J. Phys. Chem. 1985,89, 2122-2129
2122
resulted in virtually perfect attendance a t these evening poster paper sessions. Session chairman/discussion leaders and speakers were invited to submit papers to be published together in this issue. The chairman for the Sixth American Conference on Theoretical Chemistry is Professor Donald G. Truhlar with Professor William A. Goddard I11 as vice-chairman. It will be held July 26-31. 1987 at Gull Lake. MN.
Session Chairmen/Discussion Leaders Berni J. Alder (Livermore), Jan Almlof (Oslo), Rodney J. Bartlett (Florida), Bruce J. Berne (Columbia), Roy G. Gordon (Harvard), Marshall Fixman (Colorado State), Karl F. Freed (Chicago), Nicholas C. Handy (Cambridge), Robert A. Harris (Berkeley), Noel S. Hush (Sydney), John C. Light (Chicago), Rudolph A. Marcus (Cal Tech), Wilfried Meyer (Kaiserslautern), Philip Pechukas (Columbia), Peter Pulay (Arkansas), Klaus Ruedenberg (Iowa State), Isaiah Shavitt (Ohio State), Per Siegbahn (Stockholm), William A. Steele (Penn State), Andrew Streitwieser (Berkeley), Benjamin Widom (Cornell), Kent R. Wilson (La Jolla). Plenary Speakers and Their Titles Steven A. Adelman (Purdue). “The MTGLE Approach to Problems in Condensed Phase Chemical Reaction Dynamics”. Reinhart Ahlrichs (Karlsruhe). “Quantum Chemistry on the CYBER 205”. Hans C. Andersen (Stanford). “Computer Simulation of Water and Aqueous Solutions of Hydrophobic Solutes”. David Chandler (Pennsylvania). “Theory of Excess Solvated Electrons in Simple Liquids”. David P. Craig (Canberra). “Molecule-Radiation and Molecule-Molecule Interaction. The View from Quantum Electrodynamics”.
Ernest R. Davidson (Indiana). “Electronic and Geometrical Structure of Radicals and Intermediates”. Rolf Gleiter (Heidelberg). “On the Validity of Koopmans’ Theorem in Some Organometallic Compounds”. William A. Goddard I11 (Cal Tech). “Mechanistic Considerations for Transition Metal Systems”. Martin Karplus (Harvard). “Dynamics of Proteins: Theory and Experiment”. Yuan T. Lee (Berkeley). “Dynamic Resonances in the Reaction of Fluorine Atoms with Hydrogen Molecules”. Raphael D. Levine (Jerusalem). “Dynamical Symmetries”. William H. Miller (Berkeley). “Recent Developments in the Theory of Quantum Mechanical Reactive Scattering”. John C. Polanyi (Toronto). “Spectroscopy of the Transition State; Computed Absorption Spectra for H,* in H + H, H,t H2 H”. John A. Pople (Carnegie-Mellon). “Thermochemistry of Small Molecules and Ions”. W. Graham Richards (Oxford). “Quantum Pharmacology”. Bjorn Roos (Lund). “Complete Active Space S C F Studies of Molecular Properties; an Overview”. John Ross (Stanford). “Chemical Systems Far from Equilibrium”. Frank H. Stillinger (Bell Labs). “Isomeric Packing Theory for Condensed Matter Structure and Dynamics”. John C. Tully (Bell Labs.) “The Dynamics of Energy Flow at Surfaces”. Robert E. Wyatt (Austin). “Quantum Dynamics and Statistical Mechanics via the RRGM (Megabasis Dynamics?)”. Richard N. Zare (Stanford). “Review of Recent H + D2 Reactive Scattering Experiments”. Robert Zwanzig (Maryland). “Brownian Motion and Chemical Kinetics”. +
+
-
Dynamical Symmetries R. D. Levine The Fritz Haber Research Center for Molecular Dynamics, The Hebrew University, Jerusalem 91 904, Israel (Received: June 27, 1984)
Dynamical symmetries are discussed with special reference to practical, approximate procedures for their determination. It is argued that for realistic molecular problems there is an intermediate time scale which is of interest and is typically of the order of a vibrational period. Much faster variations tend to get averaged over (Le., eliminated in an adiabatic fashion) while much slower variations can be treated in the sudden approximation. The result is a self-consistent procedure which can also be derived from a variational principle. The stability of the self-consistentprocedure is closely related to the separation of time scales and to the utility of constraints in the information theory approach. Also discussed are the symmetries of bound vibrational states.
1. Introduction
Dynamical symmetries are the symmetries of the equations of motion.’** They are in general time-dependent transformations which convert one solution of the time-dependent Schriidinger equation into another such solution while leaving the Hamiltonian unchanged. (1) In a more loose way one can say that dynamical symmetries are the symmetries of the laws of nature. See, for example, E. P. Wigner, ‘Symmetries and Reflections”, Indiana University Press, Bloomington, IN, 1967. (2) There is however no agreed upon technical definition. The one given in section 3 is, as far as I know, consistent with both the spirit and content of other definitions. See, for example, C. E. Wulfman in ’Recent Advances in Group Theory”, J. C. Domini, Ed.,Plenum Press, New York, 1979, and references therein.
0022-3654/85/2089-2122$01.50/0
The practical utility of the concept hinges on our ability to determine the dynamical symmetries directly from the Hamiltonian without solving for the wave functions first. It is concluded that for realistic molecular Hamiltonians the use of approximation schemes is inevitable. A practical, self-consistent scheme for implementing such an approximation is discussed in detail. Particular attention is given to the implications of the approximation in the context of the maximum entropy formalism. The paper begins with the formal definitions and then (section 4) reviews the simpler case where exact results are possible. This is the case where one can determine the exact dynamical symmetries directly from the Hamiltonian. The inevitable price however is that the Hamiltonian has a comparatively simple structure (it is linear in the generators of some Lie group). Realistic molecular Hamiltonians are typically at least bilinear 0 1985 American Chemical Society
Dynamical Symmetries
The Journal of Physical Chemistry, Vol. 89, No. 11, 1985 2123
in the generator^.^ An interesting but open question is whether methods analogous to those used in section 4 can be used in this more general case. We do however use the results of section 4 in an essential way in the approximation procedures (section 5). The variational principle used therein is equivalent to replacing the exact Hamiltonian by an approximate, self-consistent Hamiltonian for which the method for section 4 is applicable. The rest of the paper is a discussion of the content and implications of the variational procedure. A general discussion on the limitations on the Hamiltonians for which exact results are currently available and on how these restrictions can be removed (section 10) concludes the paper.
