INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, VOL. XIX. 755-761 (1981)

Nodeless Wave Functions and Spiky Potentials

GEOFFREY HUNTER Department of Chemistry, York University, Toronto, Canada M3J 1P3

Abstract

The exact (nonadiabatic) nuclear and electronic factors of a molecular wave function are expanded in the basis of eigenfunctions of the electronic Hamiltonian according to the Rayleigh-Schrodinger perturbation theory of Born and Huang. Thus it is shown that, with rare exceptions, the exact nuclear factor (a marginal amplitude) is a nodeless function. The nodes in vibrationally excited nuclear wave functions within the Born-Oppenheimer approximation become node-avoiding minima in the exact nuclear wave function. Corresponding to each node-avoiding minimum in the nuclear wave function the exact (nonadiabatic) effective potential for the nuclear motion has a spiky barrier superimposed upon the Born-Oppenheimer (adiabatic) eigenenergy of the electronic Hamiltonian. These barriers are the result of nonadiabatic coupling between electronic states, which is strongest in the vicinity of the nodes in the Born-Oppenheimer-approximation nuclear (vibrational) wave function.

1. Introduction

The theory by which a wave function may be factorized into conditional and marginal amplitudes was first enunciated in 1975 [l]. Subsequently, Czub and Wolniewicz [2] demonstrated that the marginal factor is necessarily a nodeless function, and in more recent work some of the implications of this nodeless property were explored [3].*

This present paper has a manifold purpose arising from the nodeless property of marginal amplitudes. First of all the nodeless property will be derived by a quasiproof that is both simpler and more general than that given by Czub and Wolniewicz. Second, we will express the total wave function in terms of an expansion in eigenfunctions of an “electronic” Hamiltonian according to the well-known scheme of Born and Huang [ 5 ] , and thereby show how the exact electronic and nuclear factors of a molecular wave function differ from their adiabatic approximants in the Born-Oppenheimer theory. ThiS difference takes the form of a coupling between electronic states that is dependent upon nuclear coordinates; we show that the coupling is strongest near the nodes in the adiabatic nuclear wave function. Thence we are able to demonstrate that the effect of nonadiabatic coupling on the effective potential for the nuclear motion takes the

* It may be helpful to readers unfamiliar with this previous work to note that the factorization is closely related to the separation of nuclear and electronic motion originated by Born and Oppen- heimer [4], the conditional and marginal amplitudes being almost identical with the electronic and nuclear wave functions, respectively, in the Born-Oppenheimer theory. The Born-Oppenheimer electronic and nuclear wave functions are to be regarded as perturbative approximants to conditional and marginal wave functions, respectively. The conditional-marginal factorization theory is an exact formulation of the Born-Oppenheimer separation, and as such it is not restricted to the traditional application of this technique to molecular motion.

@ 1981 John Wiley & Sons, Inc. CCC 0020-7608/81/050755-07$01.00

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form of spiky barriers superimposed upon the adiabatic potential in the vicinity of the nodes in the adiabatic nuclear wave function.

2. Basic Factorization

The essential elements of the marginal-conditional factorization [ 11 are as follows. The whole system of interest (such as a molecule) is represented by a total wave function 4(x, y ) , where x and y represent particle coordinates in the Schrodinger representation of quantum mechanics [6]. The factorization takes the form

(1)

where f ( y ) is the marginal factor and 4 ( x I y ) the conditional factor. The coor- dinates x are those of a subsystem (such as the electrons in a molecule) and y represents the remaining, complementary, set of coordinates (those of the nuclei in a molecule).

It is apparent that the factorization depends upon the partitioning of all the particle coordinates into the two complementary subsets x and y . Actually it depends upon the partitioning of the configuration space into the part spanned by x and the complementary part spanned by y , different choices for these subspaces producing essentially different factorizations. Within each of the subspaces coor- dinate transformations may be carried out without affecting the factorization.

For a given choice of these two complement,ary subspaces factorization (1) is defined by requiring that the conditional factor &(x I y ) satisfy the normalization condition

4(x, Y ) =f(y ) 4 ( x I Y 1,

j &*(xIY)4(xlY)dx=1 (2)

throughout the space spanned by y . This kind of normalization condition, involving integration over only a subspace of the configuration space, is exem- plified by that pertaining to electronic wave functions in the Born-Oppenheimer theory [ 5 ] .

