dynamic aspects of quantum chemistry

ELSEVIER

THEO CHEM

Journal of Molecular Structure (Theochem) 330 (1995) I-15

Dynamic aspects of quantum chemistry

A. Macias Institute de Estructura de la Materiu, C.S.I.C.. and Departmento de Quimica C-IX, Universidad Autdnoma de Madrid,

28049 Madrid, Spain

Received 29 October 1993; accepted 15 November 1993

Abstract

Some contributions of quantum chemistry to the field of atomic dynamics are reviewed and it is argued that, in return, the area of application of quantum chemistry has become considerably extended and has given rise to insight into processes closely related to bond formation.

1. Introduction

Theoretical treatment of atomic collision processes yields results of similar accuracy to those obtained in experiments; for low collision energies these treatments provide a stringent test for some methods, concepts and approximations of quantum chemistry because they use the results of molecular-type calculations such as energy correlation diagrams and dynamical couplings, so that the precision of the calculated cross-sections will finally depend on the quality of the wavefunctions. From those contributions quantum che- mists have gained physical insight into the workings of electronic and nuclear interactions for states that are spectroscopically inaccessible, has enabled us to visualize such processes as hybridization taking place as a function of time, and its area of application has become extended to new problems.

As early as 1932 it was suggested [l] that a tran- sient quasimolecule is formed in slow atomic collisions, and measurements of differential and total cross-sections in slow atomic collisions were

explained by the one-electron MO model [2-41 ~- the MO promotion model, so-called because the principal quantum number of some MOs increases from the separate to the united atoms limits. How- ever, standard quantum chemistry textbooks [5] do not discuss collision theory (although they often include other time-dependent processes such as interaction of radiation and matter); only bound systems are studied (except for simple model problems) and the Born-Oppenheimer approximation is presented in a few sentences stating that the electrons (in a molecule) can be considered as moving in the field of fixed nuclei. This kind of presentation may leave the impression that the usual methods and techniques of quantum chemistry cannot be applied, or extended, to the treatment of problems involving electronic and vibrational continua, and that a different approach is needed for cases where the nuclei do not move very slowly. In particular, ionization, transitions (except those caused by interaction with radiation). dissociation and molecule formation may appear beyond the reach of quantum chemical techniques. Furthermore, it is clear that

0166-1280/95,/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0166-1280(94)03813-Z

the interaction between two neutral or ionized atoms provides the simplest example of a chemical reaction, and has the advantage that the processes involved can be studied and interpreted with the help of almost exact calculations. When the relative velocity of the two subsystems is high, their interaction is much smaller than the total energy of the system, and it can be treated as a perturbation; but for lower velocities the same forces (ionic, exchange, induction and dispersion) which are responsible for the formation of chemical bonds play an important role in the dynamics of the system.

The idea that a slow collision between two atoms can be interpreted in terms of the properties of the quasimolecule they form resulted in the so-called molecular model, which was originally based on the adiabatic theorem [6], because the system was assumed to evolve in terms of an adiabatically slow change on the nuclear positions (Born- Oppenheimer approximation [7]). Such a theorem can only be applied to the study of elastic (and resonant) processes. Extension to the case of inelastic and electron exchange collisions was achieved by including non-adiabatic coupling terms between different electronic states. The resulting close- coupling formalism, derived by Mott [S] and Born and Huang [9] for the treatment of vibrational problems, has the advantage that in the limit of a complete set of functions an exact result is obtained; therefore the approximation can, in principle, be improved as much as desired.

Since the effectiveness of dynamical couplings increases with the nuclear velocity, a large number of adiabatic states must be included in the molecular expansion, even for moderately high velocities. For that case, and in order to minimize the number of states required in the molecular expansion, it may be convenient to define a new set of functions, which are linear combinations of the adiabatic ones. In addition to simplifying calculations, the new (diabatic) states are expected to provide a simpler interpretation of collision processes.

States representing the electronic continuum have to be included in the molecular expansion to represent the final states at (moderately) high collision energies when ionization becomes

competitive, or even dominates, excitation and charge exchange processes. Also, in some systems the initial (stable) state may give rise to a resonant (autoionizing) one as the internuclear distance decreases, as in collisions of multicharged ions with helium, where the electronic energy of the quasimolecule lies above one or several ionization limits. Then, standard quantum-chemical techniques can be extended or modified to calculate the energies and couplings [lo].

2. Separation of electronic and nuclear motion

After separation of the total centre of mass, the Schradinger equation describing the collision can be written as:

HQJr, R) = &QJrl R) (1)

where r and R represent the sets of electronic and nuclear coordinates, respectively. H is the sum of the nuclear kinetic energy operator and the electronic Hamiltonian, H,, (defined for clamped nuclei). In the molecular approach, first one sol\,es:

HelQ,(rr 4 = &(R)Q,(r, R) (2)

for fixed R, and the solutions of Eq. (lj can be formally expanded in terms of the complete set of electronic wavefunctions

*,(r, R) = C xcti(R)4j(r: RI i

and substituted in Eq. (l), yielding a system of coupled differential equations for the nuclea I- functions:

where p is the reduced nuclear mass. Neglecting all matrix elements (called non-adiabatic couplings) on the right-hand side of Eq. (4) yields the single term Born-Oppenheimer approximation, m hich is

A. Macias/J. Mol. Struct. (Theochem) 330 jI99.5) l-15 3

adequate to describe molecular states when the energy curves are well separated, because contributions from the non-adiabatic couplings are then negligible. This is usually the case for the electronic ground state near the equilibrium geometry.

