INTERNATIONAL J O U R N A L OF Q U A N T U M CHEMISTRY SYMP. NO. 9. 311 315 (1975)

Ionization Potentials and Conditional Amplitudes

GEOFFREY HUNTER Quantum Theory Project, University of Florida. Gainesville, Florida 3261 I

Abstract A procedure for deriving ionization potentials is presented based upon the factorization of a wave

function into the product of a conditional amplitude and a one-electron marginal amplitude. In the asymptotic limit as one electron is removed the conditional amplitude becomes the wave function of the residual positive ion, so that successive ionization potentials can be generatcd in sequence by the same procedure.

Introduction

Some recent work on the direct evaluation of ionization potentials and electron affinities has been based upon extensions of Koopmans’ theorem [ l , 21, or upon Rowe’s equations of motion method [3,4]. A preliminary account of the relationship between these two methods has been given [5] , although it has been suggested that the theoretical basis of the former method may be unsound [6] despite some good numerical results [l]. In this paper ionization is described in terms of a formalism based upon the theory of conditional probability amplitudes in wave mechanics [7].

Mathematical Removal of an Electron

Consider an N-electron wave function Y Y ( R , xl, x 2 , ... , x N ) where x i denotes the space-spin coordinates of electron i, R denotes nuclear coordinates in case the system is a molecule rather than an atom, the subscript j denotes a specific state of the system, and the superscript N is introduced to indicate an N-electron wave func- tion when all its arguments are not explicitly shown, that is, Yy = Y Y ( R , xl, x 2 , ... , xN) . In the context of this discussion “7 is an exact eigenfunction of the hamil- tonian of the system, and it is presumed to satisfy all appropriate angular momentum, electron exchange, and possibly other, symmetry properties. It is also assumed to be normalized :

Yy* YY dR d x , dx2 ... d X N = 1 (1) s

In accordance with previous considerations [7] the wave function “7 is expressed as the product of a one-electron marginal amplitude f y ( xN) and a conditional amplitude @ ‘ ( R , x l , x 2 , . . . . X , - ~ ~ X N ) :

I

(2) y y ( R , x23 * * ’ 9 x N ) = f 7 ( x h ’ ) 4r(R, x2, ” ’ 9 X N - IIXN)

31 1 @ 1975 by John Wiley & Sons, Inc.

312 HUNTER

The modulus of the marginal amplitude is defined in terms of "7 by

(3) l fY(xN)12 = s '4'7' '4'7 d R d x , dx2 ... dXN- 1

Thus the factorization (2) is defined except for a real phase function ~ ( x , ) :

(4) S Y C X N ) = ~ X P { ~ x ( x N ) } I fj" (x,v)I

The significance, if any, of the phase function ~(x , ) is not at present understood. In applications carried out so far [S, 91 it has proved sufficient to let x = 0 or x = n. thus allowing f Y ( x N ) to be real, but to change sign wherever I~Y(x,) I has a zero. This uncertainty of phase will not be considered further in this paper. The N-electron conditional amplitude 47 may be obtained from Equation (2) by division of "7

The interpretation of fY(x,) is that it is the wave function for the quantum mechanical motion associated with x,, averaged over { R , xl, x,, ... , 1,- ,}. That is, lfrl' is the probability of an electron being in the volume element d x , in state j , averaged over all the other coordinates [lo]. The interpretation of 4 Y ( R , x l , x 2 , .... x N - ,Ix,) is that it is the wave function (probability amplitude) describing the motion associated with { R , xl, x2, ... , x,- 1 } in s ta te j for a fixed value of x,. That is, 1471' is the probability of the system being in the volume element d R d x , d x , ... dx, - , for a fixed value of x N [lo]. These marginal and conditional wave functions satisfy the normalization conditions

by fY.