From Eqs. (1) and (2) it follows that the marginal factorf(y) is defined in terms of the total wave function +(x, y ) by

l f ( y ) l ’ = f * ( ~ ) f ( ~ ) = I 4*k y ) 4 ( x , Y ) dx. ( 3 )

This equation defines the modulus of the marginal factor in terms of the known (at least in principle) total wave function for a particular state of the whole system. For bound states the argument (phase) of f(y) is unimportant; f(y) may be taken to be a real function so that it is fully defined by Eq. (3) apart from a possible ambiguity of sign. For scattering states (unbounded in the y subspace, such as those involved in atomic and molecular collisions) the phase of f(y) is determined by criteria unrelated to the factorization [7].

It is apparent that such a factorization is definable for all states of the whole system which are normalizable in the sense that the integrals in Eqs. (2) and (3)

NODELESS FUNCTIONS A N D SPIKY POTENTIALS 751

exist; i.e., for all states that are bounded within the x subspace. Thus this kind of factorization may only be possible for total states that are bounded in this sense. In application to molecules this means that processes involving absorption or emission of an electron may not be describable by a factored wave function; processes in which all the electrons remain bound to a nucleus (and in particular molecular bound states) are describable by a Born-Oppenheimer-like factored wave function. A discussion of conditions for the Born-Oppenheimer separation to be valid has been given recently by Wooley and Sutcliffe [8]. In a recent review Ozkan and Goodman summarize the traditional perturbation theory especially with regard to the case of degenerate electronic states [9 ] .

3. Nodeless Property of a Marginal Amplitude

In order to show that the marginal amplitude f ( y ) is a nodeless function we expand the total wave function $(x , y ) as a sum of products according to the scheme of Born and Huang [ 5 ] :

where the Pi(x I y ) is a complete orthonormal set of functions having the same behavior as b ( x I y ) at the boundaries of the x subspace; i.e.,

everywhere in the y subspace. Substituting Eq. (4) into Eq. (3) and using Eq. ( 5 ) , we deduce that

I f (Y ) I 2 = c I N Y ) I 2 . I

This expression for l f ( y ) I 2 is a sum of non-negative terms. Thus, for I f ( y ) l to be zero, all of the terms IF,(y) l would have to be zero at the same ualue o f y . In view of the fact that the summations in Eqs. (4) and (6) generally involve an infinite number of terms, it is, in general, extremely unlikely that all of the terms of the summation will be zero at the same value of y .

The exceptional case arises when the summation in Eqs. (4) and (6) reduces to a single term. In the Born-Huang expansion the functions F ; ( y ) are regarded as v -dependent coefficients of the complete set expansion. This expansion reduces to a single term only when the variables represented by x and y are strictly separable in the Hamiltonian of the complete system. Strict separability is exemplified by the exactly soluble problems of quantum mechanics [lo].

Apart from this exception, the summations will contain many (and usually an infinity) of terms, so that even though individual terms IFi(y)I2 may have zeros, their sum is always positive. Thus \ f ( y ) I 2 is positive and since’a zero of l f ( y ) I 2 is also

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a zero of f ( y ) [even if f ( y ) is complex], it follows that marginal amplitudesf(y) are generally nodeless functions.*

A simpler, but possibly less convincing, argument arises from the fact that the integrand in Eq. (3 ) , I$(x, y ) I 2 , being the squared modulus of a complex number, is necessarily positive or zero everywhere. For l f ( y ) I 2 to have a node iq?(x, y ) I 2 must be zero at allpoints in the x subspace for the point in the y subspace where the node occurs. Excepting the case where the total wave function $(x, y ) is expressible as a single product [i.e., x and y are separable in the Hamiltonian H(x, y)] i t is very unlikely that I$(x, y)I2 could have such a node.

4. Splitting of the Hamiltonian

In the original formulation of the Born-Oppenheimer separation [4] and in the later Rayleigh-Schrodinger perturbation theory of Born and Huang [S], the Hamiltonian of the complete system H(x , y ) is partitioned into the nuclear kinetic energy T, and the electronic Hamiltonian Ho(x, y ) :

H ( x , y ) = T, + HO(x, y ) . (7) While this partitioning is subject to a degree of ambiguity [ll], its essential

feature is that Ho(x, y ) does not contain differential operators with respect to y ; these terms of H(x , y ) are contained entirely within T,.