Even when a single Born-Oppenheimer wavefunction gives a good description of the system, the image provided by that wavefunction is, in part, contradictory: the electrons follow the motion of the nuclei, but this motion cannot be described by an electronic wavefunction that depends only parametrically on the nuclear coordinates. One can visualize a static electron cloud embedding the nuclear framework, but it is diffi- cult to see how it can be done as the nuclei vibrate, rotate or collide [l I]. The lack of correlation between electronic and nuclear motion, which can be fatal in collisional calculation, also causes difficulties in spectroscopy (e.g. the IR paradox [ 12,131).

Far from the equilibrium geometry, and for electronically excited states, there can be regions of configuration space where the energy surfaces approach closely. In that case, the molecular wavefunction must be expanded over the strongly inter- acting states [9].

3. Low- and high-velocity domains

The relative nuclear velocity in a collision A + B may be compared in units of the orbital velocity of the (active) electrons which undergo excitation or are transferred from one atom to the other. The basic unit, u. = 2.1877x lo6 m s-l, corresponds to the electron in the ground state of the hydrogen atom. For a monoelectronic ion of nuclear charge 2. the velocity of the electron in the ground state is 2~. (For more details on this section, see e.g. Ref. 14.) According to the nuclear velocity one can roughly classify collision as follows.

(1) Very low-energy collisions (~1 d 0.021~~. or impact E < 0.01 keV a.m.C’). These are the conditions for processes of interest in astrophysics, and for laboratory experiments with merged beams. The electron cloud adapts itself quasiadiabatically to the nuclear positions, and the electronic structure of the molecule can be represented by, at most, a few stationary states. Transitions between these

states occur via dynamic couplings at regions loca- lized near pseudocrossings of the energy curves.

(2) Low-energy collisions (0.02~~ d I: < 0.5~, and 0.01 < E < 6 keV a.m.u.-‘). This region corresponds to cross-beam experiments. Many more Born-Oppenheimer states than in the previous case are needed to describe the electronic structure of the quasimolecule. In addition to the couplings present in (1) there now appear (smaller but effective) ones due to delocalization and polarization effects. For this range of velocities, and for higher ones, the nuclear momentum is high enough to allow definition of the nuclear trajectory. and the nuclear motion can be treated classically.

(3) Intermediate-energy collisions (0.05~0 < 7: :S 1.5vo, and 6 < E 6 50 keVa.m.u.-‘). This is the energy region of maximum charge exchange cross-sections for collisions between multicharged ions with hydrogen and helium, and therefore is of interest in the study of fusion plasmas. In addition to requiring many stationary states in the molecular expansion, correlation between electronic and nuclear motions cannot be neglected. For ‘12 > {to ionization becomes competitive with charge exchange processes; then Born-Oppenheimer wavefunctions representing the ionized system at-e needed to describe the electronic wavefunction. However, molecular interactions are still important and cannot be treated by perturbation methods.

(4) High-energy collisions (1: > 1.5~ or E > 50 keV a.m.u.-‘). The atoms maintain their indici- duality, and perturbation theory is adequate to describe the collisional process.

4. Semiclassical approach

A detailed presentation can be found in most books on atomic collisions (see e.g. Refs 15 and 16). The following is one of the possible ways [ 171 of introducing the semiclassical approximation. Unless the relative velocity is very small (7) < 1) the linear momentum of the atoms is very large, due to the magnitude of the nuclear masses, and the solutions of Eq. (1) will present strong oscillations. For the relatively large velocities corresponding to low- and intermediate-energy collisions, part of these oscillations can be extracted by defining a

4 A. MaciasjJ. Mol. Struct. (Theochew] 330 11995) I-15

vector k:

i; = & k2 = 2,uE

and developing the solutions of Eq. (1) in a power series of p-l:

XLJ = [exp(ik.R)] [ $,‘$(‘) +.

P . . I

Substituting in Eq. (1) and setting the coefficient of each power of p equal to 0, we obtain, to order 0:

ik . V$J = pH,& (5)

which can be given a simple interpretation by introducing cylindrical coordinates of the internuclear vector R = (2, b, CJ!J), where b is the impact parameter. For fixed b and 4, Z defines a linear nuclear trajectory. Defining a velocity v = kp-’ , and a “time” t = Z/v, Eq. (5) can be re-written as:

which is the eikonal or impact parameter equation [18]. As neither b nor 4 appear in this equation, it has to be solved for each nuclear trajectory R = b + YZ.

In the molecular method, the solutions of Eq. (6) are expanded in a set of molecular solutions. There ensues the following system of coupled differential equations:

.daj(t) 27 = Cai(t) .,

J

which is generally solved by numerical integration. In these equations there appear the molecular energies, as functions of R, which form the energy correlation diagram, the electrostatic couplings due to H,,, and radial (d/dR) and angular (L,) dynamic couplings.

When the difference Ej - {jr is large, the phase in Eq. (7) oscillates quickly and cancels the couplings. In the perturbed stationary states (PSS) method the only channels included in the expansion are those

for which the energy difference with the entrance channel does not cancel the corresponding (non- negligible) coupling matrix elements. As the nuclear velocity appears in Eq. (7) multiplying the dynamical couplings, the higher that velocity, the more states have to be included in the molecular expansion. Of course, when the molecular expansion is carried out in terms of Born-Oppenheimer exact solutions of Eq. (2), the couplings due to H,, vanish.