47' @ ' d R d x 1 d x 2 . ' . d X N - , = 1 s ( 5 ) sf:* f7dXN = 1 ;

The coordinates of one electron xN consist of one distance, two angles, and a formal spin coordinate [ 111. The distance, denoted by rN, may be taken to be from electron N to the center of mass of the remaining particles in the system. In view of its interpretation &' must become a constant function of X, in the limit rN -, x. This constancy arises because the motion of the remaining particles is independent of x, as rN -, cc.

As has been shown in general terms previously [7], the marginal amplitude fY(x,) satisfies a one-electron Schrodinger equation, which in atomic units is

(6) - l/(Zp) vifr(x,N) + { U r ( x N ) - E 7 ) f r ( x N ) =

The effective one-electron potential V~(X, ) is the expectation value

(7) U ? ( X , ~ ) = s 4j " HN 4 y d R d x 1 d x z . . . d x , - l

H" being the N-electron Hamiltonian for the system, and p the reduced mass of an electron and the total mass of the remaining particles. In practice U ~ ( X , ) will com- monly be simply a radial potential UY(rN) .

In the limit rN -+ cc the Coulomb interactions ofelectron N with the other particles tend to zero, and since @' becomes a constant function of x,, the kinetic energy

IONIZATION POTENTIALS 313

part of HN# associated with x N must also tend to zero, so that in the context of Equation (7) HN effectively becomes HN- in this limit. Thus in this limit $7 becomes a state of the (N - 1)electron system with one electron (formally electron N , but in view of the antisymmetry of "7, any one electron) at rest at infinity. That is,

YY-'(R,xl,x2, . . . , x ~ - ~ ) = lim 4 Y ( R , x l , x 2 , ... rN-m

(8)

In addition to simply taking the limit rN + co, it may also be necessary in some cases to apply projection operators in order to make Y7-l an eigenfunction of the spin and angular momentum operators of the ( N - 1)-electron system [3]. This will be understood implicitly in the following discussion.

The above considerations imply the result that

E7-l = lim UY(x,)

That is the asymptotic limit of the potential U 7 ( x N ) is an eigenenergy of the (N - 1)- electron system.

In passing it may be noted that the above discussion does not depend upon x N being the coordinates of an electron, so that by choosing rN to be the distance between the centers of mass of any two molecular fragments, dissociation processes generally may be described by a similar formalism.

r N - m (9)

Excitation Followed by Ionization

Ionization is produced physically by inducing a change of state of the system from its initial bound state Yy (commonly the ground state Y!) to an excited state Y,". This change may be produced, for example, by absorption of a photon. Irrespective of how the change is produced, Yf represents the excited state. The excitation process represented by Yr + Yp is in principle an exact description, although practical calculations will usually involve perturbation techniques such as the sudden approximation [3].

For ionization to occur, the excited state Yf must mutatis mutandis be auto- ionizing. In terms of Equation (6) this means that the kinetic energy - 1/(2p) V i f : ( x , ) must be 20 as rN -+ co. The radial component of ff(x,) will be a typical sinusoidal radial scattering wave function in this limit [12]. In an actual experiment the kinetic energy will always be positive, but the idea of an ionization potential is that it is the minimum energy required to produce ionization, that is, the energy required to remove an electron with zero kinetic energy. Thus using Equation (9) the ionization potential (IP) is generally given by

I P = E:-' - E y = lim U r ( x N ) - E r rN-+ 30

where E r is the eigenenergy of the bound N-electron system in the initial state i and U r ( x N ) is the effective one-electron potential in the excited autoionizing state s.

It should be noted that in computational practice it is usually more convenient to calculate Uf(x,) by rearrangement of the differential Equation (6) since ff(x,)

314 HUNTER

is known from Equation (3), rather than by using the definition (7) [8, 91. Thus to calculate the first ionization potential only the marginal amplitude f : ( x , ) is required. In order to compute the second ionization potential the ( N - 1)electron total wave function 'Yr-' must be generated from the conditional amplitude 4; using Equation (8). By subjecting 'Y,"-' to the same process used on Yr, the second ionization potential E:-' - E r - ' will be obtained. Clearly by repetition a succession of ioniza- tion potentials can be produced by removing a single electron at each step until all the electrons have been removed. Electron affinities are in principle produced by begin- ning with the wave function of the negative ion.