2. Transformations Let T be a possibly time-dependent transformation. We do not insist that it be a unitary transformation, although in practice it often will be. The solution $ of the time-dependent Schriidinger equation i h a$/at = H$
(2.1)
is transformed by T to the solution of i h a(T$)/at = i h ( a T / a t ) r l T $ + TH$ = (ih(aT/at) [T,HIJTIT$ = KT$
+
+ HT$ (2.2)
Equation 2.2 is again Schradinger like with a transformed Hamiltonian K
+
K = THT’ i h ( a T / d t ) T 1 = H ih(dT/dt)T’ = H - ihT(dT‘/dt)
+
(2.3)
To write K i n the compact form (2.3), we have introduced the total time derivative dT/dt
th dT/dt = ih aT/at
+ [T,N
(2.4)
and d(T’T)/dt = 0 If Tis unitary and H i s Hermitian, K will also be Hermitian. Even if T i s not unitary but f l T is a constant of the motion, i.e. d($lTw)/dt = 0
(2.5)
then normalization will be conserved in time. Since T will, in general, depend on time, the condition (2.5) reads in operator form d(flT)/dt = 0 = a(TtT)/at
+ (ih)-’[flT,HI
(2.6)
The condition is immediately satisfied if T itself is a constant of the motion, dT/dt = 0. Nonunitary transformations have recently received considerable attention from Prigogine and co-workers4 as a route from the “reversible” evolution under a Hermitian Hamiltonian to the irreversible evolution which is typical of our everyday experience in the macroscopic world.
3. Dynamical Symmetries Dynamical symmetries are those transformations which leave the form of the Hamiltonian unchanged,6 K = H . That is, the transformed wave function satisfies the very same Schrodinger ~~
(3) For applications of this point of view to spectra of realistic triatomic molecules see 0. S. van Roosmalen, F. Iachello, R. D. Levine, and A. E. L. Dieperink, J . Chem. Phys., 79, 2515 (1983). (4) See, for example, R. Balescu, “Equilibrium and Nonequilibrium Statistical Mechanics”, Wiley, New York, 1975. For a more recent discussion
see B. Misra, I. Prigogine, and M.Courbage, Physica A (Amsterdam),%A, 1 (1979). ( 5 ) Any function that can be defined via its power series expansion
equation as the untransformed Only the boundary conditions can distinguish $ and TI) when Tis a dynamical symmetry. The derivation of (2.3) shows that a sufficient condition for T being a dynamical symmetry is that it is a time-dependent constant of the motion, i.e. that its total time derivative vanishes. It also follows from (2.2) i h a(T$)/at = ih(dT/dt)$
+ HT$
(3.1)
that a weaker condition is that dT/dt vanishes on the manifold of solutions $ of the equations of motion
(3.2) is to hold for solutions of the time-dependent Schriidinger equation of our particular Hamiltonian H . A . Generators. As in the case of transformations which do not depend explicitly upon time, it proves very useful to introduce the concept of a generator, G. It is defined by requiring that the infinitesimal transformation Z + ieG be the limit of T (for small values of the parameter e ) . Here Z is the identity transformation and the infinitesimal transformation is near the identity in that it induces only a small change. Under an infinitesimal transformation the operators change as
A
=A
+ i e [ ~ , +~ 0(e2) ]
(3.3)
If e is a small time step, e = 6t/h, and G is the Hamiltonian, then (3.3) is a small propagation in time of A, hence the familiar statement that the Hamiltonian generates the time evolution. The new Hamiltonian following an infinitesimal transformation is determined from (2.3) as K= H + i e [ G , a - he(aG/at) = H - t h dG/dt O(e2)
+
+ O(f2) (3.4)
The generator of a dynamical symmetry is necessarily (cf. (3.4)) a time-dependent constant of the motione6A special and familiari0 version of this result is that, for transformations which are time independent, and which leave the Hamiltonian invariant, the generator commutes with the Hamiltonian. In general, the transformation will be characterized by more than one parameter or T = Z + iCe,X, r
(er small)
(3.5)
As soon as we allow for this possibility, it should be recognized that the generators X,cannot be any arbitrary set of operators if the transformations (for different values of e, e = (el, ..., e,)) are to form a group.I1J2 Sufficient conditions are that the generators close under commutation (3.6) Le., they form a Lie algebra. The procedures to be discussed later for the construction of dynamical symmetries are based on the following arguments. If V(t,to) is the evolution operator for the Hamiltonian
ih
au/at
= HU
(3.7)
with boundary conditions (6) R. D. Levine, Chem. Phys. Lerr., 79, 205 (1981). (7) H. R. Lewis and W. B. Riesenfeld, J . Marh. Phys., 10, 1458 (1969). (8) I. A. Malkin, V. I. Man’ko, and D. A. Trifonov, J . Marh. Phys., 14, 576 (1973). (9) P.Pfeifer and R. D. Levine, J . Chem. Phys., 79, 5512 (1983). (10) E. Merzbacher, “Quantum Mechanics”, Wiley, New York, 1961. (11) B. G. Wybourne, ‘Classical Groups for Physicists”, Wiley: New York, 1974. (12) R. Gilmore, ‘Lie Groups, Lie Algebras”, Wiley, New York, 1974.