To discuss the boundary conditions to be imposed on the solution of Eq. (6), consider the simplest case: a diatomic one-electron system A-B, and assume that at t + -cc the electron is attached to nucleus A. The initial exact boundary condition is [11,19]:

Q;(rr t) I+Tooexp[-ipv.r - (p2/2)u2t]

x exp(-iEft# @a)

where p = MB/(MA + Ma), pv is the relative electron velocity with respect to the origin of coordinates (in this case the nuclear centre of mass), exp(-ipv . r) is a phase describing the motion of the electron relative to the nuclear centre of mass, p2/vt is the energy corresponding to that motion, and Elf’ is the initial electronic energy. After a non- ionizing collision, at t + cx), the electron is either on nucleus A for elastic or excitation collisions, or on nucleus B in the case of charge exchange processes. The corresponding possible final states are:

\kj(r, t, tzmexp[-ipv.r - (p2/2)~‘t]

x exp(-iEFt)c$ WI

for elastic and inelastic processes, and

Qj(r, t) ,zmexp[iqv. r - (q2/2)u2t] exp(-iE,BI)#

@cl

for charge exchange, where q = 1 -p. The i 4-j transition amplitude for each trajectory is calculated as:

tij = hl(Ql@j)

and the total cross-section is

Clij = 2-ir

A. MaciasiJ. Mol. Struct. (Theochem) 330 (1995) I-15 5

In the PSS method, Eqs. (Sa-8c) are substituted

by

*, (r: 4 - exp(-iEFt)4f r--DC: PaI

Qj(r: t) N exp(-i$t)$P f-IX Pb)

Q, (r, t) tyX exp(-iE:t)@ PC)

For the initial channel, excitation and change- transfer processes, respectively. An undesired consequence is that there can appear non-physical effects such as constant (non-zero) dynamic couplings when R --f 00. In addition, the calculated cross-sections can depend on the origin of electronic coordinates chosen to calculate the couplings. One solution to both difficulties, within the Born Oppenheimer scheme, is to modify the nuclear (or electronic) wavefunction by introducing translation factors.

5. Molecular calculations

The molecular data required in Eq. (7) cannot, in

general, be obtained using computer programs based on methods designed to describe molecules in their ground state at near-equilibrium conligura- tions; the methods need to be extended to describe excited or very excited (even autoionizing) states over the whole range of internuclear distances. Selection of the channels to be included in the molecular expansion requires a detailed study of the molecular energies and corresponding dynamic couplings as functions of the internuclear distance R, because transitions between two channels are the more likely to occur the closer their energies and the larger the coupling matrix element.

Large radial couplings appear between two states whose energies avoid crossing as functions of the internuclear distance; in the neighbourhood of a narrow pseudocrossing the shape of the molecular wavefunctions changes abruptly, and the adiabatic states are said to interchange some character in that region. Character here means any property which may differentiate the adiabatic wavefunctions outside the avoided crossing region, and does not correspond to any of the operators that commute with the Hamiltonian,

30 0 10 20 30 40

Z (target)

Fig. 1. Experimental cross-sections for excitation to the Kr34+ (1~2~) state in the reaction Kr 34+ (1 s’) +X (X = C, Si, Ar, Cu and Zr).

6 A. Macias/J. Mol. Struct. (Theochem] 330 11995) l-15

(a) Kr Ar

-1.201 I I I I I I I I 1

I 10 20 30 j 40 50 60 170 80 90

; R + 2 (a.u.)

(b) KrAr

0.160 -

0.140 R 0.120

0.100

2 z 0.080 a -H \ ^1 .~--

R + Z (a.u.1

Fig. 2. (a) Scaled energies and (b) rotational couplings for the states of (KPA~)~~+ correlated to 1CKr34+ (1s’) + .k and 1rIKr341 (1~2~‘). (c) The transition probability between the two states. (-) OEDM; (-+) RHF.

because states with different eigenvalues for any of these operators will be separated when solving the secular equation. In this sense, covalent and ionic states have different character (while having the same symmetry) which can be ascertained by exam- ination of the corresponding wavefunctions. Radial couplings may also be due to polarization effects, as in the case when the separated atom (SA) limit of two states corresponds to an ion and two

atomic states which differ only in the 1 quantum number of one occupied orbital; as R decreases, the Coulomb field of the ion causes those orbitals to form Stark hybrids, which are radially coupled. Delocalization of the electronic cloud centred on one nucleus as it takes an (exponentially) increas- ing component on the other nucleus as R decreases, also results in radial couplings.

In other words, any topological variation in the

A. Macias:‘J. Mol. Struct. (Theochem) 330 (1995j l-15

0 2 4 6 8 10 12 14 16

b l Z (a.u.)

Fig. 2 (Contd).

electronic cloud results in a dynamic coupling. A very similar reasoning to that used in the discussion of bond formation permits not only interpretation (and prediction) of the shape of correlation diagrams, but also serves to analyse the quality of the description provided by the basis set employed.

An important application of ideas from quantum chemistry comes in the study of inner-shell vacancy production (called “inner-shell chemistry”) where a simple independent-particle model is adequate to describe the quasimolecule. because the structure of inner shells is dominated by the nuclear Coulomb attraction [2-4,20,21]. The com- bination of MOs plus the independent-particle model may suggest to a quantum chemist the SCF approach. However, in practice one finds that the use of simplified SCF MOs presents serious problems: for inner-shell transitions, there are crossings of the HOMO and LUMO energies which cause dis- continuities in the MO correlation diagram [22], problems which do not appear in the general Hartree-Fock (GHF) method [23]. For outer-shell processes construction of excited-state wavefunctions from virtual orbitals is not straightforward, and in the case of neutral or negatively charged systems the energy of the resulting state may be basis-set dependent [24]. In the case of very highly charged ions, and for the same reason as in the case of inner-shell transitions, SCF calculations, even at

the RHF level, can be very useful. In collisions of Krs4- (1s’) with different solid and gas targets, the excitation cross-section to the Kr34+ (1 s2p) state calculated from the Lymann (I: line intensity increases with the nuclear charge (Zr) of the target but levels off at high Zr [25] (Fig. 1). To ascertain the influence of screening by target electrons on this saturation effect, the transition probability between those two states was calculated [26] using the independent particle model, in two sets of calculations; the energy differences and rotational couplings between those states are almost identical when calculated from the exact orbitals for a one electron diatomic system orbitals (OEDM) [27] (no screening) and from the RHF one (which includes screening), and so are, of course, the transition probabilities and the cross-sections (see Fig. 2 for an example). This simple test was sufficient to rule out screening as the main cause of the observed saturation effect.

