The Journal of
Physical Chemistry
0 Copyright, 1982, by the American Chemical Society
VOLUME 86, NUMBER 7
APRIL 1, 1982
Some New Directions of Molecular Quantum Mechanics Joseph 0. Hirschfelder Theoretical Chemishv Institute, Unlvefstiy of Wisconsin-Madison, I n Final Form: August 3, 198 1)
Madison, Wisconsin 53706 (Received: Juk 23, 198 1;
Many new theoretical techniques are being devised to respond to the fantastic precision of experimental laser-inducedstate-*state chemistry. The author (with K.-H. Yang and B. R. Johnson) is deriving the dynamics of molecular systems composed of moving electrons and nuclei in electromagnetic fields accurate through fiie-structure (semirelativistic)terms. By using center-of-mass and internal (rotational,vibrational,and electronic) coordinates, the dynamical operators are expressed in terms of molecular multipole moments. Two of the by-products of this formalism are (1) semirelativistic van der Waals forces which include a set of novel magnetic forces and (2) elimination of gauge origin errors in practical calculations of magnetic susceptibility. Having determined the operators in phase space, the question remains, what sort of wave functions should be used for practical applications. Some of the most promising techniques involve the use of (1) angle-action variables, (2) semiclassical wave functions, and (3) rotations of the coordinates into complex space.
Introduction The papers collected in this issue of The Journal of Physical Chemistry were part of a symposium whose purpose was to pinpoint some of the key problems of theoretical chemistry. Many new techniques are necessary in order to enable theoreticians to meet the challenges posed by the amazing new generation of experimental miracles! Since my promotion (?) to “Emeritus” status was the occasion for holding this symposium, please excuse me for discussing some of my past and present research before considering some of the new directions of molecular quantum mechanics.
for any realistic interatomic potential energy funckion. I knew that the Lennard-Jones 6-12 function was the best simple two-constant potential because (before World War 11) I had calculated second virial coefficients and Joule-Thomson coefficients for many gases over a wide range of temperature and had compared my calculations with experimental data.2 Thus Bird used the Lennard-Jones 6-12 potential to numerically calculate the collision cross sections. This involved the evaluation of the indefinite (hyperelliptic) integral3
I. The Kinetic Theory of Gases Curtiss, Bird, Spotz-Bunyan, and I made our reputations by making rigorous kinetic theory of gases applicable to practical problems. In 1939, Chapman and Cowling published an excellent book on the kinetic theory of gases.’ The mathematics was elegant, but their formalism could not be used because collision cross sections were not known
This was a formidable job with the 1948 IBM computer which consisted of separate key punch, sorter, additionsubtraction, and multiplication-division units! Curtiss generalized Chapman and Cowling’s formalism to apply to multicomponent gas mixtures. This required
(1) S. Chapman and T. G.Cowling, “Mathematical Theory of NonUniform Gases”, Cambridge University Press, London, 1939. 0022-365418212086-1045$01.25/0
1
+
d y [ l - D2y2 (4/k)(y6 - y12)J-1/2
(1)
(2) J. 0. Hirschfelder and W. E. Roseveare, J. Phys. Chem., 43, 15 (1939); J. 0. Hirschfelder, F. T. McClure, and I. F. Weeks, J. Chem. Phvs.. 10. 201 (1942). 13)’J.0. Hirkhfeider, R. B. Bird, and E. L. Spotz, J. Chem. Phys., 16, 968 (1948).
0 1982 American Chemical Society
1046
The Journal of Physical Chemistiy, Vol. 86,No. 7, 1982
a deep insight into the underlying physics. Spotz then calculated the coefficients of viscosity, diffusion, and heat conductivity for a large number of gases and compared the theoretical values with experimental data. Thus, the book, “The Molecular Theory of Gases and Liquids” (MTGL),4awas born! There were many novel features in this book including the bipolar expansion of l / r i ,and the expressions for the local electric and magnetic fielh intensities in terms of the radial distribution function. Subsequently, Bird, Stewart, and L i g h t f ~ osimplified t~~ our treatment of transport properties and adapted it to many kinds of engineering applications. After the publication of MTGL, I worked on a number of problems such as the theory of flames and detonations, the heat conductivity of reacting gases, the molecular coordinates which diagonalize the kinetic energy, the general formulation of molecular collision theory, and the hypervirial theorem. My discovery of the hypervirial theorem is curious: In 1932 when I was a graduate student, I doodled with derivatives of the Schriidinger equation and obtained a variety of seemingly useless relations which I carefully saved in my files. Then 28 years later, when I had to give a paper at a symposium in honor of Jack Kirkwood, I studied these doodles and found that I had discovered a generalization of the virial theorem! However, intermolecular forces have been my principal interest for the last 44 years. Thus, after the publication of MTGL, I studied all kinds of intermolecular forces and their relativistic corrections and Born-Oppenheimer deviations. In order to calculate the interaction energies, I worked on variational and perturbation techniques applied to nondegenerate, degenerate, and almost-degenerate problems. However, I found that perturbation theory applied to practical electron exchange problems is a mathematical whirlpool so that I started to go around in circles and got sucked in, ever deeper! Thus,I decided to make a big change in my research and study the dynamics of molecules with moving nuclei either in the presence or in the absence of external electromagnetic fields. 11. Present Research For the last 4 years, Yang, Johnson, and I have been doing research on the state-to-state dynamics of molecules in electromagnetic fields.s,6 In order to cope with laser resonance experiments, our object is to develop a formalism which is intermediate in complexity and precision between covariant quantum field theory and the model Hamiltonian techniques now being used to solve molecular problems. The special features of our treatment are as follows: ( 1 ) It is semirelativistic or accurate through terms of the order of 1 / c 2 ;that is, we include all fine-structural terms with the exception of the Lamb Shift. (2) We have established term-by-term correspondence between all classical dynamical properties (including spin) and the quantum mechanical dynamical operators. (3) The operators for all dynamical properties are gauge invariant (as they should be). (4) The nuclei are considered to be moving; that is, we use center-of-mass and internal coordinates (corresponding (4)(a) J. 0. Hirschfelder, C. F. Curtiss, and R. B. Bird, ’Molecular Theory of Gases and Liquids”, Wiley, New York, 1954. (b) R. B. Bird, W. E. Stewart, and E. N. Lightfoot, ”Transport Properties”, Wiley, New York, 1960. (5) K.-H. Yang and J. 0. Hirschfelder, J. Chem. Phys., 72,5863(1980). (6)K.-H. Yang, J. 0. Hirschfelder, and B. R. Johnson, J.Chem. Phys., 75, 2321 (1981).