2124
The Journal of Physical Chemistry, Vol. 89, No. 11, 1985 U(t0,to) = I
(3.8)
c(t) = U(t,toMru(t,to)
(3.9)
then the operator G(t) is a time-dependent constant of the motion i h X / d t = [H,G]
(3.10)
with specified boundary conditions, G(to) = X,. The face of it, (3.9) is not much help since determining the evolution operator is equivalent to solving the problem. If however U is (or, as in section 5, can be approximated as) a member of the Lie group of which X r is a generator, then (3.9) is a linear combination of the generators UX,vt = EgS(tJo)Xs
(3.11)
S
Both the exact and approximate (self-consistent) methods discussed below aim a t providing an equation of motion for the g matrix defined by (3.1 1). In mathematical language,1’,12(3.1 1) is known as the automorphism of the algebra. B. Initial Conditions. To see explicitly that $ ( t ) and T$(t) differ in their initial conditions, we write the dynamical group element as T = UToU
(3.12)
where U is the evolution operator (cf. (3.7)) of the Hamiltonian. To is the group element at the initial time to (note the boundary conditions (3.8)). Now v t $ ( t ) = $(to),where $(to) is the initial state out of which $ ( t ) evolved, $ ( t ) = U$(to). Hence T$(t) = UTo$(to) (3.13) that is, To$(to)is the initial state out of which T$(t) evolved. $ ( t ) and T$(t) will be distinct states only when $(to) is not invariant under To. In, say, a collision experiment or in an experiment where an external perturbation is applied, let to be an initial time when the system is yet unperturbed. Then To is the symmetry due to those initial conditions which cannot be distinguished in that experiment. Invariance of a state under a dynamical symmetry implies a set of distinct initial conditions that will not be resolved by the dynamics. To actually distinguish them, one would have to use an initial state which is not invariant. In general, an initial state will not be invariant under the entire group. It is therefore useful to consider the particular subgroup under which the initial density matrix of the system is invariant. (Of course, it may be an improper subgroup containing only the identity.) It is suggestive to refer to this subgroup as a stability group, Bo. Only those generators X under which Bo is not invariant can induce a change in the state. The larger the stability group is, the fewer will be the details revealed by the dynamics. An extreme example is stability under the entire group. A simple illustration is the tossing of a coin. By assumption, the way such an experiment is usually done, one cannot distinguish such tosses where the coin is originally heads up or tails up. Hence, the final state should also be invariant (assuming a perfect coin so that the gravitational field does not distinguish the two sides). In section 7 we shall seek to generalize this notion, arguing that the stability group should in practice include not only those generators under which the initial state is truly invariant but also such generators which defacto cannot be distinguished because of their extreme instability. 4. Exact Results A simple but nontrivial case for which exact results are available
is when the set of generators of a Lie algebra is closed under commutation with the Hamiltonian” (4.1) ~
~~
(13) Y. Alhassid and R. D. Levine, Phys. Rev. A, 18, 89 (1978).
Levine Here the a’s are a set of numerical coefficients which are independent of the state of the system. In section 5 we derive a self-consistent approximate Hamiltonian, denoted there by h, for which (4.1) remains valid (except that the a’s will then depend on the state). Later in this section we discuss the need for a generalization of (4.1). The reason why such a generalization is necessary is that, as it stands, (4.1) allows for only a restricted set of eigenfrequencies of the Hamiltonian. This will be shown as part of the discussion of stability, later in this section and again in section 10. To see how (4.1) is useful, consider solving for a time-dependent constant of motion G by a separation of variable^^^,'^ G ( t ) = Ef(t)Xr r
(4.2)
Here the g’s are numerical coefficients which depend on time only and the Xis are (time independent) generators. From dG/dt = 0 we have the set of coupled equations Eg‘WXs = (i/h)Cf[HJ,l
(4.3a)
r
S
and using (4.1) we obtain Eg‘(t)Xs = (i/h)C&Cg‘(t)as S
S
(4.3b)
I
Here the dot denotes the time derivative. Equating the coefficients of each generator on both sides of (4.3), we see that (4.1) is indeed a sufficient condition and that there will be up to m linearly independent constants of motion given by the solutions of
g‘ = ( i / h ) C f a S r
(4.4)
The independent solutions arise because there are up to m possible independent initial conditions for the m-component vector g (given, say, by taking g = (1,0,0, ...), (0,1,0, ...), etc.). Since eq 4.4 is linear, any desired particular initial condition can be constructed as a linear combination of the independent solutions. Consider now first the case of the compact Lie group. By definition,11J2the range over which the group parameters can vary is finite. A simple example is the rotation group in three dimensions where the group parameters are essentially the Euler angles. The eigenvalues of the matrix a (cf. (4.1)) will then have to be real so that the coefficients f ( t ) which serve as the group parameters will be either constant (if f = 0) or oscillatory functions of time. For those group parameters whose oscillation is very rapid, the transformation matrix elements will be rapidly varying. Consider a finite-dimensional (A’ X N), irreducible unitary representation of the transformation. If one uses i , 1,and k to label the matrix elements, unitarity implies N cTIJTl= k*
i=1
(4.5)
Now if the matrix elements are rapidly varying, then on the average we have thatI5 ( TIJTlk*)= s J k / N
(4.6)
Such a transformation is “statistical” in that it tends to erase details. To see this explicitly, consider an observable A which is diagonal in the basis employed. Then, on the average (A]k)=
=
(CTjiAiiTki*)
ca,,(
TjtTkl*)
6,k
Tr ( A ) / N
(4.7)
The matrix A averages out to a multiple of the unit matrix (vanishing off-diagonal element; equal magnitude of all diagonal elements)! (14) R. D. Levine, Adv. Chem. Phys., 47, 239 (1981). (15) See the corresponding discussion for the special case, that of the scattering matrix, in R. D. Levine, ‘Quantum Mechanics of Molecular Rate Processes”, Clarendon Press, Oxford, 1969, section 3.6.