In general. the success of the independent- particle picture is unrelated to the SCF method and it can be traced to the fact that most quasimol- ecular states can be approximately built from MOs that describe the motion of each electron in the presence of screened nuclear charges [28].

Returning now to the many-electron problem. calculations should be carried out at the CI level, although “chemical” accuracy (errors less than 10 3 a.u. or 1 kcal mall’ in the energies) is generally

8 A. Macias:J. Mol. Struct. (Theochem) 330 (199.5) l-l.5

not required. Approximations on energies and couplings are not independent, and once a set of molecular wavefunctions has been constructed, dynamic couplings should be evaluated from these wavefunctions as exactly as possible [17].

Correlation energy, and its variation as a function of nuclear geometry, is important in the study of molecule formation; for the ground state of H2 it varies from zero in the SA limit to 24% of the binding energy at the equilibrium distance [29]. In the case of atomic collisions, even small changes in the amount of correlation introduced can be very important; for example, in the neutralization of H+ by H- one possible exit channel is H(ls) + H(n = 3). At large internuclear distances, the energy of the covalent exit channel varies little with R, while that of the ionic entrance channel varies inversely with R. The energy lowering achieved by introducing angular correlation (xx’ + .v,v’ + zz’ configurations) in the basis set is negligible for the covalent channel and about -0.01 a.u. for the ionic one. This results in changing the crossing region between these states from about 28.6 (no angular correlation) to about 36.5a.u. (including angular correlation) [30]; as the corresponding radial coupling decreases exponentially with R, it is negligible in the second case. For the nuclear velocities considered, the dynamic situation corresponding to both cases is quite different and the calculations without correlation predicted transitions between the ionic and the covalent channel at this avoided crossing, which is not the case.

Regarding the calculation of dynamic couplings, in the case of rotational couplings it involves evaluation of matrix elements of Ly between the states included in the molecular expansion, and presents no particular difficulties because the monoelectronic Ly operator acting on one spheri- cal harmonic yields another, and the coupling matrix element between two orbitals is given by an overlap integral. However, calculation of radial couplings between approximate (e.g. variational) SCF or wavefunctions means evaluating expressions of the form:

where r represents the set of electronic coordinates

defined with respect to a common origin. For CI expansions

Qi(y, 4 = c cin(RMn(r, R) (10) n

the expression for the radial couplings is:

where c, represents the ith column vector of the coefficient matrix in Eq. (1 l), and S is the overlap matrix; the (n, m) element of the matrix B is defined

:I I> da ; dm

and involves evaluation of standard overlap integrals [31]. The difficulty appears in the differentiation of the coefficients of Eq. (lo), especially when they vary rapidly as functions of R, and several methods [32] have been proposed for this purpose.

A finite-difference method requires the CI expansion coefficients at closely spaced internuclear distances. While it is simple to program, the com- putational effort involved is considerable. and the appropriate value for the step size may depend on the particular states involved [33]. The Cartesian coordinate Hellman-Feynman expression [34] is very sensitive to the quality of the wavefunctions near the nuclei; and the accuracy of the Hellman Feynman expression in elliptical coordinates [28], at large internuclear distances, is related to that of the molecular wavefunctions in the SA limit.

In the analytical method, based in the formalism developed by Bratos [35] to calculate harmonic force constants, the first term in Eq. (1 1) can be written [36] as

and involves differentiation with respect to R of all expressions corresponding to the molecular integrals, which can be carried out in analytical form when the configurations are expressed in terms of Gaussian type orbitals (GTOs), or exactly calculated using numerical techniques (OEDMs). The equations for SCF wavefunctions, and for expressions of second derivatives, can be found in

A. Mucias:J. Mol. Struct. (Theo&em) 330 (1995) l-15 9

Refs. 17 and 36. Radial couplings can be used either in the solution of Eq. (7) or to define a new set of states, along the lines presented in the next section.

6. Diabatic states

Electron density maps for the ground state of the Cl + Na system, calculated from Born- Oppenheimer wavefunctions at different values of the internuclear distance R, show the system to be made up of two neutral (almost unpolarized) atoms down to R z 17 a.u., and then it suddenly becomes Cll + Na+. The probability of finding the 3s electron of sodium at such a distance of the nucleus. and the overlap between chlorine and sodium orbitals, are negligible, so the mechanism for electron transfer is not intuitively obvious. The explanation is simple when one looks in detail at the correlation energy diagram, which presents an avoided crossing (around R, = 17 a.u.) between the covalent and ionic states correlated to the neutral atoms and the ions, respectively. The analysis of the Born-Oppenheimer solutions indicates that the ground state of Cl + Na has covalent character for distances greater than R,: in a very small region about that R value it takes more and more ionic character until there is a complete exchange of character between the two (adiabatic) states. As the wavefunctions representing the ionic and covalent states do not change abruptly with R. their energies cross and the radial coupling between them is negligible. Outside the crossing region both sets of states are practically identical; for very low nuclear velocities the lowest energy BornOppenheimer state describes the Cl + Na system, while at high nuclear velocities the covalent state provides a better description.

Different names, all of them including the word “diabatic” have been proposed in the literature for molecular states which have a smoother depen- dence on R than the adiabatic ones, and which are usually expressed as linear combinations of the. latter, so that the accuracy achieved in the quantum-chemical calculations will be passed onto the new functions. Clearly, the diabatic states do not diagonalize H,r and they will present

electrostatic couplings (which are easier to calculate than radial ones).