In the Born and Huang scheme [ S ] , the basis functions P,(x I y ) of Eq. (4) are chosen to be eigenfunctions of Ho(x, y):

H?x, y)Pz(xIJJ)= [ J , ( y ) P , ( 4 y ) , (8)

where U,(y) is the y-dependent eigenvalue (an electronic energy in the appli- cation to molecules).

The wave functions of the complete system $(x, y ) are, of course, eigen- functions of H(x, y ) :

H(x, Y ) $ ( X , Y ) = E$(x, J-1, (9)

where E is a particular stationary state energy of the system. In the application to molecules the kinetic energy T, contains an inverse mass

factor of the order of and as a result the expansion of $(x, y ) , Eq. (4) contains one dominant term (say i = k). Thus Fk(y) is about 1000 times larger in magnitude than the other components of the expansion. The product Fk(y)pk(x I y ) is approximately an eigenfunction of H(x , y ) [12].

5. y Dependence of Coupling Between Electronic States

The exact electronic wave function (conditional amplitude) 4 ( x 1 y ) in terms of the expansion in eigenfunctions of H"(x, y ) becomes [from Eqs. (l), (4), (6)]

* At the boundaries of the y subspace f ( y ) and all the F , ( y ) will tend to zero asymptotically; this, however, is not regarded as a node-it is merely the boundary condition that applies to the y subspace.

NODELESS FUNCTIONS A N D SPIKY POTENTIALS 759

Throughout most of the domain of y the dominant component, Fk(y) is much larger than the other components F , ( y ) (i # k), and hence it follows that Eq. (10) may be approximated by

(1 1) 4(x I Y ) -- fPk(X I Y), since the numerator in Eq. (10) is dominated by Fk(y)pk(xy), and the denomina- tor by [IFk(y)/2]1'2 = IFk(y)I. Equation (11) expresses the approximate equality of the exact electronic wave function 4 ( x I y ) (i.e., the conditional factor of the exact total wave function) with the appropriate eigenfunction of the electronic Hamil- tonian.

This approximate equality pertains everywhere in the domain of y excepting those regions (points, surfaces) where the dominant component Fk (y ) has a zero. At, and adjacent to, the zeros of F k ( y ) the i = k term effectively disappears from the summations in both the numerator and denominator in Eq. (lo), and hence instead of Eq. (1 1) we obtain

The denominator in Eq. (12) is typically 1000 times smaller than that in Eq. (10) [i.e., away from the nodes of Fk(y)], and it is this feature of Eq. (12) that effectively increases the coefficients of nondominant electronic wave functions Pi(x 1 y ) ( i # k ) in the expansion of 4(x 1 y ) .

In summary, the exact electronic wave function & ( x 1 y ) is approximately equal to the appropriate eigenfunction pk (x I y ) of the electronic Hamiltonian H"(x, y ) except in the vicinity of the nodes in the dominant nuclear wave function Fk(y), where 4 (x 1 y ) becomes a linear combination of nondominant electronic wave functions. In particular if the dominant electronic eigenfunction is the ground electronic state, the exact electronic wave function becomes a linear combination of excited electronic wave functions near the nodes in the nuclear wave function

It is thus apparent that the exact electronic wave function 4(x I y ) has a dependence upon the (vibrational) state of the nuclei. For the lowest-energy state of nuclear motion (zero-point vibrational state) the exact electronic wave function 4(x I y ) is everywhere dominated by the appropriate eigenfunction pk(x I y ) of the electronic Hamiltonian H*(x, y ) . This dominance also pertains to excited states of nuclear (vibrational) motion, excepting the domains of nuclear coordinates adjacent to the nodes in the dominant nuclear wave function Fk(y), where 4(x I y ) becomes a linear combination of nondominant (usually excited) electronic wave functions pi(x I y ) ( i # k ) .

F k ( y ) .

6. Spikes in the Effective Nuclear Potential

It was shown previously [l] that substitution of Eq. (1) into Schrodinger equation (9) followed by integration over the subspace spanned by the electronic coordinates x, leads to a reduced Schrodinger equation for the motion of the nuclei [Eq. (13) of Ref. 11 in which the effective potential energy for the motion of

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the nuclei U ( y ) is given by

U(Y 1 = I 4 (x I Y ) H ( x , Y 14 (x I Y ) dx. (13)

This equation expresses the fact that the effective nuclear potential is the average value of the total Hamiltonian H(x , y ) weighted by the exact electronic wave function C#J (x I y ) , the integration (i.e., averaging) being over the subspace spanned by the electronic coordinates x.