Dynamical Symmetries
The Journal of Physical Chemistry, Vol. 89, No. 11, 1985 2125
One can isolate the source of this statistical behavior by changing the basis set of generators. This is most readily seen by introducing a new set by the linear transformation A, =
CaFr r
(4.8)
The matrix a is now chosen as to diagonalize the matrix a. Then [ H A , ] = ZX,Z:a;aS S
(4.9)
I
but by construction
Gaia: = wa', r
(4.10)
[ H A , ] = wA,
(4.1 1)
and hence There will be no more than N distinct eigenfrequencies (and this is the prime limitation on the Hamiltonians which satisfy (4.1)). For a compact group they will come in pairs (and if N is odd, at least one eigenvalue is zero). In this way one can avoid from the very start including such generators which are rapidly varying. This will not always be an exact procedure because Ap's are not necessarily exact constants of the motion. The reason is that if the Hamiltonian H does depend explicitly on time, then so will the matrix a! and hence the matrix g. Only if we neglect the time dependence of the a's is it true that A,(t) = exp(-iwt/ h)A, (4.12)
+
of ( H - i a/&)+ = 0. Another route is to subject the Hamiltonian to a transformation which makes the new Hamiltonian K (cf. (2.3)) linear in the ge11erat0rs.l~ 5. A Self-consistent Procedure A practical procedure20for determining approximate dynamical symmetries (and their stability) will be discussed in this section. The procedure will be motivated first on intuitive grounds of self-consistency and derived later from a variational principle.21-22 The procedure will also determine an effective Hamiltonian, linear in the generators, for which the dynamical symmetry is exact. To achieve a compact derivation, it is convenient to work in a notation where the state of the system is described in terms of a density matrix p. This is however not essential. In any case, p need not be a mixture but can represent a pure state, and such a pure state will remain a pure state under our approximation. In general, when p evolves under an approximate Hamiltonian, it will not be true that
where His the true Hamiltonian of the system. In other words, time propagation of the state and time propagation of the observables need not yield the same results. Our requirement is that they do, i.e., that (5.1) be correct at least for the generators of the algebra. A sufficient condition is that a Hamiltonian h, linear in the generators h = ZhrXr
is an exact constant of the motion d-4,(t)
ih -- wA,(t) + [A,(t),HI = 0 (4.13) dt If the a's do depend on time, then the A,(t)'s are only adiabatic constants of the motion.I6 If the algebra is noncompact, it may happen that the eigenvalues of a are complex. (This does depend on the magnitude of the coupling constants in H and is not bound to happen.I6) As is evident from (4.12), complex eigenvalues imply true instability since small changes in the initial values of the group parameters will exponentially grow with time. In general, to satisfy (4.1) the Hamiltonian need be linear in the generators. But as we have just seen, this puts a severe restriction on the eigenfrequencies of the system. It is for this reason that algebraic Hamiltonians found only limited applications in elementary prticle physics.17 But one can go for Hamiltonians which are of higher order (e.g. bilineaP) in the generators. For such a case, the prescription Gr(t) = uxrU (3.9) where U is the exact evolution operator, is formally still correct. In particular, since U is unitary, the time-dependent generators will close under commutation (cf. (3.6)) [Gr(t),Gs(t)I =
CWu(t) U
However, we no longer have a simple yet exact route to evaluating G,(t) in terms of the time-independent generators. Another way of stating this problem is that if H is bilinear and G ( t ) is linear in the generators, then [H,G(t)]will, in general, be bilinear in the generators and so must therefore be dG/dt (cf. 3.10)). One way out is to note that we do not have to have
[
a41
H-i--,G
(5.2)
r
can be defined such that Tr ( P [ H J r l ) = Tr ( P [ h J r I )
(5.3)
It should be noted that (5.3) is only required to hold for the generators and that, in general, h will depend on the state p of the system. To determine the coefficients hr in (5.2), one introduces the matrix23 u i C T r (~[xrJslI#' = art
(5.4)
S
so that from (5.4) and (5.3) hr = iCdr Tr ( p [ H J , ] ]
(5.5)
S
Introducing the generator G as in (4.2), G = Z.g'(t)X, we see, using (5.5) and (5.3), that the condition that G is a constant of the motion of h for the state p
is equivalent to the self-consistency condition (5.1). Indeed a variational principle21-22can be readily based on this condition. Introduce a Lagrangian L
L = Tr ( p ( t ) dG(t)/dt]
(5.7)
where both p and G are evaluated at time t . The corresponding action integral is
3 = ] " L ( t ) dt - Tr
(p(tl)
G(t,))
(5.8)
10
=O
Since both
p
and G satisfy equations of motion which are first
It is only necessary that the commutator annihilate any solution (16) C. E. Wulfman and R.D. Levine, Chem. Phys. Lett., 84, 13 (1981). (17) See, for example, J. Dothan, Phys. Reu. D, 2, 2944 (1970). (18) For diatomic molecules see (a) F. Iachello and R.D. Levine, J. Chem. Phys., 77, 3046 (1982) or (b) the simpler case considered in R. D. Levine, Chem. Phys. Leu., 95, 87 (1983). See also ref 3.
(19) See, for example, C. E. Wulfman and R. D. Levine, Chem. Phys. Lett., 97, 361 (1983), and references therein. (20) N. 2.Tishby and R. D. Levine, Phys. Rev. A, 30, 1477 (1984). (21) R. Balian and M. Veneroni, Phys. Rev. Lett., 47, 1353 (1981). (22) N. Z. Tishby and R.D. Levine, Chem. Phys. Lett., 98,310 (1983). (23) The matrix u will not always have a well-defined inverse. See discussion in ref 20.