The absence of radial couplings would simplify solving Eq. (7) especially in the case of sharp avoided crossings. when the radial couplings present needle-like peaks. Furthermore, diabatic states are very convenient to discuss processes at nuclear velocities such that the probability of traversing an avoided crossing is very close to unity. and the character of the electronic state is maintained during the collision, instead of changing in the way described by the adiabatic wavefunctions. Then, the system is said to behave diabatically [4], i.e. it evolves following a diabatic energy correlation diagram.

In the first general definition of diabatic states. Smith [37] proposed a linear transformation of the adiabatic basis {&}:

with the condition that

which leads to a set of differential equations for the elements of the matrix U.

to be solved with an initial condition such as U;,(m) = h;,, so that both sets of functions are identical far from the crossing region.

This definition of diabatic states requires calculation of the radial couplings, but there is a more serious problem: it can only be exactly imple- mented for diatomic systems [38], and even then it is not free from difficulties [39]. Approximate ways of constructing diabatic states, without cal- culating radial couplings, are based on the fact that smooth changes in the wavefunctions with the internuclear distance result in small radial couplings. “Intrinsic” diabatization methods are based on the evaluation of expectation values and off- diagonal elements, in the avoided crossing region, of molecular properties, to construct the adiabatic- to-diabatic transformation matrix [40].

10 A. Macius:.J. Mol. Struc.t. (Theochem) 330 (1995) I-15

“Extrinsic” diabatization techniques assume that diabatic states are known in an intuitive manner, and attempt to invest then with the accuracy of adiabatic wavefunctions. The fact that adiabatic states have. outside the crossing region. different character, can be used to construct approximate diabatic states. This character may be quasi-g or quasi-u symmetry [41]. ionic and covalent structures [42], or configuration structure with approximate quantum numbers [43]. In practice, a constraint may be introduced in the linear variational treatment of H,, that causes the adiabatic wavefunctions to have a particular character; this is equivalent to block-diagonalization [ 17j of the Hamiltonian matrix, and solving the secular equation for each of the diagonal blocks sepa- rately. The “maximum overlap” method [44] starts from a reference basis of approximate diabatic states, and transforms the adiabatic functions so that the new ones resemble as much as possible

those in the reference set. For a recent review on diabatization, see Ref. 45.

Note that states represented by wavefunctions with different characteristics may not be diabatic; for instance, CI solutions obtained from two sets of configurations, corresponding to ionic and covalent valence bond structures, may present non-negligible radial couplings. For H + H- it was found that the overlap between ionic and covalent states, obtained by the block-diagonalization scheme, is large and varies rapidly, outside the adiabatic energies crossing region [30] (Fig. 3); this means that both radial couplings between these states cannot be simultaneously zero because:

0.6

R (a.u.)

Fig. 3. Radial couplings between the ‘C, (ionic) H- (Is’) -H- and (covalent) H(ls) - H(3d) (a), H(ls) - H(3s) (b). and Hjls) - H(3p) (c) channels. They are diabatic in the region where the energies of the adiabatic states avoid crossing (R = 36 a.~.), but not at smaller

‘4. Macias:J. Mol. Struct. (Theochem) 330 (1995) l-15 11

In addition to couplings due to (total) character exchange, others due to polarization or delocalization may be effective in causing appreciable transitions. For total character exchange, the correspondence between adiabatic (qA) and diabatic (qD) states is: !I$ = S’F, qFD= @ on one side of the crossing region and 9i = +@, !I!f = r@ on the other; for polarization and delocalization the mixing is of the form qD = 2-‘12 (e;’ & @) and imposing the initial condition O;,(m) = ~ij results in diabatic states which are different from the adiabatic ones outside the crossing region. and present strong

electrostatic couplings. One can conclude that, while the term “adia-

batic” is well defined in the sense that sufficiently low nuclear velocities, such a state describes the evolution of the system, “diabatic” is not the high velocity equivalent because in many cases such a state cannot be defined. The lack of a mathematical definition. however, does not prevent “approximate” diabatic states from having a physical sense, or from being very useful in the treatment of some collisions involving more than one active electron.

and some discretization method; moreover, the calculation of dynamic couplings will be very laborious. In many cases this will not be necessary, as the variational treatment used to solve Eq. (2) for the bound wavefunctions will also yield discretized approximations to the continuum ones. These discretized wavefunctions are L,’ integrable functions and they represent, within the region of configuration space spanned by the basis set used [46], and up to a normalization factor, the exact non-L2-integrable exact continuum solutions. A practical solution [47] to the calculation of the normalization factor is easily imple- mented when the atomic orbitals used in the discretization procedure are chosen according to some simple rules. With the basis sets normally used for bound state calculations, the continuum solution corresponding to a given energy value [48] can be reproduced in a domain of molecular dimensions. This technique has been applied suc- cessfully to the calculation of resonance positions and lifetimes in atoms and molecules [49]. and to model potentials of the type involved in pre- dissociation mechanisms [50].

6. Ionization

Continuum states have to be included as exit channels in the molecular expansion when the nuclear velocity is of the order of that of one electron in the system. which may be detached in the collision

The use of very large basis sets to represent the ionization continuum and the infinite series of Rydberg states converging to it is very cumber- some, but it is not necessary if one is interested only in the calculation of total cross-sections, including that corresponding to ionization.

A+B-+A+B++e-

with or without simultaneous excitation or de- excitation of the atoms. A case where ionization cross-sections can be important even at low nuclear velocities is that of Penning ionization

There are two mechanisms which couple discrete states to the continuum. The resonant one is that of a state which becomes autoionizing (lies in an electronic continuum). In the Feshbach theory [51], the projection operator Q defines the Hamil- tonian QHQ [52] with eigenfunctions a’, whose eigenvalues ER are interpreted as the energy positions of the resonant states, and the continuum states satisfy the equation

A*+B+A+B++e (PHP- EC)$ = 0

where P = 1 - Q. when the excitation energy of A* is greater than the ionization potential of B. which means that the initial state also lies in the continuum of A + B+. Including the continuum wavefunctions in the molecular expansion means having to deal with an infinite set of integro-differential equations which must be approximately solved by truncation

Accurate values of resonance parameters can be read directly from stabilization graphs [53], in the kind of calculation that can be carried out on the “back of an envelope”. The stabilization method can be applied to cases where the projection operators cannot be defined a priori as in the case

12 A. Ma&s/J. Mol. Struct (Theochem) 330 (199jj 1-15

of shape resonances [54]. Another approach involves the use of Hamiltonians containing a pseudopotential; this method can be placed on the same footing as the Feshbach theory when a generalized Phillips-Kleiman formalism [55] is used.