Hirschfelder
to molecular rotations, vibrations, and electronic modes). (5) All dynamical properties are expressed in terms of electric and magnetic molecular multipole moments. For most practical applications, our “nonrelativistic” approximation (containing all terms which vary as l / c ) should suffice. Even this approximation contains many novel features. Since our formulation is very general, it should be applicable to many types of experiments. Our semirelativistic corrections may help to explain observations of ( 1 ) radiation trapping by lasers;’ (2) magnetically induced optical dichroism;s ( 3 ) the moving Stark effect and the (avoided and nonavoided) energy level crossing^;^ (4) isotope shifts, positronium, etc.;’O ( 5 ) coherence spectroscopy;” and (6) collision- and field-induced transitions between fine-structural levels (microwave spectra).I2 The derivation of our formulation is the biggest challenge I have ever had. Although the underlying physics is simple, the derivations are fantastically complicated. A younger or less-established person could not have tackled this problem because of “publish or perish” considerations. I was indeed very fortunate to have the help of both Yang and Johnson. Yang is a theroetical physicist who established himself as an expert on electromagnetic dynamics and gauge transformation^'^ before he came to Wisconsin in 1977. Johnson is a theoretical chemist with lots of mathematical ability and an amazing knowledge of the research literature. The three of us have been working extremely hard and expect to attain our goal this year. Then, we are looking forward to applying our techniques to practical problems. When I proposed this research, some critics complained that our formalism would be too complicated to apply to practical problems. My reply is that before making approximations we want to thoroughly understand the underlying physics. Only then can we construct model Hamiltonians, potentials, and wave functions which faithfully predict those features which are important to explain a particular type of experimental phenomena. However, we should stress that our complicated formalism is only the first step in obtaining a solution. I maintain that the rigorous formulation is frequently a necessary first step. Now that we have explained our intentions, let us describe some of the interesting by-products of our efforts. 111. Intermolecular Forces’* First, let us consider a new approach to intermolecular forces and van der Waals energies which leads to a family of magnetic (current-current) interaction terms which have not been considered previously. However, before discussing the dynamics of a pair of interacting molecules, (7)K. A. U.Lindgren and J. T. Virtamo, J. Phys. B, 12,3465(1979); B. G. S.Doman, ibid., 13,3335 (1980). (8)L. Seamans and A. Moscowitz, J.Chem. Phys., 56,1099(1972);P. J. Stevens, Annu. Reo. Phys. Chem., 25,201 (1974). (9)T.A. Miller and R. S.Freund, J.Mol. Spectrosc., 63, 193 (1976). See also, M. Rosenbluh, T. A. Miller, D. M. Larsen, and B. Lax, Phys. Rev. Lett., 39,874(1977);G. C. Neumann, B. R. Zegarski, T. A. Miller, M. Rosenbluh, R. Panock, and B. Lax, Phys. Reu. A, 18,1464(1978);R. Panock, M. Rosenbluh, B. Lax, and T. A. Miller, ibid., 22, 1041, 1050 (1980);W. A. Blumberg, W. M. Itano, and D. J. Larson, ibid., 19,139 (1979). (10)G.Wunner, H. Ruder, and H. Herold, J.Phys. B, 14,765(1981). (11)A. H. Zewail, “Coherent Optical Spectroscopy of Molecules”,in ’Laser Spectroscopy”, J. H. Hall and J. L. Carsten, Eds., Springer, Heidelberg, 1977. (12)I. V. Hertel, Adu. Chem. Phys., 45,Part 2, 341 (1981). (13)KUO-HOYang, Ann. Phys., 101,62,97 (1976). (14)J. 0. Hirschfelder, Kuo-Ho Yang, and B. R. Johnson, J . Chem. Phys., in press.