2126 The Journal of Physical Chemistry, Vol. 89, No. 11, 1985
order in time, one can only specify their boundary values at one time point. The simplest choice is p at to and G a t t , . Subject to these mixed boundary conditions the stationary action 63 = 0 condition yields Tr ( 6 p ( t ) dG(t)/dt] = 0 (5.9) and using the invariance of the trace to cyclic permutation and integrating by parts, we obtain (5.10) Tr (6G(t) dp(t)/dt] = 0 If the variation 6 p ( t ) a: p ( t ) is allowed, it follows from (5.9) that Tr ( p dG/dt) = 0. Of course, if G is an exact constant of the motion, then (5.9) vanishes for all possible variations and the action is at its minimum (and equal to Tr ( p ( t ) G ( t ) )which is time independent). We shall vary G by varying the g‘ coefficients in the form (4.2). Using 6G = CGgX,, one readily verifies that for independent variations of the coefficients 6g‘ we recover the self-consistency conditions (5.3) that were imposed on intuitive grounds. As in section 4, the generators are closed under commutation with the self-consistent Hamiltonian h
Levine
Cof(t) a r / a t = hs r
(6.5)
where hs is defined by (5.5). The coefficients hs are themselves functions of the group parameters. To see this and to introduce a suggestive way of writing (6.5), define
W(l7 = (+oluHvl+o) where
+ = U+o is the current state of the system. a % / W = i(+oIU[HX,IU(+o) = iCoS(+oI [HJrI Vl+o )
(6.6) Then (6.7)
Defining a%/a3. =
C D ; ~a~% / a r S
= i(+oIU[HJrI~ l + o )
(6.8)
it follows from (6.7) and (5.5) that
hs =
~
8
a %s / a P
(6.9)
or from ( 6 . 5 ) except that here the coefficients
cy
do depend on the state
ap/at =
EO,
a%/ar
(6.10)
S
= Chfqr
cy;
(5.12)
I
The very same arguments regarding the stability as were introduced earlier can again be applied. Indeed one sees from (4.4) that a small difference 6 g ( t ) in the group parameters propagates also under the matrix a dbg‘/at = (i/h)CGgPcy:
(5.13)
S
and hence exponential divergence of “group orbits” will result when the matrix a has complex eigenvalues.
Having brought the equations of motion of the group parameters to a Hamiltonian form, the only thing left is to compute the matrix elements a%/aS. Now if H is linear in the generators, that is immediate:4 using the g matrix introduced in (6.3). But suppose His not. Say it is bilinear or even higher order in the generators. The order of the commutator [ H A will be the same as the order of H. One can now invoke the group automorphism (3.9) to conclude that u [ H , X l U will also be of the same order. As a specific example, consider the bilinear term XJs; then
uxJsu = rrtx,mxsu = CG;GtX&
6. Group Dynamics
(6.1 1)
a,B
To implement the self-consistent procedure of section 5 , we write the generator as U X , u (cf. (3.9)), where U is the evolution operator for the self-consistent Hamiltonian h (5.2). Since this Hamiltonian is linear in the generators, its evolution operator is an element of the Lie group and can be written as13
u(t)= exp(iCS(t)X,)
(6.1)
To prove that U is the evolution operator, differentiate it with respect to the group parameter^'^^^^
au/ar = i2,U
(6.2)
Here 9,= X , only for an Abelian group and is, in general, a linear combination of generators, reflecting the need to take care of the special noncommutation of the Xis. To determine the necessary linear combination, one can, for example, note the relation (6.3) valid for any algebra, and differentiate both sides with respect to {. We shall be particularly concerned here with the situation where the Hamiltonian H i s given by H = Ho + V(r),where the perturbation v(t)vanished prior to some initial time, say to. If one chooses +o as an eigenstate of Ho, $ = U+o is a solution of the Schriidinger equation with the evolution operator satisfying the boundary condition (3.8). Since the time dependence of U is via the group parameters
For higher powers, similar results apply. The only matrix elements over q0 that need be computed are thus of products of generators. If the Hamiltonian is only bilinear, then only bilinear terms, i.e. (+ocyolx,~+o), need be computed. All else is done by the group automorphism (6.3). Of course, the final stage is the (numerical) solution of the “classical” equations of motion (6.10).
7. Sum Rules and Slow Variables In sections 5 and 6 we have introduced an approximate evolution operator U (6.1), which is an exact evolution operator for the self-consistent Hamiltonian h. It follows that, for any observable A,, one can define Ar(t) A r ( t ) = UA,U
(7.1) which is an exact time-dependent constant of the motion for the self-consistent Hamiltonian h. For the state p , p = Upout, used in the definition of h (5.3), Tr ( p ( t ) A , ( t ) ) is, by the cyclic invariance of the trace, the mean value of A, at the initial time to (under the evolution generated by h ) . Consider now the special case where A, is a generator, X,, of the group. Then there is an explicit expression for U X r U given by the automorphism (6.3) of the group. Hence (Xr)m 2 Tr ( p d r ) = Tr ( p X r ( t ) ) = Tr (pCgS(t)Xs) S
= E N ) Tr
S
)
= E&) (4) (t) S
I
= iE(Co”%Y/at)x,u s r
W
S
au/at = iE(ay/at)&u (6.4)
where 9, = Co”Js and the matrix D(t), determined via (6.3), is time dependent. The equation of motion of the group parameters is therefore
(7.2)
or by inverting (24) Explicit examples and many practical details regarding this special case will be found in ref 13.
The Journal of Physical Chemistry, Vol. 89, No. 11, 1985 2127
Dynamical Symmetries (xs)(t) =
CG:(xr)in r
(7.3)
where G is the inverse of the g matrix. Equation 7.3 is the explicit statement that the mean value of a time-dependent constant of the motion is time independent. It has the same value initially ( t = to) as at any other time t . (Note from (3.8) that &=to) = 1 and hence G(t=to) = 1.) The practical point is that eq 7.3 can be verified by experimental or computational data and hence serves as a flag for the presence of a dynamical symmetry. If the actual Hamiltonian H i s not linear in the generators, then A,(t) is not strictly an exact constant of the motion of the actual problem. But it will be a very slow variable. In other words, we have three time scales. Very fast variables, if present, are being considered as part of the stability group and are assumed to be averaged out on the time scale of interest. Ordinary variables are the generators of the group such that the self-consistent Hamiltonian is linear in these variables. Their frequencies are the eigenvalues of the matrix (cf. (5.11)). Very slow variables are those which are constant under h. The variation with time is due to the actual Hamiltonian containing bilinear (and higher order) terms in the generators. In realistic Hamiltonians such terms reflect, e.g., the anharmonicities of the internal vibrations. Hence, their rate of change is slower by the ratio of the anharmonicity parameter (w,x,) to the harmonic frequency (to,), Le., by about 2 orders of magnitude. Realistic perturbations are often predominately linear in the generators, and hence the self-consistent Hamiltonian will typically fully account for the perturbation. It is primarily the unperturbed, Nopart of the Hamiltonian which makes h but an approximation. Hence, using the constants of motion of h as exact ones is a sudden-type approximation but better than the ordinary sudden approximation. We do not regard the internal energy levels as degenerate but rather as having a quasi-harmonic spectrum. It is the anharmonicities which are being neglected. Thus, we have an approximation which is sudden with respect to the very slow motions and adiabatic with respect to the very fast motion (since we effectively average over the very fast variables, if such exist). The elimination of the very fast variables as an adiabatic approximation was already discussed in section 4. Here we pointed out that regarding U as an exact evolution operator is also a sudden-type approximation with respect to the very slow variables.