In order to express U ( y ) in terms of the Born-Huang expansion we make the simplifying approximation that the operator H ( x , y ) in Eq. (13) can be replaced by Ho(x, y ) . This is justified by the small inverse mass factor in T,, and while in actual calculations T, must be retained in Eq. (13), we believe that our conclusion regarding the nature of U ( y ) will not be affected by leaving it out. Thus substitution of Eq. (12) into Eq. (13) using Eqs. ( 5 ) and (8) yields the following approximate expression for U ( y ) :

i.e., the effective nuclear potential is a weighted average of the electronic eigenvalues, the weighting factors being the nuclear components IF, ( y ) I 2 .

For typical molecular states this expression for U ( y ) will be dominated by one component ( i = k) producing

U ( y ) = U& ( y 1, (15)

i.e., the effective potential for nuclear motion is approximately equal to an eigenvalue of the electronic Hamiltonian. This is what is usually understood by the notation “the Born-Oppenheimer approximation.”

This dominance of the exact effective nuclear potential U ( y ) by one electronic eigenvalue will pertain throughout the domain of the nuclear coordinates y except in the regions where the dominant nuclear component F k ( ) ’ ) has a node. In the vicinity of such a node Eq. (14) simplifies to

i.e., U(y ) becomes a linear combination of nondominant electronic eigenener- gies.

The denominator in Eq. (16) at a node in F k ( y ) [as in Eq. (12)] is small compared with the denominator in Eq. (14) [as in Eq. (lo)] away from the nodes in Fk(y ) . Thus although the F ; ( y ) ( i f k) do not vary greatly as Fk(y) passes through a node, the rapidly decreasing denominator has the effect of greatly increasing the contributions of nondominant electronic eigenenergies Ui(y) ( i # k) to the effective nuclear potential U ( y ) .

In the usual case where the dominant electronic eigenvalue is that of the ground electronic state, U ( y ) becomes a linear combination of excited electronic energies at, and adjacent to, the nodes in the dominant nuclear component F k ( y ) .

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Thus the effective potential U(y) near a node in Fk(y) becomes large and more positive than U ( y ) -- Uk(y) away from the nodes.

Thus overall, the exact effective potential for the nuclear motion U(y) is describable as approximatly equal to an eigenvalue of the ele.ctronic Hamiltonian Uk(y), with spiky barriers superimposed upon it at each node in the Born- Oppenheimer approximation nuclear wave function Fk (y ). These barriers are the result of nonadiabatic coupling between electronic states [i.e., between eigen- functions of Ho(x, y)] which is y dependent, being strongest at the nodes in Fk(y). This description of the effective potential accords with our inference in Section 5 , that at these nodes the exact (nonadiabatic) electronic wave function q5(x I y ) becomes a linear combination of excited electronic eigenfunctions.

Bibliography

[ l ] G. Hunter, Int. J. Quantum Chem. 9, 237 (1975). [2] J. Czub and L. Wolniewicz, Mol. Phys. 36, 1301 (1978). [3] G. Hunter, Int. J. Quantum Chem. 17, 133 (1980). [4] M. Born and J. R. Oppenheimer, Ann. Phys. (Leipzig) 84, 457 (1927). [5] M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Oxford U. P., London, 1954),

[6] P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed. (Clarendon, Oxford, 1958). [7] D. Rapp, Quantum Mechanics (Holt, Rinehart, and Winston, New York, 1971). [8] R. G. Wooley and B. T. Sutcliffe, Chem. Phys. Lett. 45, 393 (1977). [9] I. Ozkan and L. Goodman, Chem. Rev. 79, 275 (1979).

Appendixes VII and VIII.

[ lo] R. A. Buckingham, “Exactly Soluble Bound State Problems,” in Quantum Theory, D. R. Bates,

[ l l ] G. Hunter, B. F. Gray, and H. 0. Pritchard, J. Chem. Phys. 45, 3806 (1966). [12] G. Hunter and H. 0. Pritchard, J. Chem. Phys. 46, 2153 (1967).

Ed. (Academic, New York, 1962).

Received July 31, 1980 Accepted for publication November 6 , 1980.