The Journal of Physical Chemistry, Vol. 86,
New Directions of Molecular Quantum Mechanics
we generalize the Lorentz force on a charged particle in an external electromagnetic field so as to apply to a molecule which has electric and magnetic multipole moments+ FB(ext)= external Lorentz force on molecule a = QaE(r,,t) + (l/C)Ia X B(r,,t) (2) Here E(r,,t) and B(r,,t) are the external electric and magnetic field intensities, respectively, at r,, the center of mass of a, at time t. Furthermore, Q, is the effectiue charge and I, is the erfectiue current of a. If e, is the true charge of a, pa and ma are its electric and magnetic dipole moment vectors, and qa is its electric quadrupole moment tensor, then Q, = e, pa.V, qa:VaV, + ... (3) I, = eara+ ra(pa.Va) + pa + cm, X V, + ... (4) Note that if the molecule had no electric or magnetic multipoles, eq 2 would reduce to the usual expression for the Lorentz force on a charged particle. Also note that both Q, and I, are classical operators so that when they are used in quantum mechanical contexts the gradients V, do not operate on the wave functions. The structure of the effective current is somewhat complicated: eara= the true current r,(pa.Va) = the ionic current pa = the electric polarization current cm, X V, = the magnetic polarization current
+
+
Furthermore, the magnetic dipole moment has two components ma = mii) - (1 c ) r a X p a (5) internal doentgen magnetic magnetic dipole dipole Thus, mii) is the magnetic dipole which the molecule would have if the velocity of its center of mass vanished, and the Roentgen term corresponds to the fact that a moving electric dipole behaves like a magnetic dipole. In 1902, Lorentz showed that a charged particle produces an electromagnetic field (we call it an internal field) due to its charge, velocity, and a ~ c e l e r a t i o n . ~Then ~ in 1920, Darwin used these internal fields to derive the interaction of charged particles and the resulting dynamics.16 We generalized the internal fields so as to apply to molecules with multipole moments. Thus, we found5i6that the internal electric and magnetic field intensities produced by a molecule b at an arbitrary point ro (outside of the charge distribution of molecule b) are
B(i)(ro;b)= -1b 1
X
(2)
C
(7)
where rbis the center of mass of b and rob = ro - rbis the vector whose absolute value is rob. In addition, the abth component of the tensor Zob is r
No. 7, 1982 1047
The force on molecule a produced by the presence of molecule b is then the same as the external Lorentz force given by eq 2 with the exception that the external field intensities are replaced by internal field intensities
+
Fii)(b) = &,E(')(r,;b) (l/c)I,
X
B%,;b)
(9)
Or, substituting eq 6 and 7 in eq 9, it follows that the force on molecule a produced by molecule b is
(provided that the charge distribution of a and b do not overlap). The total force on molecule a in the presence of both molecule b and an external electromagnetic field is Fa = F P t ) F,c')(b). Note that Newton's action-reaction law does not apply to semirelativistic dynamics because the internal electromagnetic fields have momenta of their own. Thus, from eq 10 it can be seen that the force on a due to b plus the force on b due to a does not vanish
+
The usual nonrelativistic expression for the force on a due to b is just the first term in eq 10 r
[r,bs+r,b5 eb
3(k'b'rab)
+
... ]pa-
Note that although all of the force terms in eq 12 are not radial, since rab= -qat Newton's action-reaction law is satisfied by the nonrelativistic intermolecular forces. All of the other terms in eq 10 are semirelativistic corrections to the intermolecular force which vary as 1/c2. The Ia*Ibterms is the usual force between two currents and the Q,(d/dt)(Ib.Zab)term is the force that makes an electric motor work! As will be explained in section V, in order for F,(')(b) to have the full semirelativistic precision, it is necessary that QaQb(rab/rab3) be expressed in terms of Kracjik-Foldy coordinates which leads to a semirelativistic form of the electric dipole. Since the velocity of light in atomic units is c = 137, it follows that the semirelativistic correction terms tend to be smaller than the corresponding nonrelativistic terms by a factor of IV. The van der Waals Energy Operators14 In most molecular interaction problems, the interaction energy operator is more useful than the intermolecular force since intermolecular force is not directly observable. The Hamiltonian for the two molecule system is H = Ha Hb + N a b (13)
+
(15) H.A. Lorentz, Encycl. Math. Wis., (5) 2, 200 (1904). (16)C.G.Darwin, Phil. Mag., 39,537 (1920).
