figure 1 and eqs 17-18 for 0. The strategy is calculate the states,

5. BERATAN


97

polarization constraint is reflected in both the energies of the states and in the wave functions. Plotting the energy of this polarized state vs. its polarization defines the effective oscillator. The energy of the state is < #(A)|if |\I>(A) >. (The states are equivalent to those that would be found in a finite field calculation, but the energy is calculated using H in zero field.) This is a standard method of including external constaints in wave function calculations [23]. It is important to note that this energy function is summed over the occupied states and that this effective potential that defines the anharmonicity constants is distinct from the interaction potential in the hamiltonian. For the two state system (as in eq 18c), the equation to solve with the multiplier A in the two orbital system is [23] ( A - E - X


V

*

t

-A-E

) ( C D \

+

=

(19a)

0

X)\c ) A

Again, t is the coupling between the two orbitals that are located at ± 1 in our distance units, 2A is the relative ionization potential of the donor and acceptor orbitals and the c's are the orbital coefficients. Solutions of this equation for E are the variationally minimized energies subject to the polarization constraint. The A dependent wave functions give the energy of the polarized state and its polarization V(X). We will define a unitless polarization V as (—c + c ). The analytical result for the energy as a function of polarization is 2

D

^

2

A

= ~ < *±|tf|*± >= 7 < * ± l # l * ± >= ( 7 ) P + Vl-V

2

(196)

H is the molecular hamiltonian in the absence of the field. This anharmonic energy profile is plotted in Figure 2 for three choices of 2A/t. A taylor series expansion of this equation around the equilibrium polarization, Po, gives the effective cubic anharmonicity in the potential, where V replaces the classical position x V(P)

+1

-Vo/2 2

(l-ô )

(V-VoY

+ •'•

(19c)

5 / 2

Interestingly, this peaks for total polarization on the donor or acceptor (V = ±1) showing that the cubic anharmonicity is related to the ground state polarization in this two-orbital case. When the square of the ground state polarization is not too large, the anharmonicity is simply proportional to the polarization of the ground state so from eqs 9b and 17 in this simple example 0 oc anharmonicity x linear polarizability

(20)

Determination of the generality of this sort of expression and assment of whether analo gous relations for 7 are valid await further study. Related strategies for partitioning the hyperpolarizability have been discussed in the literature [24-29]. Structure-activity correlations for 7 in pi-electron systems While we have identified some of the design principles for /3 in a two-state model, design criteria for molecules with enhanced 7 appear more difficult to state succinctly because single large matrix elements do not necessarily collapse the 7 expression to




98

POLARIZATION

Figure 2. Effective anharmonic oscillator potential for a two-orbital donor-acceptor system with A/t = 2.0, dashed line; A/t = 1.0, solid line; A/t = 0.0, dot line.



5. BERATAN


99

a small number of terms. For some model hamiltonians, the hyperpolarizability can be calculated unambiguously, and qualitative features of its structure understood. This section is devoted to such simple models. Refs. 24-29 also discuss the relevance of few state models for the calculation of the second hyperpolarizability. Hydrogenic atoms (one electron bound by a nuclear charge Z) have 7 proportional to the seventh power of the orbital radius [29]. Square well 1-D potentials with infinitely high walls and an appropriate number of filled states give 7 proportional to the 5th power of the well width [29]. There is clearly a rapid increase expected in the second hyperpolarizability with system size for delocalized systems. Square well models are not adequate to describe the length dependence of the spec troscopic properties in very long bond alternated systems since these molecules actually retain a nonzero HOMO-LUMO gap even for very long chains. This bond alternation leads to a hyperpolarizability (per repeat unit) that saturates as the chain length is extended beyond a critical length. Spatially varying 1-D potentials that exhibit this behavior (e.g., sine function potentials [29]) have been used to model long unsaturated hydrocarbons, as have tight-binding or modified Hiickel hamiltonians [30-34]. The latter calculations are parameterized from spectroscopic data and predict the order of magnitude and sign of 7 r x x z correctly, as well as its dominance over the other tensor elements. These mod els predict a rapid rise of 7 (per repeat unit) with increasing chain length followed by saturation as the chain becomes very long. Flytzanis predicted the value of the length normalized 7 for long bond alternated polymers [34]. Finite chain length tight-binding calculations [30], shown in figure 3, demonstrate saturation to the Flytzanis value at long chain length following a rapid rise of 7/iV for short chains. In polyenes and polyynes, the saturation is predicted to occur at roughly the chain length where the HOMO/LUMO