Pseudeadiabatic Ionization Potentials

In the case that

lim U:(x,) = lim U r ( x , )

the ionization may be described as adiabatic, since it apparently takes place by the electron receiving just enough kinetic energy to escape from the attractive potential U r ( x , ) in the initial state. In fact this description is probably artificial, since it is unlikely (in view of earlier work [8]) that U:(x,) = Ur(x , ) throughout the whole range of I,. Nevertheless the equality of the asymptotic limits of these two potentials means that the corresponding first ionization potential'can be computed from a knowledge of the initial bound state wave function 'Y? alone. For a practical calcula- tion of this kind the wave function would have to have the correct asymptotic form in the limit as an electron is taken to infinity, as well as being a good wave function in the sense of the variation principle. Examples of such asymptotically correct wave functions are those computed by Pekeris for two-electron atoms [13].

I N + w I N + w

Relationship to Other Theories

Although there is no immediately obvious relationship between the above formalism and that used in the other work already referred to [l-41, there is one point of contact. The propagator people [14, 3, 41 have noted that the overlap amplitudes g:(x,), defined below, are important in the theory of ionization :

g ,"xN) = y ~ - 1 ( R , ~ 1 7 x 2 , " ' , X N - ] ) y r ( R , X 1 , X 2 , "',xN) s (11)

X d R d x , dx2 ... dXN- 1

Substituting fr 4; for 'Yr from Equation (2), and limrN+w@ for Vr- ' from Equa- tion (1 l) becomes

g;(x,) = f r ( x , ) lim 4;' #(R , xl, ... Ix,) d R d x , d x 2 ... d x , - , J{ I N + w I (12)

In the adiabatic case, where in effect s -+ i as rN -+ co, the integral in Equation (12)

IONIZATION POTENTIALS 315

becomes simply the normalization integral of Equation ( 5 ) as f N -+ 00, so that in this limit g r ( x N ) -+ f?(xN). The significance of this relationship, as well as the overall problem of relating the different theoretical methods, must be left for future investi- gation.

Acknowledgment

The author is grateful to Robert G. Parr and Yngve o h m for making their recent papers available prior to publication.

Bibliography

[I] M. M. Morrell, R. G . Parr, and M. Levy, J. Chem. Phys. 62. 549 (1975). [2] 0. W. Day, D. W. Smith, and C. Garrod. Int. J . Quant. Chem. s8, 500 (1974). [3] B. T. Pickup and 0. Goscinski. Mol. Phys. 26, 1013 (1973). [4] J . Simons and W. D. Smith, J . Chem. Phys. 58,4899 (1973). [5] 0. W. Day and D. W. Smith, Int. J . Quant. Chem. s8, 51 I (1974). [6] V. H. Smith and Y. Ohrn, Proc. Density Matrix Conf., Kingston, 1974. [7] G. Hunter. Int. J. Quant. Chem. 9, 237 (1975). [8] G. Hunter, Int. J . Quant. Chem. S8. 413 (1974). [9 ] D. M. Bishop and G. Hunter, Non-Adiabatic Potentialsfor Hi,Mol . Phys. (in press).

[lo] A. W. Drake, Fundamentals of Applied Probability Theory (McGraw-Hill. New York. 1967) [ I I] P. 0. Lowdin. Phys. Rev. 97, 1474 (1955). [I21 E. H. S. Burhop. in Quantum Theor,v, D. R. Bates, Ed. (Academic Press, New York, 1961). [I31 C. L. Pekeris. Phys. Rev. 112, 1649 (1958): 115, 1216 (1959); 126, 143. 1470 (1962). [I41 0. Goscinski and P. Lindner, J . Math. Phys. 11, 1313 (1970).

Received March 4, 1975. Revised March 21. 1975.