where Habis the interaction energy operator and Hais the Hamiltonian for molecule a. Similarly, the Lagrangian for the two molecule system is
1048
The Journal of Physical Chemistry, Vol. 86, No. 7, 1982
where La is the Lagrangian for molecule a and L,b(s is the interaction Lagrangian. In addition, is the Lagrangian of the cross-internal field produced by a and b 1 Lab("' = - -~d3ro[E(i)(ro;a).E(i)(ro;b) - B(')(ro;a). 87r
In eq 15, the integral is taken over all space. It is well-known in quantum field theory that Habis almost equal to -Lab('". Furthermore, through terms of the order of 1/c2 Hab= -L ab(ifl (16) is exactly true both for the interaction of electrons according to the Briet-Pauli Hamiltonian and for the interaction of Dirac particles according to the Breit equation.14 Landau and Lifshitz's exp1anation:l' "For small changes of L and H , the additions to them are equal in magnitude and opposite in sign." The physics involved can be seen from considering a simple system where the Lagrangian depends on some parameter X in addition to the coordinates and velocities. Then, by the chain rule dL = p-dr + (dL/dr)i,A.dr + (dL/dX),,, dX (17) and since H = p-r - L dH = p-dr + r*dp - dL = r-dp - (dL/dr),,x.dr - ( ~ L / C ~ XdX ) , , ~ (18) Thus (dH/dX)p,r = -(dL/W+,, or (JH),, = -(JL)i,r
(19)
From eq 15 and 16 it follows that the interaction energy operator is Hab
= QaQb(1/Tab) - (1/c2)I$b:zab
(20)
Here, the first term is the usual nonrelativistic van der Waals energy operator QaQb(1/Tab) = eaeb
.
Hirschfelder
QaQb(l/rab)in terms of the Kracjik-Foldy coordinates and use the semirelativistic expression for the electric dipole moment. V. T h e Semirelativistic Electric Dipole Moment6 The usual nonrelativistic electric dipole moment of molecular a (relative to its center of mass r,) can be expressed in the dipole length form. If molecule a consists of N particles (electrons/nuclei) designated by the subscript j , then [~lalnonrel= Cejrja
(22)
I
where rja.=rj - ra. In addition, an electron or nucleus in the laboratory reference frame has both a characteristic electric dipole moment l j=
[ej/2mjc2](gi - l)rj x sj
(23)
and a characteristic magnetic dipole moment mj = [e,gj/2mjc]sj = Pisj
(24)
Although eq 24 is well-known, very few of our chemist friends are aware of eq 23. The characteristic electric dipole contributes semirelativistic components to the molecular dipole moment. However, in order to obtain the full semirelativisitic precision, it is necessary to make relativistic corrections to the definition of the center of mass. Kracjik and Foldyl* worked for 7 years before they discovered a nonsingular coordinate transformation which led to a semirelativistic Hamiltonian expressed in center-of-mass and internal coordinates. Thus, the semirelativisitic molecular electric dipole moment which we obtaineds has three parts
Nr
1
l " ' [ej --ra(ra.rj;) 1 - (ra*rja)rj;+ -(ka 1 c2 j=1 2 2mj
X
sj')
n
Notice that minus the gradient (with respect to the coordinates of a) of the nonrelativistic van der Waals energy operator is equal to the force on a due to b as given by eq 12. The current-current or magnetic interaction energy is given by the second term in eq 20. When the effective currents are expanded, -(1/C2)I&z& can be expressed as the sum of ten different types of terms involving the velocities, electric and magnetic dipole moments, and rate of change of the dipole momenta of the two molecules. Most of these magnetic or current terms are novel. They are of the order of loe4 times the magnitude of the electrostatic terms. That is, they are small but still observable. For an array of molecules, they are pairwise additive and long range. Also, their expectation values are generally first-order (rather than second-order) perturbations. Thus, they could be important for condensed media or biological materials. In order to obtain the interaction energy operator with semirelativistic precision, it is necessary to express (17)L. Landau and E. Lifshitz, 'Classical Theory of Fields", 2nd ed, Addison-Wesley, Reading, MA, 1959,p 184.
Here the primes indicate that the property is expressed in Kracjik-Foldy coordinates. The second summation in eq 25 is a purely relativistic term which results from the new definition of the center of mass. Of course, eq 25 is easily expressed in terms of canonical momenta in Hermitian form suitable for use as a quantum mechanical operator. VI. Transition Probabilities The most important contribution of our present research may be the derivation of gauge invariant transition probabilities based upon Yang's definition of a molecule in an oscillating electromagnetic field.13 He defined a time-dependent energy operator Hs so that its rate of change is equal to the rate at which the field does work on the molecule dH,/dt = 1.E
(26)
Since H , is Hermitian, it has a complete set of time-dependent eigenstates HE(^) Xn(t) = e n ( t ) Xn(t) (27) (18) R. A. Kracjik and J. L. Foldy, Phys. Reu. D, 10,1777 (1974);12, 1700 (1975);Phys. Rev. Lett., 24, 545 (1970);32, 1025 (1974). See also, K.J. Sebastian and D. Yun,Phys. Reu. D , 19,2509 (1979).
