Electrical Polarization in Diatomic Molecules Clifford E. Dykslra University of Illinois, Urbana, IL 61801 Quantum chemistry is rapidly providing new tools to understand molecular properties, and in particular, to understand electrical polarization. High-level calculations can now show how all the multipole polarizahilities, hyperpolarizabilities, and beyond depend on molecular geometrical parameters for anv eiven molecule chosen for calculational study (I). From results on real molecular species, the models that might be introduced in beginning physical chemistry courses to explain features of electrical polarization can be checked. This is important because the models implicitly offer a physical picture of polarization. Furthermore, elect& cal properties of molecules are being used today to help explain hydrogen bonding, to understand aspects of surface chemistry, to construct potentials for biomolecules, and to interpret various optical effects and spectroscopic measurements. Thus, the basic ideas that students first encounter on this subject can profoundly affect how well they will grasp future developments coming from a number of contemporary research areas. Quantum chemistry's best approximation, the Born-Oppenheimer approximation, allows one to consider a molecule as if the nuclei were frozen for an instant. The lighter electrons move so quickly that i t is the actual positions of the nuclei, not their motion, that is importanr~indetermining the electronic wavefunctions. Thus, the electronir wavefunction or electron distribution is most often calculaced at specific geometrical arrangements of nuclei, or in the case of a diatomic, at sperific internurlear distanres. The quantum mechanical electron distribution, liken classical charge distribution, is polarizahle. An external uniform electric field will shift the distribution along the field direction, and this can be determined by adding a field interaction operator to the Harniltonian. The rexrhook quanriries rhat are used to define this response to an electric tield are terms in a power series expansion of the molecular energy, E, with respect to the field strength, F.
separation. T o the extent that tbis change is mostly repositioning partial atomic charges, the dipole will be linear in the separation coordinate. The electronic polarizability and hyoernolarizahilitv tend to he verv sensitive to the electron . . distribution in the fringe regions, at the extremes of the molecular volume because this is where the effect of an external field will be felt the strongest. Because the electron distribution even in the frinee reeions chances with internuclear separation, these properti& are also-sensitive to the molecular geometry. Also, electronic polarization may be a ,shift of charge from one valence region to another, and that too will have a dependence on the molecular geometry. As Figure 1illustrates for the H F molecule, a and 8 are found to vary with the geometry or the internuclear separation distance in almost a linear manner. Vibration of molecules plays a definite role in the molecular electrical properties, and one may speak of a "vibrational polarization" given that the "electronic polarization" is that associated with just the electron distribution. Basically, the way the nuclei vibrate is actually affected by the presence of a field. That, in turn, means that the molecular vibrational state energy changes withthe application of a field. Since the electrical properties p, rr, 8, etc. (eq 1) are derivatives of the energy, then a vibrational energy dependence on fields implies a vibrational contribution to these properties. Calculations on H F are used here to illustrate how tbis vibrational polarization arises. Harrnonlc Oscillator Model A simple rule for the harmonic oscillator is that perturbation with a potential that is linear in the stretching coordinate, x , will not change the oscillation freauencv or the ener&spacing between states. (A linear potential term leaves the force constant unchanged.) It will, however, change the equilibrium point and the energy of the potential
Eo is the electronic energy a t the specific nuclear geometry if there is no field. The parameters introduced are w, the permanent dipole moment, a, the dipole polarizability, 8, the dipole hyperpolarizability, y, the second hyperpolarizability, and so on. Of course, in general F may he a vector and then the parameters in the energy expansion would be tensors of rank corresponding to the power of F in the energy expression term. Also, were the field non-uniform, there would be additional terms from higher multipoles in the enerrv (I). -.expansion . From quantum mechanical calculations, it is possible to obtain the derivatives of the electronic enerm with respect to an electric field strength (or other perthations) a n d thereby obtain p , a, 8, etc. This can be done analytically to any order of differentiation (2).Figure 1shows thecurves for these properties for the H F molecule that are generated by repeating the calculations a t a number of internuclear senaration dGtances (3).The curves tend to be linear near equilibrium. Approximating them to be strictly linear while approximating the vibrational potential to be harmonic is referred to as the "doublv harmonic" a ~ ~ r o x i m a t i oorn model (4). A dipole moment ihanges with Gernuclear separation because of how the electronic wavefunction changes with 198
Journal of Chemical Education
Figure 1. Calc~latedl3l d pole moment. polarlzabll~ty,an0 nyperpolarmoll~ty 01 hydrogen Iluorlos from thee actron c ravsl.ncttan asa l~nc18onolthe h-F distance. The dipole moment is given in Debyas (thescale on the right vertical axis), and me polarizability and hyperpolarirabillty are in atomic units (the scale on the len vertical axis).
the slopes of the property functions, one sees that there will be no vibrational polarization effect all the way through B if the dipole moment function is constant (if gl = 0). Thus, within the doubly harmonic model, the vibrational state dipole polarizahility and hyperpolarizability of a nonpolar diatomic such as Nz will be identical with the corresponding equilibrium (electronic) properties. An ab initio harmonic potential curve for HF (3) was obtained by a fit of electronic energies to a harmonic potential, while the properties in Figure 1were fit to linear functions. Then, the doubly harmonic values given by eqs 7-10 were calculated. These are given in Table 1.The vibrational motion effect, which is the result of how the equilibrium would shift in the presence of a field, alters the dipole polarizability by less than 10%. Interestingly, i t has a relatively large effect on the hyperpolarizability.
