2974
J. Phys. Chem. 1996, 100, 2974-2984
Spectral Theory of Physical and Chemical Binding†,‡ P. W. Langhoff Department of Chemistry, Indiana UniVersity, Bloomington, Indiana 47405-4001 ReceiVed: September 8, 1995; In Final Form: NoVember 30, 1995X
A spectral method which provides unified quantum mechanical descriptions of both physical and chemical binding phenomena is reported for constructing the adiabatic electronic potential energy surfaces of aggregates of atoms or other interacting fragments. The formal development, based on use of a direct product of complete sets of atomic spectral eigenstates and the pairwise-additive nature of the total Hamiltonian matrix in this basis, is seen to be exact when properly implemented and to provide a separation theorem for N-body interaction energies in terms of response matrices which can be calculated once and for all for atoms and other fragments of interest. Its perturbation theory expansion provides a generalization of familiar (Casimir-Polder) secondorder pairwise-additive and (Axilrod-Teller) third-order nonadditive interaction energies, expressions which are recovered explicitly in the long-range-dipole expansion limit. A program of ab initio computational implementation of the formal development is described on the basis of use of optimal (Stieltjes) representations of complete sets of discrete and continuum atomic spectral states, which provide corresponding finite-matrix representations of the Hamiltonian. The widely employed pairwise-additive approximation to nonbonded N-body interaction energies is obtained from these implementations in appropriate limits. Additionally, the development clarifies and extends rigorously diatomics-in-molecules approaches to potential-surface construction for bonding situations, includes the effects of state mixing and charge distortion missing from semiempirical and perturbation approximations commonly employed in theoretical studies of collision broadening and trappedradical spectroscopy, and encompasses and demonstrates equivalences among these apparently dissimilar approaches in appropriate limits. Large non-pairwise-additive contributions to the lowest-lying potential energy surfaces are found in illustrative studies of the structure and spectra of physically bound Na-ArN cryogenic clusters.
1. Introduction Adiabatic electronic (Born-Oppenheimer) potential energy surfaces provide a well-known basis for computational studies of chemical reaction rates and of the attributes of molecules, clusters, crystals, and other stable aggregates of atoms. Methods for obtaining such energy surfaces have been categorized by Professor Martin Karplus as “purely theoretical”, “semi-theoretical”, and “purely empirical” in nature, in accordance with the information employed in their construction.1 It was my good fortune and great pleasure to assist Martin in the late 1960s, when he first moved to Harvard University, in the development of methods spanning these general categories in the particular case of the long-range (dipole dispersion) limits of potential energy surfaces. In our work we employed the so-called Casimir-Polder2 and Axilrod-Teller3 expressions, which represent dispersion forces in terms of the dynamic (frequencydependent) polarizabilities of the interacting species, to devise bounds on nonbonded pairwise-additive and nonadditive threebody interaction energies.4 I have long felt it would be highly desirable to devise related methods for the chemical binding interactions among atoms and other fragments, methods which would specifically provide representations of electronic potential energy surfaces generally in terms of properties of the isolated atoms or other fragments comprising the aggregate of interest.5 In this approach, electronic structure calculations would be performed of atomic “response functions” of some type, † Work supported in part by a grant from the Air Force Office of Scientific Research to Indiana University, administered by Research and Development Laboratories, Inc., and by contractual support from Phillips Laboratory, Edwards AFB, through Hughes STX Corporation. ‡ This paper was originally submitted for the Karplus Festschrift [J. Phys. Chem. 1996, 100 (7)]. X Abstract published in AdVance ACS Abstracts, January 15, 1996.