8. Coherent States An interpretation of the self-consistent scheme which helps show that it accounts for the coherent part of the dynamics and only neglects the slow "dephasing" or "incoherent" motion is discussed in this section. An explicit example, that of a perturbed Morse oscillator, is used as an illustration. The formal definition of coherent states25 reflects the requirement that they remain coherent, Le., do not dephase, for some Hamiltonian. Specifically, let a Hamiltonian have a dynamical symmetry. Then time propagation from t to t'is simply a change in the set of group parameters g U T ( g ) f l = T(g') (8.1) Given some state we define the corresponding coherent state by T$. Then by (8.1)
+
UT(g)+ = T(g')U+ (8.2) It follows that a time evolution of a coherent state always yields a coherent state. Implicit in (8.1) is the assumption that the Hamiltonian is linear in the generators. While this can be somewhat relaxed, we do not do so here; for even if the actual Hamiltonian is not linear, the self-consistent Hamiltonian is. The coherent states under the present discussion are coherent under the time evolution generated by h. Given some initial state $o, we form $(g) = T(g)fi0,a state (25) A. M. Perelomov, Common.Math. Phys., 26,222 (1979); R. Gilmore, Rev. Mex. Fis., 3, 143 (1974).
parametrized by the group parameters, and the time evolution of$(g) under h is simply a change in the numerical values of the group parameters. As an example, consider the coherent states for a Morse oscillator. Using the notation of ref 18b, we obtain the Hamiltonian for an unperturbed oscillator H = hwo(A+A- + l/zZO) (8.3) Here oo= o,(l - x,) and x, = W ~ ~ / with W , xo = l/N. N/2 or N/2 - 1 is the quantum number of the highest bound state. The creation and annihilation operators in (8.3) are represented by A+ = N-'/2,?/3 A- = N-'/2/3t, 10 = N-'(Bt/3
- at,)
(8.4) CY and /3 satisfy the usual harmonic oscillator commutation relations. A suitable basis (the ordinary Morse bound states) is IN,u) = [u! ( N - ~)!]-'/~(a~)"(/3t)~-"lN,O) (8.5)
- -
The harmonic limit is N m (or xo 0) with wo remaining finite. The commutation relations are [A-,A+I = Io
-
[Io,A+I = 7 2 x d * (8.6) Note that Io is the identity operator only in the harmonic N limit. The Hamiltonian (8.3) is thus bilinear in the generators. The dynamical symmetry for a Hamiltonian linear in the generators is T(a) = exp(aA+ - a*A-) = exp[(aN-'/2)J+ - (cu*N-'/~)J-] (8.7) The second form in (8.7) is in order to write the dynamical symmetry in terms of angular momentum operators26for which one can readily write (8.7) in an alternative form T ( a ) = exp(S;I+) exp(-Jo(N/2) In (1 + exp(-[J-) (8.8)
In2))
with ( = (a/lal) tan (lal/N1/2). From (8.4) and (8.5) A-IN,u) = ([1 - XO(U-~)]U)'/~~N,U-') (8.9) and so A- annihilates the ground state and Io acts as the identity operator on the ground state ZOIN,U) = (1 - ~XOU)IN,U) (8.10) Hence, it is easiest to apply T(a)(8.8) to the ground state T(~)IN,o)= (1 + I ~ I ~ ) - ~ ~ ~ ~ ( ~ J + ) I N , o )
- -
In the limit N m, [ c~N-'/~, the binomial coefficient tends to NVI2/v! and the Morse coherent states tend to the harmonic oscillator onesz7 IN,,) T(a)lN,O) e x p ( - l a 1 2 / 2 ) ~ ( ~ ! ) - ' / 2 a u l (8.12) ~)
-
N-..
u
The harmonic oscillator coherent states are coherent for the group whose generators are 1, a, at, and at,. Hence, they are coherent not only for an unperturbed motion but also, say, when the oscillator is linearly forced, v(t)= f(t)(a at), since the perturbation is linear in the generators. A linear forcing of the Morse potential is also linear in the generators (8.4), but the unperturbed Hamiltonian (8.3) is quadratic. Indeed operating with exp(-iHt/h), Hgiven by (8.3), on the coherent states shows that not only is the phase of (changing by wot/h but each term
-
(26) J. Schwinger in "Quantum Theory of Angular Momentum", Academic Press, New York, 165, p 229. (27) I am indebted to Dr. E. B. Stechel for pointing out an incorrect normalization in the version of (8.1 1) given in ref 18b. Without this correction the harmonic limit (first introduced in R. D. Levine and C. E. Wulfman, Chem. Phys. Lett., 60, 372 (1979)) does not obtain.