10° I 10°

1

1

i

i

I

i

i

10 NUMBER OF DOUBLE BONDS 1

i

i

i i i

11

10

2

Figure 3. jxxxx/Nyo calculated from a Huckel-like bond alternated chain as a function of the number of sites (N). The ratio of the coupling between p-orbitals in single vs. multiple bonds determines the saturation of the 7 / ^ 7 0 plot (here the ratio is 0.79 to model a polyene). 70 is the hyperpolarizability of an isolated double bond.


100


gap energy also has stabilized as a function of chain length ~ 10 — 30 double bonds). Numerically intensive methods are beginning to access the long chain length regime in which saturation occurs [35-38]. For chains with an inversion center, 7 contains contributions from two terms of opposite sign [13]. Sign changes in 7 can occur when there is a switching of dominance between these terms and strategies to force an unsaturated system into one limit or another have been discussed [31]. Noncentric materials are of increasing interest for 7 studies [15].


Concluding remarks The origin of molecular hyperpolarizability can be described in the off resonant limit using time-independent ortime-dependentperturbation theory. We have emphasized the time-independent theory here for simplicity. Applications to the first hyperpolarizability, /?, were described in some detail. The two-state limit was discussed and the structure dependence of each term in the two-state expression was analyzed in a two-orbital frame work. Anharmonic oscillator descriptions of nonlinear phenomena are very common and form a useful starting point for theoretical studies of nonlinear optics. We showed how molecular orbitals can be reduced to their effective nonlinear oscillator potentials. Such analysis should allow determination of how "nonlinear" a particular charge cloud is com pared to another. It may also allow separation of the molecular design problem into one of planning molecules with large anharmonicities together with large overall (linear) polarizabilities. The development of design guidelines for molecules with large second hyperpolar izability, 7, is more difficult because of uncertainty in whether few or many state models are appropriate [24-28]. Some effects, such as saturation of 7 with chain length, can be addressed with one-electron hamiltonians, but more reliable many-electron calcula tions (already available for /?) are just beginning to access large 7 materials [24,35-38]. Theoretical and experimental work in this area should hold some interesting surprises. The theoretical problems associated with calculating nonlinear polarizabilities is closely linked to the field of charge transfer spectroscopy and reactivity as well as the field of multi-photon and excited state spectroscopy. It is likely that theoretical methods from these fields will contribute to a deeper understanding of nonlinear optical phenomena in organic, inorganic, and organometallic compounds. Acknowledgments I am grateful to the referees for their thoughtful comments on the manuscript. Thanks are due William Bialek (UC Berkeley) for suggesting the LaGrangian multiplier method to link molecular orbital and oscillator models. I also thank Joe Perry and Seth Marder for their enthusiastic collaboration on these problems over the last few years. This work was performed by the Jet Propulsion Laboratory, California Institute of Technology, as part of its Center for Space Microelectronics Technology which is sponsored by the Strategic Defense Initiative Organization Innovative Science and Technology office through an agreement with the National Aeronautics and Space Administration (NASA). It was also supported by the Department of Energy's Energy Conversion and Utilization Technology Program through an agreement with NASA.


5. BERATAN

Electronic Hyperpolarizability and Chemical Structure 101

References


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Electronic Hyperpolarizability and Chemical Structure - American

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