0022-3654/96/20100-2974$12.00/0
generalizations of the dynamic polarizabilities appearing in the Casimir-Polder and Axilrod-Teller expressions, to which they would reduce in the appropriate (long-range) limits. In this way, a unified theoretical description of both physical, or nonbonded, and chemical, or bonded, interactions would be obtained which would be suitable for interpretive and computational applications. The possibility of devising such a unified responsefunction description of physical and chemical interactions is suggested, among other considerations, by early secondquantized descriptions of atomic interactions6 and by related formulations of the long-range interactions between macroscopic objects.7 Additional motivation is provided by the remarks of John Slater, among others, who indicated his belief “that there is no very fundamental distinction between the van der Waals binding and covalent binding” on the basis of virial and Hellmann-Feynman theorem considerations,8 discussed by him in the context of the so-called XR method.9 In the present work, as a tribute to Martin on the occasion of his 65th birthday and in recognition of the influence he has had on my studies of atomic interactions, I report the development of a response-function-based description of physical and chemical binding.5,10 In particular, I describe a “separation theorem” for interaction energies which expresses the adiabatic electronic potential energy surfaces of aggregates of atoms in terms of spectral response properties. The latter quantities can be calculated once and for all for atoms or other fragments of interest, with interaction energy surfaces for a particular spatial configuration of atoms obtained from diagonalization of an Hamiltonian matrix that is constructed by a three-dimensional quadrature which does not involve the electronic degrees of freedom of the aggregate. Accordingly, in this method, there is an end to calculations of a quantum chemical nature required in studies of potential energy surfaces. Moreover, there is the © 1996 American Chemical Society
Spectral Theory of Physical and Chemical Binding possibility to understand aspects of physical and chemical binding in terms of the properties of the non-interacting constituents. By contrast, contemporary ab initio methods for construction of accurate abiabatic electronic potential energy surfaces are generally limited to simple systems11 and are arguably not suitable for applications to larger atomic aggregates, since the computational effort involved in the commonly employed approaches scales with some power (≈4-5) of the number of electrons treated.12 The formal theoretical approach is described in section 2 employing methods which are seen to be largely pedestrian in nature but which are analytically rigorous. The development borrows freely from earlier related work,13-18 extending and clarifying these studies where necessary. A direct product of complete sets of atomic spectral eigenfucntions13,14 is employed to formally represent the total adiabatic electronic Hamiltonian matrix, which is seen to be rigorously pairwise additive in the individual atomic interaction matrices in this basis.14-16 Fourier representation of the electrostatic potentials serves to separate the individual interaction matrices in terms of atomic properties alone, which are identified as (infinite-dimensional) response matrices over the individual complete sets of atomic spectral states.5,10 In accordance with the early observations of Eisenschitz and London,13 as emphasized more recently by Musher and Amos,17,18 the exclusion principle is correctly accommodated in the limit of spectral completeness in the absence of adoption of “explicitly antisymmetrized” product states in the development. It is this recognition, and use of recently devised Stieltjes methodology described below for achieving spectral completeness in finite-basis representations of discrete and continuum eigenstates,19 that largely distinguishes the present development from earlier attempts to devise atomic-13,14 and pair-based15,16 theories of chemical interactions. A program of computational implementation of the formal development is described in section 3, which includes both atomic- and pair-based methods for, in Martin’s classifications,1 ab initio, semiempirical, and purely empirical applications. The ab initio approaches are based on optimal finite-basis spectral representations of the atomic eigenspectra appearing in the formal development,19 which methodology makes computationally viable both atomic- and pair-based approaches to chemical and physical binding as well as the separation theorem for interaction energies indicated above. A perturbation theory development of the atomic-spectral-based version on the method extends and generalizes the familiar second-order CasimirPolder2 and third-order nonadditive Axilrod-Teller3 expressions, which are recovered in the dipole-expanded limit of long-range interactions.5 In the pair-based implementation of the spectral development described in section 3, individual pair-interaction matrices are expressed in terms of diatomic potential energy curves and atomic-product eigenstates represented in the Stieltjes spectral basis set by suitable transformation, employing rotation matrices20 to achieve the (angular) geometric arrangements of interacting pairs in the particular aggregate of interest.16 The spectral product basis in this variant of the development provides a scaffold, rather than a computational basis, with which to represent diatomic pair functions obtained by other means. Experimentally determined potential curves can also be employed and the required diatomic-to-atomic product transformation matrix constructed following semiempirical or entirely empirical approaches if appropriate information is available. This implementation of the development, when appropriately renormalized, is seen to contain the familiar pairwise-additive approximation to N-body nonbonded interaction energies in all orders, to further clarify and extend diatomics-in-molecules