The non-resonant ionization mechanism appears at high nuclear velocities, when a discrete state is closely coupled to the continuum. In the framework of the impact parameter formalism, the interaction between the states included in the close coupling expansion and those left out of it may be accounted for through the use of Feshbach- type optical potentials [56]. Alternatively, a selected set of continuum states, such that the transition probability from the discrete to these states will be high, is included in the molecular expansion. Ioniza- tion channels have been represented by packets states constructed from exact continuum ones [57]. They can also be introduced as discretized states, called probability absorbers [58], by defining a pro- jector operator

p = c IQ!4 (dkl k

onto the manifold of discrete states {dn}, and another operator Q = 1 - P onto its complement. Then, to any given function ~$k in the P subspace, an absorber state belonging to Q

can be ascribed. Addition of these states to the close coupling basis may be considered as a “best augmentation” procedure to select from the Q subspace those (non-adiabatic) functions that are more closely coupled to those in P. A limitation of this technique is that information is lost on the particular continuum states involved, because absorbers are superpositions of discrete and continuum v, ave- functions, and this prevents calculation of ionization cross-sections as functions of the energy of the detached electron. However, the total ionization cross-section can be obtained without much com- putational effort.

Unless ionization is accounted for, cross-sections for other processes, at intermediate and high nuclear velocities, will be grossly overestim;lted,

I I I111111 I

10 20 50 100 200

Proton energy (kev)

Fig. 4. Cross-section for the reaction He-( 1 s) + H+ + He2+ + H( 1s). (. .) Basis set of 10 discrete HeH2+ states [58] (see text 1. (-) Basis set augmented with four absorber states [58]. (---) Ref. 59; (0) Ref. 60; (0) Ref. 61; (x) Ref. 62: (V) Ref. 63; (A, V) Ref. 64.

A. Macias/J. Mol. Strut. (Theochemi 330 11995) I-15 1 -3

or even be totally wrong. As an example, Fig. 4 presents plots of the total cross-sections calculated [58] for the charge exchange reaction He-(1s) + H+ -+ He2- + H( 1s) using the basis set of (discrete) HeHI+ MOs {Isa, ~s(T, 2~0, 2p7r, 3pa, 3da: 3d7r, 4fa, 5gu, 4f7r) and with the same basis set augmented with the set of absorber states { Isail), 2p7r(‘), 4f7r’11, 5g0(‘)}. Clearly, inclusion of these states in the close coupling expansion considerably extends the validity domain of the molecular method. The use of absorber states has permitted an explicit study of the mechanisms of non-resonant transitions to the continuum. Direct transitions from a bound molecular state to the continuum are caused by radial couplings between the states involved. due to the strong variation of the discrete wavefunction in the small R region, where it passes from being two-centred to being one-centred and back, as the nuclei approach and then separate. At high velocities, the electron cloud cannot contract and expand quickly enough, and non-adiabatic couplings cause promotion of the electron to continuum states. Indirect transitions at large R values are due to radial couplings between a bound molecular state and a higher excited one, until the ionization continuum is reached. Physically, this mechanism represents the process by which part of the electron cloud is left in the region between the nuclei as they separate, and it resembles Wanier two-electron threshold ionization [65]. It can be visualized as the electron occupying higher and higher Rydberg orbitals as the nuclei separate. The two processes are not independent. because each of the states involved in the indirect process is also directly coupled to continuum states.

7. Electron translation factors and the electronic continuum

The continuum spectrum plays an important role in the theoretical treatment of atomic collisions, even at relative nuclear velocities for which ionization is not likely to occur. This is due to the fact that the molecular expansion (Eq. (3)) cannot, in general, fulfil the boundary conditions (Eq. (8)) except in the limit of a complete basis set. The reason is that the exact solutions for Eq. (6)

when R + 30, given by Eqs. (8a-8c), include a term of the type exp(-iv -r), corresponding to a plane wave and describing the motion of the electron as it follows the nuclei. This type of term is not present in the basis sets normally used in quantum- chemical calculations, and in order to reproduce a plane wave using such orbitals an infinite expansion over the discrete and the continuum of the Hamiltonian is required:

q$ exp( -iv * r) = C c,b,A

where the sum extends over the discrete (including Rydberg) states and the integral over the continuum ones, and

c, = (&+exp(-iv.r)/f&)

are the complex coefficients obtained in solving the system of equations Eq. (7).

The standard molecular model substitutes the boundary conditions (Eq. (8)) by the approximate ones given by Eq. (9) which are satisfied by expansion (3) but then we are dealing with essentially incorrect boundary conditions which do not satisfy the Schrodinger eqn. (Eq. (6)) in the limit t + &x. This approximation has two important consequences: first, a truncated expansion is not an eigensolution of the coupling operator of Eq. (7). that is:

and in Eq. (7) there may appear non-zero radial couplings between the wavefunctions representing the different channels as R + 00, and rotational couplings that decrease like R-’ instead of Rp2. The second consequence is due to the fact that the boundary conditions (Eq. (9)) do not contain information on the origin of electronic coordinates; then, the reference to the nuclear centre of mass can be coherently removed from Eq. (6), and an origin of electronic coordinates be freely chosen. Then the dynamic couplings in Eq. (7) may be origin dependent, and so will be the calculated cross-sections. which is awkward to say the least.