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The Journal of Physical Chemistry, Vol. 89, No. 11, 1985
is acquiring an additional phase w G 0 u 2 t / h , This dephasing will only be slow for low-lying states. 9. Relevant Variables and Maximum Entropy The maximum entropy f ~ r m a l i s m ~ assumes ~ - ’ ~ - ~that ~ the state of the system can be well described as depending on the mean values of m relevant variables. Given that such observables can be identified, one can show28that consistency conditions require that the state by constructed via the maximum entropy formalism. The real question therefore is to identify the relevant variables. The purpose of this section is to argue that the relevant variables are the stable constants of the motion of the self-consistent Hamiltonian h. That is, they are the slow variables of the exact Hamiltonian. Consider an initial state for the collision. It will have definite values for the m expectations ( A , ) (r = 1, ..., m ) of the relevant variables. Other possible initial states will have the very same m expectation values but will differ in that additional variables will have different expectation values. The stability argument of section 4 suggests however that these differences between the initial states will be erased during the collision. All the different initial states which have the same set of values for the m expectations of the relevant variables will therefore yield essentially the same final state. A specific illustration of such a behavior was recently discussed.29 It follows that the final state will be characterized by the expectation values of the m relevant observables. General consistency arguments can then be used to show that such a state is one of maximal entropy subject to the m expectation values as constraints.28 It may however be worthwhile to examine in some more detail the following question: why is the maximum entropy procedure appropriate under the stated circumstances? Entropy measures the (logarithm of the) number of quantum states or the (logarithm of the) volume in classical phase space. By using maximum entropy (subject to constraints), one identifies the largest possible volume in phase space or the largest number of quantum states consistent with the given (final) value of the m relevant observables. Every initial state which gives rise to a final state consistent with the values of the m relevant observables is therefore represented. A final state of maximal entropy is thus a “common denominator” to all these initial quantum states, which, by assumption, are not to be distinguished, after the collision. If we choose any different final state, then, by construction, it will have a lower entropy. Some of the possible final quantum states are thereby necessarily excluded. That means that the corresponding initial quantum states can be distinguished from those initial quantum states whose final states are included. But by assumption that is not the case. What is the initial state corresponding to the final state of maximal entropy? Within the self-consistent scheme discussed in section 5, this is also a state of maximal entropy. Indeed it remains one throughout the collision.20 As mentioned earlier, there are circumstances when the self-consistent propagation gives exact results, and then it is a rigorous result that an initial state of maximal entropy remains one throughout its time ev01ution.I~ Here too we can appeal to entropy as a measure of volume in phase space to understand what distinguishes such an initial state. An initial state of maximal entropy is the largest set of initial quantum states which have common values for the m relevant observables. Choosing another initial state (consistent with the values of the relevant observables), but of lower entropy, will not give a final state which can be distinguished from the one we are after but will require a mor,edetailed computation, generating details which will get averaged over. 10. Structure Recently algebraic Hamiltonians, bilinear in the group generators, have been extensively a ~ p l i e d ~to. vibrotational ~~,~~ states (28) Y. Tikochinski, N.Z. Tishby, and R. D. Levine, Phys. Rev. Letf., 52, 1357 (1984). (29) P.M.Agrawal, N. C. Agrawal, R. Viswanathan, and L. M. Raff, J. Chem. Phys., 80,760 (1981).
Levine of molecules. To examine the underlying idea of such an approach, consider a time-independent generator A. If [H,A] = 0 (10.1) then one readily verifies that if $ is an eigenfunction of H , then so is A$ which is necessarily degenerate with $. The group whose generators satisfy (10.1) is the familiar symmetry group of the Hamiltonian. It suffices to provide quantum numbers for the degenerate eigenstates but cannot be used to compute the spectrum (or the states). A generalization of (10.1) is [ H , A ] = wA (10.2) Then, if $ is an eigenfunction of H with eigenvalue E , A$ is an eigenfunction with eigenvalue E + w . The condition (10.2) is a special case of a dynamical symmetry, when the Hamiltonian can be expressed as a linear combination of generators. However (cf. the discussion of (4.1 l)), the number of eigenfrequencies w is restricted so that (10.2) can give rise, at most, to a quasi-harmonic spectrum. One (a posteriori, obvious) way to generalize (10.2) is to take the Hamiltonian H to include bilinear (or even higher order) terms in the generators. But then, if H is bilinear, so will [H,A] be, and hence one cannot satisfy (10.2). It is then no longer obvious as to what role the group plays or in simpler words ”who is invariant”. One could think of taking A itself to be bilinear in the generators, but then the commutator of two distinct A’s will be trilinear so that such A’s do not form an algebra. The proper procedure under such circumstances is to go back to the definitions and solve for the time-dependent generators G(t), ( H - ia/at)G(t) = 0. For any but the simplest Hamiltonian this is a tall order. Hence, the recent a p p r o a ~ hhas ~ , ~been ~ to choose a group, on the basis of both physical and formal arguments, introduce a Hamiltonian bilinear in its generators, and follow the consequences. The question of “whose group” is thus even more pressing since a particular Hamiltonian (with given numerical coefficients in front of the terms bilinear in the generators) will not be an invariant under such a group. Rather (cf. (3.10)), the group will generate an entire class of possible Hamiltonians bilinear in the generators. (If the original Hamiltonian is a polynomial of degree n, then again other polynomials of the same degree will be generated.) Take as an example a single anharmonic oscillator. It will have a finite number of bound states. This number is determined by the well capacity parameter3’ 2mD/h2a2.Here m is the mass, D an energy scale parameter (e.g. the dissociation energy of the Morse oscillator), and a a length scale parameter. If (“law of corresponding states”) the anharmonic potential can be characterized by two parameters (e.g. D and a ) , the spectrum will be fully characterized if we specify an energy scale parameter. We shall measure energies in units of A = h2a2/2m. ( A is the anharmonicity, A = a $ ~of , , the Morse oscillator.) We now consider all anharmonic (say, Morse) potentials irrespective of the number of bound states. (That is, we allow the well capacity to be variable.) Any particular oscillator will, of course, have a definite number of bound states, but our group is a symmetry group of that larger problem, to which we assign a super Hamiltonian2 7 f . Now 7f is to be bilinear in the generators but needs to be invariant under the group. There is only one way. 7f has to be the Casimir operator of the The eigenvalue of 7f identifies the number of bound states, Le. specifies the spectrum in dimensionless units. All the eigenfunctions of 7f (generated via the group) will be degenerate. But this is as it should be. For a given physical problem all eigenfunctions have the same number of bound states. The eigenfunctions are thus obtained by using the group generators, but how is the physical spectrum to be determined? (30) 0. S.van Roosmalen, I. Benjamin, and R. D. Levine, J . Chem. Phys., 81, 5986 (1984). (31) C. E.Wulfman and R. D. Levine, Chem. Phys. Left., 104,9(1984). (32) The Casimir opcrator commutes with all the generators. The familiar Casimirs are bilinear in the generators, but one can introduce Casimirs of higher order if required. Groups can have more than one bilinear Casimir operator corresponding to the possibility that the physical system requires more than one parameter for its identification.