Figure 2. Effect of a linear potential, tlx, on a harmonic oscillator.
Complete Vlbratlonal Analysls
minimum as illustrated in Figure 2. In the doubly harmonic model, the electrical properties are taken to be linear in x.
This means the influence of an external field of strength F amounts to adding a linear potential, V'(x) = a' b'x, where
+
o r = -(&+'/~a@+'l6BaP+...)
(4)
Following Figure 2, this added potential changes the minimum potential energy by a' - b'V4c. (This comes about after choosing x = 0 to be the minimum of the unperturbed potential, making x a displacement coordinate. Then b in Figure 2 must be zero.) The change in the spacing of the harmonic oscillator energy levels is unaffected by the linear potential, and so each state's energy will be changed by a like amount that is just the change in the potential minimum energy (Fig. 2). Thus, using eqs 4 and 5, it becomes clear that for all harmonic oscillator levels, the field dependence of a state energy, E'"), is a simple function of F.
T o go beyond the doubly harmonic model, anharmonicity must be allowed for in the stretching potential, and nonlinearity must be allowed for in the property functions. With calculations, it is possible to do each separately, of course. Allowing the property functions to be quadratic, for instance, while keeping the potential harmonic can be analyzed by adding a term in eqs 2 and 3. This implies a change in the force constant when a field is applied, and unlike the doubly harmonic model it will mean a change in the vibrational frequency. Consequently, higher state energies will be affected by a field more than lower states. This type of calculation has been carried out for H F and the results are given in Table 2. They are only slightly different, for the u = 0 state, from the doubly harmonic model values, and that means that the realistic jt, a, and fl curves show little curvature. The simplicity of formulas such as eqs 7-10 is entirely the result of approximations. T o account for anharmonicity properly, the quantum mechanical details need to be devel-
Table 1.
Harmonlc Oscillator Model Resuns for HFa VlbratioMl States
Equllibrium Electronle VBIYBS
Doubly Harmonic Vibrational Slate Values
0.784 5.915 Po = 10.096
pvib= 0.784 a* = 5.675 = -0.104
po =
a. =
(n= 0 only) Values wiU? Quadratic Functions
for P, a, 0 +b
a
&
10.784 - 5.674 = 0.173
.VaIws are in atmic unns. For dipole moments. 1 a.u. = 2.541 Debye.
Within this doubly harmonic model, the derivative8 of eq 6 vield the vibrational state electrical properties, and, again, because of the model, those properti& are identical for each state.
Table 2.
Complete Anharmonlc Analysis ot the Vlbrallonal PolarlzatlonIn HF
n=o En- 6 (em-') calc'd (3)
PO, ao, Bo, etc., are the equilibrium (x = 0) values of the properties. How the vibrational state properties differ from these is the "vibrational polarization". I t reflects how the vibrational motion is influenced, rhrough shifting the equilihrium, by the application nf a field. Sincegl,fiz,8s.etr., are
e m v l (bl ~d~ebye) calc'd (5) expt'l(8) a. (a,".) calc'd (5)
B.
(a.u.) calc'd (5)
Vibrational States n= I
n=z
-
3961 3961
7757 7751
1.843' 1.828
1.980 7.872
1.937
8.115.
6.612
7.160
-0.47'
-1.07
-
-1.99
~ h a an 11-r fit is used fn me property tunctlona, the valves are po = 1.999, ao = 5.659, a d 80= 0.50.
Volume 65 Number 3 March 1988
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oped. One type of development, and it is one often used for electronic polarizabilities, is a "sum over states" or a perturbative expansion in terms of a basis of field-free wavefunctions. Another development, direct differentiation ( I ) , is almost the same but does not lead to any confusion that somehow, many "states" (as opposed to basis functions) contribute to one state's polarizahility. The Hamiltonian for an oscillator in a field is where H(0)is the zero-field Hamiltonian and p(x), a(x) . . . are the property functions developed from the purely electronic wavefunctions. The Hamiltonian may be differentiated repeatedly with respect to F, and the derivatives evaluated at F = 0 are
The Schrodineer eauation can be differentiated as well, and evaluated a t I: = 0: This leads to a set of differential equations with inhomoneneous terms that depend on lower-order derivatives of the wavefunction. ( P O I - E(OI)+(OI= 0
In principle, these can be solved order by order, and a computational procedure for doing this by converting to difference equations has been developed recently (5). The energy derivatives (which become the pdb, ru,ih,. . .) are obtained by integration of the derivative Schrodinger equations with
@".