From the presentation of the problem. there are two obvious ways out of these difficulties: one is to use an infinite expansion, including the continuum

14 A. Ma&as/J. Mol. Struci. (Theocheml 330 1199.5) I-15

(not a very appealing solution); and the other is to define new basis sets made up of products of (ordinary) atomic orbitals and some type of phase linking the electronic and nuclear motions, that is to say: define a new basis set which has the characteristics of the exact solutions. Introduction of these phase linking terms, called “electron translation factors” provides a solution consistent with all the ideas and images based on the Born-Oppen- heimer approximation; for instance all information on molecular structure will still be contained in the electronic wavefunction, because the introduction of a multiplicative phase leaves invariant all topological constructions [66].

A practical advantage of this method is that all the integrals needed to study a dynamic process can be calculated without major difficulties in the new basis with translation factors [67]. Details about the different translation factors proposed in the literature are beyond the scope of this paper, and the particular choice, from the practical point of view, is on the same level as the choice of basis sets, because it will mainly depend on the computer programs available.

As boundary conditions are trivially imposed for stable molecules, and a single-term Born-Oppen- heimer solution provides, in general, a good description of the system, the relation between phase linking terms and the so called IR paradox is not obvious [68]. Walnut and Nafie [13] showed that correlation between electron and nuclear positions is reasonably well described by single-term Born-Oppenheimer wavefunctions, and the elec- tric dipole E. r matrix element (which depends on this correlation) is well approximated at that level. However, the vector potential A. p matrix element (which depends on the correlation between electronic and nuclear velocities missing in Born-Oppen- heimer solutions) is not. For a more detailed discussion see Ref. 14.

8. Concluding remarks

Quantum Chemistry appears often to the out- sider either as methodology and mathematics bear- ing little relation to physical reality, or as more or less involved computations in an applied version of

Inorganic or Organic Chemistry. This paper tries to show how ideas and methods belonging to the domain of Quantum Chemistry are very useful in the field of atomic dynamics, since the atomic collision can, to a large extent, be described in terms of the electronic structure of the (quasi) molecule. In this way, Quantum Chemistry provides not only numerical results but interpretation of the phenom- ena in terms of mechanisms. Whether this interpretation is to be preferred to the results obtained using statistical techniques is an open question. From treating dynamical processes Quantum Chemistry benefits by being able to visualize the forces that are responsible for the formation of chemical bonds in action, and to quantify them in the form of measurable quantities, such as transition probabilities.

Acknowledgements

1 want to thank Profs. L. F. Errea and A. Riera for many discussions and their critical reading of the manuscript. This work was partially supported by the DGICYT project No. PB 90-0213.

References

[I] W. Weizel and 0. Beeck, Z. Phys., 32 (1932) 1x6. [2] U. Fano and W. Lichten, Phys. Rev. Lett.. l-1 (1965)

627. [3] W. Lichten, Phys. Rev., 131 (1963) 229; 139 (1965) A27:

164 (1967) 131. [4] M. Barat and W. Lichten. Phys. Rev. A, 6 (1972) 211. [5] See, e.g. A. Szabo and N.S. Ostlund. Modern Quantum

Chemistry, MacMillan, New York, 1982. [6] A. Messiah. Quantum Mechanics, North-Holland.

Amsterdam. 1958. [7] M. Born and R. Oppenheimer. Ann. Phys., X4 (1927) 457. [8] N.F. Mott, Proc. Camb. Phil. Sot., 27 (1933) 553. [9] M. Born and K. Huang, Dynamical Theory ol‘ Cristal

Lattices, Oxford University Press, London, 1954. [lo] A. Macias, R. Mendizabal, F. Pelayo, A. Riera and M.

Yaiiez, Phys. Rev. A, 33 (1986) 242. [ll] A. Riera, J. Mol. Struct. (Theochem), 259 (1992) X3. [12] N.V. Cohan and H.F. Hameka, J. Chem. Phys.. 45

(1966) 4392. [13] T.H. Walnut and L.A. Nafie. J. Chem. Phys.. 07 (1977)

1491. [14] A. Riera, J. Mol. Struct. (Theochem), 169 (19X8) 79.

A. Mucius:J. Mol. Struct. (Throchemj 330 (1995) l-15 15

[15] C.J. Joachain, Quantum Collision Theory, North-Hol- [41] A. Macias, A. Riera and M. Yaiiez, Phys. Rev. A, 23 land, Amsterdam, 1975. (1981) 2941.

[16] B.H. Bransden and M.R.C. McDowell. Charge Exchange and the Theory of Ion-Atom Collisions, Oxford University Press, Oxford, 1992.

[ 171 A. Macias and A. Riera, Phys. Rep., 90 (1982) 289. See also A. Riera, NATO ASI Ser. B: Phys., 299 (1992) 331.

[42] F. Borondo. A. Macias and A. Riera, J. Chem. Phys., 74 (1981) 6126.

[18] T. Adachi and T. Kotani, Prog. Theor. Phys., Suppl. Extra Number (1965) 316; 39 (1968) 430. 785.

[19] N.F. Mott and H.S.W. Massey, The Theory of Atomic Collisions, Oxford University Press, Oxford, 1965.

[20] R. Anholt, Rev. Mod. Phys.. 57 (1985) 995. [21] U. Wille and R. Hippler. Phys. Rep., 132 (1986) 129. [22] A. Macias and A. Riera, J. Phys. B, 9 (1976) 497. [23] A. Hammes-Schiffer and H.C. Andersen. J. Chem.

Phys., 99 (1993) 1901. [24] A. Macias and A. Riera, Gazz. Chim. Ital., 108 (1978)

329.

[43] V. Lopez, A. Macias, R.D. Piacentini. A. Riera and M. Yiiiez, J. Phys. B, 1 I (1978) 2889.