New Dlrections of Molecular Quantum Mechanics
The Journal of Physical Chemistry, Vol. 86, No. 7, 1982 1049
Here &,,(t)is the energy of the nth state. In an oscillating external electromagnetic field, the Hamiltonian H(E,B;@,A)(expressed in arbitrary gauge) is different from the energy operator. We proved (through the semirelativistic approximati~n)~ that H6 = H(O,B;O,A) so that the energy operator has the same functional form as the Hamiltonian would have ifE(t) = 0 = @(t)but both B ( t ) and A ( t ) correspond to the actual external electromagnetic field. Of course, the molecular wave function \k(t)satisfies the Schrijdinger equation
H\k
a* = ih-
at and $(t) can be expanded in terms of the energy eigenfunctions
*(t) =
Cn W ) X n W
(29)
where the b,(t) are the expansion coefficients. Thus, it is apparent that the energy eigenfunctions are mathematical constructions and the rate of change of bn(t)corresponds to the absorption and emission of photons. In the radiation dipole approximation, we find the rate at which the molecule a makes the transition from the nth to the mth energy state is 1 -(30) i f i (xn(t)lpa.E(r,,t)lxm(t)) Thus the radiation interaction Hamiltonian is
as it should be. It is important to note that these expressions for the transition probabilities and radiation interaction Hamiltonian do not depend upon the gauge which is used in constructing the Hamiltonian H. In contrast, as Power has stressed,lDthe transition probabilities for two-photon processes calculated according to standard procedure depend upon the gauge which is used.
VII. The Dynamics in the “Center of Mass” Formulations The center of mass (which henceforth we designate as r,) is chosen as the molecular reference point because it makes the dynamics simpler than it would be with any other choice of reference point. In the absence of an external electromagnetic field, the Hamiltonian separates into one part that involves the motion of the center of mass (r, and ita conjugate canonical momentum p , ) and a second part that involves the internal coordinates. However, in the presence of an external field the two parh are weakly coupled by the moving Stark effect. In order to derive the Hamiltonian H,for an N particle molecule expressed in terms of the center-of-mass and the internal coordinates, it is customary to start with the familiar minimal coupling Hamiltonian
frame) and their conjugate canonical momenta p . . Furthermore, A ( r j , t )is the vector potential expressedin any arbitrary gauge. The “center-of-mass” Hamiltonian H,is derived by a Power-Zienau-Woolley type of canonical transformationm
where
x =
[M,
+ q,.V, + ...l.A(r,,t)
(34)
Here p, and q c are the electric dipole and quadrupole moments of the molecule referred to r,, and A(r,,t) is the vector potential at r, in the same gauge as the A(rj,t)which appears in Hmin.coup. Goeppert-Mayer discovered this transformation in 1931 and Lamb used it in 1952 in his analysis of ,the fine structure of the hydrogen atom.z1 H, is expressed in terms of the center-of-mass rc and ( N - 1)-independent internal coordinates Rk. There are many possible choices of the internal coordinates if N > 2. We call the coordinate system ( R l ,&, ...,R(N-~): and RN = r,) which we use the “general linear diagonalzzing” or GLD coordinates.6 They are the most general linear combinations of the particle coordinates which are orthogonal, which diagonalize the (nonrelativistic) kinetic energy, and which have r, as a member of the coordinate set. The GLD are defined by
where M = C g l m j is the mass of the molecule and d is any realm unitary or orthogonal matrix (which means that d is its own inverse transpose, d = d-lT) subject to the condition that
dNj
[mj/&fl‘J2
(36)
so that RN = r, The GLD are generalizations of the Smithzzor the Jacobi coordinates. For N particle molecules, d contains ‘ l 2 ( N - 1)(N- 2) arbitrary parameters or hyperspherical angles. The canonical momentum conjugate to Rk is
where P N = p,. However, it follows from differentiating the Lagrangian with respect to the GLD velocities Rk that
+ e,C
1
p c = Mi., -A(r,,t) - - [p, C
+ qc.V, + ...I
X
B(r,,t)
(38) where e, is the total charge on the molecule and A(r,,t) is the vector potential expressed in arbitrary gauge. Also N
Pk = A4$tk + E ( M / r n j ) d k j ( e j / c ) A j , 1 5 k
IN
-1
j-1
(39)
(or the Breit-Pauli Hamiltonian which is its semirelativistic counterpart). Here Hmhcoupis expressed in terms of the particle coordinates rj (in the laboratory reference (19)E.A. Power and T. Thirunamachandran, Am. J. Phys., 46,370 (1978).
(20)See P.W. Atkins and R. G. Woolley, h o c . R. SOC.London, Ser. A, 319,549 (1970),for the history of the Power-Zienau-Woolley transformation. (21)W. E. Lamb,Jr., Phys. Rea, 86,259(1952).See also, W. E. Lamb, Jr., and R. C. Retherford, ibid., 79,549 (1950);81,222 (1951). (22)Felix Smith, J. Math. Phys., 3, 735 (1962);J. Chem. Phys., 31, 1352 (1959);Phys. Rev., 120,1058 (1960). See also, R. C. Whitten and F. T. Smith, J. Math. Phys., 9, 1103 (1968).