function, second derivatives, and so on. This means that one has obtained power series expansions in F for the wavefunction and the energy. The energy derivatives, eqs 18-20, can he associated with different vibrational effects. The first term, the only term for E") or the dipole moment, is just the vibrational average (vihrational expectation value) of the pure electronic propertv function. For the dipole ~olarizabilitvthere is a term that depends on the dipoie moment, f?", the dipole moment function, It'", and the first derivative wavrfunction. This is the teri t h a t ' c ~ r r e s ~ o nto d sthe second term in the expression for a v i b in eqs 7-10. I t gives the effect of lowering the potential through interaction with the dipole. A like association between eqs 7-10 and eqs l a 2 0 follows for the third derivative. Using the difference equation method (5) for solving the derivative Schrodinner eauations. completelv ah initio values have been calcurated 'for hydrogen fluorde. The results are in Table 2. The agreement with measured dipole moment values is quite good, hut what is interesting from the perspective of this report is that the vihrational properties are quite similar to those obtained with the doubly harmonic model. The significant vibrational polarization effect on @ was first reported by Bartlett and co-workers (6). Conclusions The doubly harmonic model yields a vihrational effect on electrical properties that is entirely due to how the equilibrium of the stretching potential would change in energy from the ap~licationof an external field. This simply presented modei; tested against rigorous quantum chemicil calculations on a realistic system, does seem to account for most of the vihrational effects. In more advanced developments, the vibrational effects due to averaging over anharmonic wavefunctions and subtle chanpes in thebotential function can be obtained from direct dcferentiaGou of the Schrodinger equation. Literature Cited
An important point to be recognized is that the derivative Schrodinger equations are generated by differentiation with respect to parameters embedded in the Hamiltonian, not differentiation with respect to x . Solution of derivative Schrodinger equations yields first derivatives of the wave-
1. Dykatra,C.E.;Liu,S.-Y.:Malik,o.J.Ad". Cham.Phys., i n p r e s . 2. Dyksfrs,C.E.;Jssien,P. G. Chem.Phys. Lett. 1984,109,388. 3. 4 x 1 and@(x)curves mere cslndatedvifh bssisseUdeseribFd in: Liu, S.-I.:Dyk8tre.C. E. J. Phys. Cham. 1988, in press. p ( x ) rvas calculated with the inelusion of electron correlatian effecU ss described in ref 1. The calculated HF ntretchmg potential is fmmB~rnholdt,D.E.;Liu,S.-Y.;Dybtra,C.E. J.Phys. Chem. 1986,85,31ZL 4. Herzhere. G. Moieulor S m l m and Molaculor Structure I; Van Nmtrand Reinhold:
..... ....., .....
5. Dykstra,C.E.;Malik,D.J. J.Chem.Phys., 1981.87. 2806. 6. Sekino,H.:BarUelt.R.J.il Chem.Phys. 1986,84,2726;Adamowin,L.;B~rt11tt,R~J.J. Chrm. Phys. 1986.84.4988. 7. Guelachuili, G. Opt. Commun. 1376.19.150. 8. Gough. T. E.; Miller, R.E.: Seolea, G. Forodor Dlaeuss. 1981.71,77.
Short Course on Modern Industrial Spectroscopy The 33rd annual program in Modern Industrial Spectroscopyto be offered by Arizona State University, August 1-12, 1988, is designed for chemists and others from industrial laboratories that make use of spectrographic eauivment (ohotoera~hic.direct reading, and ICP). This intensive course of lectures and practical laboratory work sew& to trah perankel to staff these i&tallations. The . oroeram .. includes basic theoretical considerations. hands-on instrumental trainine, . .and ~ the inter~retationof results. F w r hours 181 lecture each morning will serve lo prrscnt thr theory, instrumentation, and applications of a l spend variety of optical ernisviun techniques including arc, spark, and indurri\ely ronpled plasma. Each i l ~ d r nwill every afternoun working in the laboratory under the direct gu~danreand suptrviiim uf experiewrd trchnicnl The instructional staff includes members of the Department of Chemistry at Arizona State University augmented by guest lecturers from industrial laboratories. Enrollment in the course is limited, and sufficient equipment is is available to insure each student adesuate time for personal operation of the instruments. The cost for the -program $1300.
For complete information,including descriptive brochure, please write Jacob Fuchs, Director, Modern Industrial Spectroscopy,Department of Chemistry, Arizona State University, Tempe, AZ 85287.
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Journal of Chemical Education