J . Phys. Chem. 1985,89, 2129-2138
Let 9 be the group and C ( 8 ) its Casimer operator. Consider a proper subgroup 9’ of the group 8. It too will have a Casimir operator which is denoted here by C(0’). Now, C(9) and C(9’) commute and thus will have a common set of eigenfunctions. Then, either the eigenvalues of C(9’)suffice to assign a unique label to each one of the N degenerate states of C ( 9 ) or some of the states are still degenerate. In the former case we identify AC(9‘) as the algebraic Hamiltonian for the physical problem. In the latter case there will be a proper subgroup 9” of 9’ (itself a proper subgroup of 9 ) . We then write the algebraic Hamiltonian as a linear combination of C(9’) and C(9”). If necessary, the procedure is continued until one has a complete set of commuting Casimir operators. To reiterate, the super Hamiltonian is the Casimir operator of the full group. It therefore commutes with all the generators (and all functions thereof). Starting with an eigenfunction of the super Hamiltonian, the action of the generators will yield the other (degenerate) eigenfunctions. The eigenvalue of the Casimir operator of the group is not the energy but identifies the physical system. As an example, in the Morse oscillator problem (section 8), N ( N - 1) is the eigenvalue of the Casimir operator of U(2) while m, m = N - 2v, is the eigenvalue of the Casimir operator of the proper subgroup U(l), U(2) 2 U( 1). The physical energy is Am2. States are thus labeled by N a n d m (or N and v). When the physical Hamiltonian is a linear combination of Casimir operators of a chain of subgroups 9 3 9’ 2 9” (10.3) 1..
then the eigenvalues of the Casimir operators of the subgroups serve as good quantum numbers. One can conceive however of a more general case where the Hamiltonian is a general linear combination of linear and bilinear (and even higher order terms) in the generators of the group 9’. By construction, such a Hamiltonian will still commute with the Casimir operator of the group 9 (Le. with the super Hamiltonian 7f will continue to have common eigenfunctions. Under such circumstances however the subgroups will no longer be symmetry groups of H. In this fashion one can build up algebraic Hamiltonians with any “amount” of desired symmetry. A very practical way of doing this is as follows.
2129
Let both 9’ and 9” be distinct proper subgroups of 9 and let 9“’ be a proper subgroup of both 9‘ and 9’‘ (10.4)
Then the Hamiltonian H = A’C(99
+ A”C(9’9 + A’”C(,’’’)
(10.5)
necessarily commutes with the Casimir operator, C(S”‘), since 9”‘ is a proper subgroup of both 9‘ and 9“. Hence, the eigenvalues of C(9”’) remain good quantum numbers. Another scheme is when 8’ has two distinct proper subgroups (10.6)
Then for a Hamiltonian of the form (10.5) only the eigenvalues of C(9’) are good quantum numbers. In this fashion it is possible to generate spectra which are neither regular (complete set of good quantum numbers) nor completely irregular (no good quantum numbers) but are of intermediate character (some good quantum numbers). It has generally been our experience30 that most observed molecular spectra are of the latter type. Finally, we need to mention that the Rayleigh-Ritz variational principle can be used to determine the spectrum of a Hamiltonian bilinear in the generators. Given some reference state q0,the class of trial wave functions is taken to be T(g)q0,where Tis an element of the group, and the set, g, of group parameters is to be used as the variational parameters. This yields33a self-consistent Hamiltonian h which is the very one introduced on dynamical grounds in section 5 . Acknowledgment. The work reported herein reflects many useful discussions with N . Z. Tishby, S . Alexander, and C. E. Wulfman. I thank the members of the Physics Department of Drexel University for their kind hospitality, while this paper was written, and the Stein Foundation for support of my visit there. (33) N. 2.Tishby and R. D. Levine, Chern. Phys. Letr., 104, 4 (1984).
Photoelectron Spectra of Transition-Metal-1-Methylborinato Complexes in the Outer Valence Region. Semiempirical Model Calculations Based on Green’s Function Formalism‘ Michael C. Bohm,+ Rolf Gleiter,* Institut fur Organische Chemie der Universitat, 0-6900 Heidelberg, West Germany
Gerhard E. Herberich, and Bernd Hessner Institut fur Anorganische Chemie der Technischen Hochschule, 0-5100 Aachen, West Germany (Received: August 8, 1984) The electronic structures of the 1-methylborinato complexes of V(CO)4 (l),Mn(C0)3 (2), CO(CO)~(3), and Co(C4H4) (4) have been studied by means of INDO calculations and their He I photoelectron (PE) spectra. The calculated rotational barriers and the orbital sequences of 1-3 have been compared with their isovalent (C,H,)M(CO), species. The measured ionization energies are discussed on the basis of those calculated by means of the Green’s function formalism. It is found that Koopmans’ theorem is only valid in the case of 1. Significant differences between the filling scheme of the canonical MO’s in the ground state and the sequence of the ionic states are demonstrated for 2-4. It is shown that large relaxation energies are partially compensated by pair-relaxation contributions (Le., many-body corrections).
I. Introduction The electronic structures of various transition-metal carbonyl derivatives have been rationalized in the past decade by means
of semiquantitative models that were developed on the basis of simple one-electron calculations based on the Wolfsberg-Helmholtz (WH) Or extended HUckel (EH)aPProximation.3’4 Mo-
Present address: Institut far Physikalische Chemie, PhysikalischeChemie I11 der TH Darmstadt, D-6100Darmstadt, West Germany.
(1) Part 28 in the series: Electronic Structure of Organometallic Compounds; for part 27 see ref 2.
0022-3654/85/2089-2129$01.50/0 0 1985 American Chemical Society