The Journd of Physical Chemistry, Vol. 86,No. 7, 1982
1050
where Ai, is the vector potential of j expressed in multipolar gauge with r, taken as the gauge origin
Aj, = -[yzrjc+ Y6rjc(rjC-Vc) + ..I
X
B(r,,t)
(40)
It follows that the (nonrelativistic) kinetic energy of the molecule is
There are two points of interest with respect to the kinetic energy (or the Hamiltonian as expressed in terms of the center-of-mass and internal coordinates). 1. The Moving Stark Effect. The motion of the center of mass is coupled to the internal dynamics by the moving Stark effect term. From eq 38 and 41,this coupling term is c
. 1 = -k,.(~, C
+ qc.V,) X B(r,,t) = -(PC + qc*Vc)*Emotion (42)
where 1
Emotion = -k, C
X
B(r,,t)
(43)
is the additional electric field acting on the molecule due to ita velocity. Furthermore, the additional energy which the molecule has as the result of being in the motional electric field is +c + qc*Vc).Emotion (44) which is the true motional Stark effect. MillerB has shown that the motional Stark effect is important in laser-induced (forbidden) singlet-triplet transitions in the vicinity of anticrossings. 2. The Elimination of Gauge Origin Uncertainty in Magnetic Susceptibility Calculations. The center-of-mass formulation eliminates errors in the calculation of magnetic susceptibilities due to uncertainty as to the proper choice of the gauge and gauge origin since (as can be seen from eq 39-41) the vector potential associated with the internal dynamics of the molecule is Ai,, which is necessarily expressed in the multipolar gauge with ita gauge origin located at the center of mass. Of course, the exact expectation values of Hermitian operators for all dynamical properties are gauge invariant. However, the use of Hartree-Fock and other approximate wave functions can lead to gauge variance problems." On this account, Ditchfield developed what he calls gauge invariant atomic orbitals (and which Ep(23)See ref 9. (24)D. L.Lin, Phya.Rev. A, 17,1939(1978);16,600 (1977),discusses the gauge problems associated with the Hartree-Fock approximation.
Hirschfelder
steinz5facetiously called gauge variant) in order to minimize the difficulties. All of the practical calculations of the magnetic susceptibility of molecules use the minimal coupling Hamiltonian and assume that the nuclei are clamped. In this formulation, the gauge and gauge origin of the vector potential A(rj,t)are arbitrary. Generally, the gauge origin is taken to be located at one of the nuclei. Then they calculate different values of the suceptibility for each choice of gauge origin. The problem persists even when multiconfiguration self-consistent wave functions with large basis sets are used.26 Generally, it is desirable to formulate molecular problems in terms of the center-of-mass and internal coordinates. Then, because the masses of the nuclei are so large compared to the masses of the electrons, the Hamiltonian can be approximated by a Hamiltonian similar to the clamped nuclei minimal coupling Hamiltonian by letting the ratio of the electronic to the nuclear masses approach zero.
VIII. T h e Future-Where Do W e Go f r o m Here? Theoretical chemistry has made great progress during the last 50 years and fantastic changes are taking place right now. As a result of the fantastic precision attained in experimental state-to-state chemistry, the fine-structure (semirelativistic) energy and deviations from the BornOppenheimer approximation have attained a new importance in the theoretical research. Some of the kinds of Born-Oppenheimer deviations are well-known, like the Jahn-Teller effect; whereas, others like the Mead and Truhlarz7 molecular Aharanov-Bohm effect are quite surprising. In our research on the interaction of molecules with electromagnetic fields, it is necessary to express the dynamical operators in terms of the center-of-mass and internal coordinates. However, when we apply our formulation to practical problems, the question will arise, "What is the best way of coping with the moving nuclei?" The usual technique (which is used in scattering theory) is to employ many-channel Born-Oppenheimer wave functions. However, this is difficult and has limitations on its applicability. The question of how to treat moving nuclei is closely related to an observation of Wheeler.
IX. Wheeler's Analysis o f Quantum Mechanics Wheelerz8has pinpointed one of the outstanding problems in molecular quantum mechanics: formal quantum mechanics is so mathematically elegant that it appears deceptively simple. It appears that all you need to solve a problem is to write a simple program and feed it to a powerful computer. In truth, the elegance and simplicity conceal the detailed physics. The quantum mechanical equations give no hint as to the regions in parameter space where minor changes in the input data can make drastic changes in the output! There is great interest in these regions of instability. Thus, Eno and RabitzZ9are developing techniques for analyzing the sensitivity of the quantum mechanical equations for molecular collisions. Furthermore, Marcus, (25)R. Ditchfield, J . Chem. Phys., 56,5688 (1972). (26)C. T. Corcoran and J. 0. Hirschfelder, J. Chem. Phys.,72,1524 (1FIRO). \____,.
(27)C. A. Mead, Chem. Phys.,49, 23 (1980);C. A. Mead and D. G. Truhlar, J. Chem. Phys., 70,2284 (1979). (28)J. A. Wheeler in 'Studies in Mathematical Physics", E. H. Lieb, B. Simon, and A. S. Wightman, Eds., Princeton University Press, Princeton, NJ, 1976,p 35f (29)L. Eno and H. Rabitz, J . Chem. Phys., 71,4828(1979).