[44] M. Persico. in F. Rostas (ed.), Spectral Line Shapes, Vol. III, Walter de Gruyter, Berlin, 1985.

[45] R. Cimiraglia, NATO AS1 Ser. B: Phys., 299 (1992) 1 I. [46] D.F. Gallagher and L. Willets, Phys. Rev.. 169 (1968 I

139. For an elementary example illustrating the method see A. Rierd, Am. J. Phys., 59 (1991) 62.

[47] A. Macias and A. Riera, Phys. Lett., 103A (1984) 3774. [48] A. MaciasandA. Riera,Chem. Phys. Lett., 117(1985)42. [49] A. Macias, F. Martin, A. Riera and M. Yanez, Phys.

Rev. A, 36 (1987) 4187; 36 (1987) 4203. [50] SC. Tucker and D.G. Truhlar, J. Chem. Phys., 86

(1987) 6251.

[25] M. Chabot, P. Nicolai, K. Wohrer, A. Chetioui, J.P. Rozet. M.F. Politis, A. Touati, D. Vernhet, C. Stephan and R. Gayet, Nucl. Inst. Meth., 56157 (1991) I,

[26] M. Chabot, K. Wohrer, A. Chetioui. J.P. Rozet. A. Touati, D. Vernhet, M.F. Politis, C. Stephan. J.P. Grandin, A. Macias, F. Martin, A. Riera, J.L. Sanz and R. Gayet, J. Phys. B, 27 (1994) 111,

[27] A. Salin. Comp. Phys. Commun., 14 (1978) 121. [28] L.F. Errea, J. M. Gomez Llorente. L. Mendez and A.

Riera, J. Phys. B, 20 (1987) 6089. [29] W. Kolos and C.C.J. Roothaan, Rev. Mod. Phys., 32

(1967) 219.

[51] H. Feshbach, Ann. Phys., 19 (1962) 287. [52] T.F. O’Malley, Adv. Atom. Mol. Phys.. 7 (1971) 223; J.

Chem. Phys., 51 (1969) 322. [53] A. Macias and A. Riera, J. Chem. Phys., 96 (1992) 2877.

For a thorough study of the stabilization method see A. Riera. J. Phys. Sot., 97 (1993) 1558; A. Riera, J. Mol. Struct. (Theochem), 284 (1993) 175.

[54] M. Cartes, A. Ma&as, F. Martin and A. Riera, J. Phys. B, 25 (1992) 83.

[55] F. Martin, 0. MO, A. Riera and M. Yafiez. J. Chem. Phys.. 87 (1987) 6635.

[30] F. Borondo, A. Macias and A. Riera. Chem. Phys., 81 (1983) 303.

[56] H.J. Liidde, H. Ast and R.M. Dreizler, J. Phys. B, 21 (1988) 4131; Z. Phys., 15 (1989) 141.

[57] G. Bandarage and W.R. Thorson, Phys. Rev. A. 37 (1988) 692, 716.

[31] A. Riera and A. Salin, J. Chem. Phys., 62 (1975) 1195. [32] See e.g. I.L. Cooper, J. Phys. B, 24 (1991) 1517. [33] L. Mendez, I.L. Cooper, A.S. Dickinson, 0. MO and A.

Riera, J. Phys. B, 23 (1990) 2797. [34] R. Cimiraglia. M. Persico and J. Tom&i. Chem. Phys.,

53 (1980) 357. P Habitz and C. Votava, J. Chem. Phys., 72 (1980) 5532.

[58] L.F. Errea, L. Mendez and A. Riera, Phys. Rev. A, 4.3 (1991) 3578.

[59] W. Fritsch and CD. Lin, in J. Eichler. W. Fritsch. I.V. Hertel, N. Stolterfoht and U. Wille (eds.). XI1 ICPEAC, Gatlingburg, 1982, North Holland, Amster- dam, 1983. p. 502.

[60] T.G. Winter, G.J. Hatton and N.F. Lane, Phys. Rev. A. 22 (1980) 930.

SE. Butler, Phys. Rev. A. 23 (1981) 1. [35] S. Bratos, Coll. Int. Centre Natl. Rech. Sci. (Paris), 82

(1958) 287.

[61] M. Kimura and W.R. Thorson. Phys. Rev. A, 24 ( 198 1) 3019.

[36] A. Macias and A. Riera, J. Phys. B, IO (1977) 861: 11 (1978) 1077.

[62] T.G. Winter, Phys. Rev. A, 35 (1987) 3799. [63] J.F. Reading, A.L. Ford and R.L. Becker, J. Phys. B, 15

(1982) 625. [37] F.T. Smith, Phys. Rev.. 179 (1969) 111. [38] C.A. Mead and D.G. Truhlar, J. Chem. Phys., 77

(1982) 6090.

[64] B.H. Bransden. C.J. Noble and J. Chandler, J. Phys. B. 16 (1983) 4191.

[39] H. Gabriel and K. Taulbjerg. Phys. Rev. A, 10 (1974) 741. J.B. Delos and W.R. Thorson, J. Chem. Phys., 70 (1979) 1774.

[65] G. Bandarage and A. Parson. Phys. Rev. A, 41 (1990) 5878.

[66] L.F. Errea, L. Mtndez and A. Riera, J. Phys. B, I5 (1982) 101.

[40] A. Macias and A. Riera, J. Phys. B, 11 (1978) L849; Int. J. Quantum Chem., 17 (1980) 181. C. Petrongolo, Mol. Phys., 40 (1990) 825.

[67] R.F.W. Bader, Atoms in Molecules. A Quantum Theory. Clarendon Press, Oxford, 1990.

[68] A. Riera, J. Chem. Phys., 99 (1993) 2891.

dynamic aspects of quantum chemistry

Documents