The Journal of Physical Chemlstry, Vol. 86, No. 7, 1982 1051
New Directions of Molecular Quantum Mechanics
Reinhardt, Miller, Wyatt, etc. are all using the simple two-dimensional Henon-Heiles potential to study the Kalmagoroff-Arnold-Moser conditions where the classical trajectories are nonergodic and the quantum mechanical energy levels have avoided crossings. From this work may come an answer to Pechukas' question, "What happens to the quantum mechanical constants of motion when a separable N-particle system is perturbed and becomes nonseparable?" Usually, chemists make use of configuration space in dealing with their quantum mechanical problems. Configuration space is very convenient because it is the only space in which the Coulomb interactions (such as e1e2/rI2) are local operators. For example, in momentum space, the Coulombic interaction is represented by the nonlocal operatorm
Note that the space [...I is intended to be used by @ ( p 1 i ) when e1e2/r12 operates on * ( p l 2 ) . However, configuration space has the serious disadvantage that the concept of probability in configurational space representations is quite different in quantum mechanics than it would be in classical mechanics. If we really want to understand the significance of molecular dynamics, it is necessary to use a representation in phase space. In phase space, the velocities of the particles are not implicit or concealed as they are in configuration space. Furthermore, in phase space, the concepts of probability are very nearly the same in quantum and in classical mechanics. Thus, Fermi31made use of this correspondence of probabilities in phase space to develop Fermi-Dirac statistics, the Fermi gas, the Fermi-Thomas atom, etc. Similarly, Wigner32used this correspondence to derive his distribution function, F(p,r,t)such that the expectation value for an operator G(p,r)is the integral over phase space
&,r) = S S F ( p , r , t )G(p,r)d p dr
(46)
In discussing the Wigner correspondence principle, mention should be made of a serious outstanding problem. We still do not have a unique prescription for determining a correct quantum mechanical operator which corresponds to a given classical function that involves products of powers of variables which do not commute in quantum mechanics.33 The Wigner distribution function is based upon the Weyl correspondence rule34which does not always give physically acceptable results. There is a very extensive literature on the subject.% However, Shewell% has the best simple analysis of the problem and its solution. When in doubt, use Yvon's rules of simplicity37 which correspond to playing around until you find a reasonable Hermitian form to represent the operator! The Moyer exact quantum mechanical distribution function% is the alternative to using the Wigner function. ~~
(30)H.A. Bethe and E. E. Salpeter, "Quantum Mechanics of One- and Two-Electron Atoms", Academic Press, New York, 1957,pp 37 and 38. (31)J. C. Slater, "Quantum Theory of Matter",2nd ed, McGraw-Hill, New York, 1968. (32)E. P . Wigner, Phys. Rev., 40,479 (1932). (33)W. H.Miller, J. Chem. Phys., 61,1833 (1974). (34)H.Weyl, 2.Phys., 46,1(1927).See also N. H. McCoy, R o c . Natl. Acad. Sci. U.S.A., 18, 674 (1932). (35)B. Leaf, J. Math. Phys., 9,769 (1968);R. Kubo, J. Phys. SOC. Jpn., 19, 137 (1964);T.F. Jordan and E. C. G. Sudarshan, Reu. Mod. Phys., 33, 515 (1961);N. Mukunda, Ann. J. Phys., 47, 182 (1979); Parmiina (India),11, 1 (1978);J. V. Lepore, Phys. Rev., 119,821(1960). (36)J. R. Shewell, Am. J.Phys., 27, 16 (1959). (37)J. Yvon, Cah. Phys., 33,25 (1948).
However, in order to form the Moyer distribution function it is necessary to know the exact wave function for the system. has derived the time-dependent response and autocorrelation functions4 and their semiclassical limits which will be useful in calculating line shapes, band widths, etc. In contrast to formal quantum mechanics, the natural habitat of semiclassical quantum mechanics is in phase space. Thus, semiclassical quantum mechanics deals directly with the underlying physics. However, it tends to be more complicated than quantum mechanics because the Hamilton-Jacobi equation is nonlinear. Furthermore, in its primitive form, it deals with the individual trajectories which are hopelessly complicated. Thus, many techniques are being developed which simplify the formalism for both pure and semiclassicalquantum mechanics in phase space. Of course, model Hamiltonians are frequently used. Recently, Miller and Ore141have developed a semiclassical modification of the old Dirac-Van Vleck-Serber-EyringWalter-Kimball first-order perturbation valence bond treatment42 which enables them to make semiclassical estimates of both the rotational-vibrational and electronic-transitional probabilities in molecular collisions. Surely, this is an exciting new development! Transformations to action-angle variables appear to be one of the most promising techniques for determining wave functions in phase space. Action-angle variables have been used to a long time in problems where the Hamiltonian is separable. In such cases, the action, S = p dq, is a constant of the motion and the angle variable is periodic in time. Then Moser at the Courant Institute taught Marcus how to use action-anglevariables for nonseparable problems (provided that the energy is sufficiently small). Marcus taught the theoretical chemists. Soon Miller, George, and many others were using action-angle variables in treating scattering problems. Augustin and R a b i t ~ have ~ ~made a careful analysis of the use of action-angle variables in quantum mechanics. The quantum mechanical formulation differs from the classical in requiring the use of projection operators to ensure that the Hamiltonian is Hermitian (for example, to rule out the negative kinetic energy states which otherwise would cause difficulties). Their treatment of the atom-diatomic collision problem is really elegant. Furthermore, they point out that for problems involving a complicated Hamiltonian, H, the action-angle wave functions can be determined for a zeroth-order Hamiltonian, Ho,and used as basis functions for determining the wave functions for H by using a perturbation procedure.
X. Rotation of Coordinates into the Complex Plane Of course, there are many exciting new techniques which are being developed which do not involve representations in phase space. For example, the rotation of coordinates into the complex plane is a procedure which converts wave functions for unbound continuum states of a molecular system into functions which are square integrable (or L2 normalizable) and which leads to well-defined complex (38)J. E.Moyal, R o c . Camb. Phil. SOC.,45,99 (1949). (39)R.Paul, Phys. Rev. A , 23, 999 (1981). (40)B. J. Berne end G. D. Harp, Adu. Chem. Phys., 17,63 (1970). (41)W. H.Miller and A. E. Orel, J. Chem. Phys., 74,6075 (1981). (42)J. H.Van Vleck and A. Sherman, Rev.Mod. Phys., 7,167(1935). See also, H. Eyring, J. Walter, and G. E. Kimbd, "QuantumChemistry", Wiley, New York, 1944. (43)S. D. Augustin and H. Rabitz, J. Chem. Phys., 71,4956 (1979); S.D. Augustin, M. Demiralp, and H. Rabitz, ibid., 73,268 (1980). See also, P. Carruthers and M. M. Nieto, Rev. Mod. Phys., 40,411 (1968).
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The Journal of Physical Chemlstry, Vol. 86,No. 7, 1982
energy eigenvalues for determining the energies and lifetimes (or line widths) of quasi-bound resonance state^.^,^ This formalism is based upon the theorem of Balslev and Combes& and a suggestion of Simon:47 If the Hermitian Hamiltonian for the system is H(p,r), form the rotated Hamiltonian He = H(eiBp,ear) by replacing p by eiBp and r by eierwhere 8 is an arbitrary constant. Then solve the rotated-wave Schrodinger-like equation45 He'kj = We'k, (47)
HirschfeMer
Chu and their associates have extended the method to apply to ionizations in static electric and magnetic fields52 and in oscillating electromagnetic fields.53 The method seems to be especially effective in multiphoton processes in very intense fields.54
XI. Discussion
My manuscript may appear to be somewhat disjointed since I have discussed a number of unrelated problems and techniques to illustrate the great changes which are taking place in theoretical chemistry. Both the problems and the to obtain the complex energy eigenvalues We which give techniques are new. It is hard to predict what theoretical both the real energy E and the line width r of the resochemistry will be like even 10 years from now. However, nance there is one thing certain: As we continue to tackle proW, = E - i ( r / 2 ) (48) gressively more difficult and more complicated problems, we shall increase our dependence upon high-powered If 28 is greater than the arctangent of (I'/2E),then We is mathematics and theoretical physics. Furthermore, we will independent of the value of 19. For bound states I? = 0 so use MACSYMA (which is available from MIT on the that We is real and independent of the value of 8. It is ARPA network) or similar symbolic manipulatiue algorimportant to note that He is not Hermitian.48 However, variational procedures have been d e v e l ~ p e for d ~ ap~ ~ ~ ~ ithms to assist us in our mathematical derivations as well as to determine the numerical solutions. proximating $e and W p This complex rotation technique has been applied to Acknowledgment. I was completely overwhelmed by many different kinds of problems. For example, Moiseyev, this Symposium and I want to take this opportunity to Certain, and Weinholdml5lhave used it to calculate the thank all of the speakers and the other participants who energies and widths of two-electron autoionizing resomade this a very memorable occasion. I especially thank nances. ChuS2used it to study the rotational predissociPhil Certain and Patty Spires-Merkel for their dedicated ation of van der Waals molecules. Then Reinhardt and effort and tireless work over a long period of time to make this possible. Of course, we are very grateful to the Na(44)N.Moiseyev, P.R. Certain, and F. Weinhold, Mol. Phys., 36,1613 tional Science Foundation, Office of Naval Research, John (1978). Wiley and Sons, the University of Wisconsin Graduate (45)P.R. Certain, Chern. Phys. Lett., 65, 71 (1979). School, and the University of Wisconsin Department of (46)E. Balslev and J. M. Coombes, Cornrnun. Math. Phys., 22, 280 Chemistry McElvain Fund for having provided the nec(1971). (47)B.Simon, Cornmun. Math. Phys., 217, 1 (1972). essary financial support. Of course, I express my gratitude (48)F.Weinhold, University of Wisconsin Theoretical Chemistry Into all of the students, associates, secretaries, and friends stitute Report WIS-TCI-590, 1978. (in addition to my wife) on whose shoulders I have climbed. (49)F. Weinhold, J.Phys. Chern., 83, 1517 (1979). (50)N. Moiseyev and P. R. Certain, Mol. Phys., 37, 1621 (1979). (51)N.Moieeyev and F. Weinhold, Phys. Rev. A , 20,27 (1979). (53)S. I. Chu, Chern. Phys. Lett., 64,367 (1978). (52)W. P.Reinhardt, Int. J. Quantum. Chern., S10,359 (1976);J. J. (54)A. Maquet, S. I. Chu, and W. P. Reinhardt, unpublished. Wendoloski and W. P. Reinhardt, Phys. Rev. A , 17,195 (1978).