J. Phys. Chem., Vol. 100, No. 8, 1996 2975 approaches to chemical potential-surface construction15,16 and to demonstrate equivalences between these latter approaches and apparently unrelated perturbative approximations to interaction energies commonly employed in collision broadening situations21,22 and in studies of degenerate state splittings in trapped-radical spectroscopy.23-27 Dilute (Na) metal radicals trapped on physically bound cryogenic (ArN) rare-gas clusters are studied employing available potential energy curves and related information28-30 in an illustrative application of the spectral method reported in section 4. The effects of state mixing and charge distortion missing from commonly employed semiempirical and perturbationtheory studies of trapped-radical spectra23-27 are examined on the basis of the spectral development employing matrix partitioning methods31 to display the explicitly nonadditive contributions to the potential surfaces. The predicted adiabatic electronic energy surfaces are seen to depend more sensitively upon the detailed angular arrangement of rare-gas perturbers around the radical metal than do results inferred from perturbation approaches due to angular-dependent caging effects present in the spectral development that are missing from earlier approaches. The calculations show large nonadditive contributions to the lowest-lying potential surfaces, as manifested in the Na-Ar radial distribution functions and optical absorption spectra obtained for low-temperature NaAr6 aggregates. The relevance of these results to interpretations of experimentally determined trapped-radical spectra more generally is reported separately elsewhere.32 2. Theoretical Development The Hamiltonian operator for an aggregate of N atoms can be written in the general form33,34
H ˆ (i,j,...) )
N
N
R)1
β)1
∑ {Hˆ (R)(i) + ∑(β > R)Vˆ (R,β)(i;j)}
(1)
where H ˆ (R)(i) is the Hamiltonian operator for the atom R containing nR electrons labeled i and Vˆ (R,β)(i;j) is the interaction Hamiltonian between atoms R and β. The electron label j runs over nβ electrons distinct from those labeled i, with a semicolon employed in eq 1 and in the sequel to separate (i;j) distinguishable groups of electrons (i and j). The assignment of particular electrons (i or j) to atomic sites (R or β) is made arbitrarily. 2.1. Coulomb Hamiltonian. In the Coulomb approximation the operators of eq 1 take the specific forms35 nR
{
H ˆ (i) ) ∑ (R)
i
p2
∇i2
-
2m
ZRe2 riR
}
n
R e2 + 1/2∑(i′ * i) rii′ i′
(2)
and
Vˆ
(i;j) )
(R,β)
ZRZβe2 RRβ
nR
Zβe2
i
riβ
-∑
nβ
ZRe2
j
rjR
-∑
nR nβ
+ ∑∑ i
j
e2 rij
(3)
employing conventional notation. The electronic eigenstates of the Hamiltonian H ˆ (R)(i) in this case, written as a row vector (R) Φ (i) of functions ΦΓ(R)R (i), are (Russell-Saunders) eigenfunctions of energy, orbital and spin angular momentum, and parity {ΓR ≡ (E, L, ML, S, MS, P )R}36 and transform irreducibly as the totally antisymmetric representation of the permutation group for the electrons in the set i.37 These atomic spectral states ΦΓ(R)R (i), which include the photoionization continua, are quantized in a set of N individual coordinate systems all having
2976 J. Phys. Chem., Vol. 100, No. 8, 1996
Langhoff
axes oriented parallel to the laboratory-frame axes. The locations of the N atomic centers are specified in the latter frame by position vectors RR, and RRβ (tRβ - RR) gives their relative positions. Since the atomic spectral states Φ(R)(i) form a complete (antisymmetric) set for representations of L2 functions of the indistinguishable electron coordinates i,37 the ordered direct product spectrum
Φ(i;j;k;...) ) {Φ (i)XΦ (j)XΦ (k)X...}O (R)
(β)
(γ)
H(R) )
∑ {H R)1
(R)
N
+ ∑ (β > R)V(R,β)(RRβ)}
{
E(R) 0 H(R) ) 0 · · ·
(4)
is sufficient for representation of antisymmetric eigenstates of the full set of electrons i, j, k, ....13,17,18 The brace symbol {...}O in eq 4 and elsewhere below implies the choice of a particular ordering rule for the indices of the direct product (X) functions, which rule, although arbitrary, must also be adhered to in matrices constructed in the spectral basis and in vector and matrix multiplications. Although the (row-vector) product set of eq 4 does not constitute a symmetrical representation of the electrons in the aggregate37 and individual members of the set are not antisymmetric with respect to exchange between electrons in the distinct sets i; j; k; ..., the basis can nevertheless be employed to represent wave functions which are antisymmetric in all electron labels. Use of an “explicitly antisymmetrized” product basis, although commonly employed to avoid the appearance of unphysical states corresponding to other irreducible representations of the electron permutation group in limited basis set calculations,14-16 is unsuitable for the present development and can lead to a linear dependence (“overcompleteness”) in the limit of unrestricted closure unless appropriate measures are devised to counter this eventuality.13,17,18 The spectral basis of eq 4 is expected to be particularly appropriate in the absence of significant electron delocalization or transfer among aggregate atoms, although the latter circumstances can also be described in the product spectral basis. 2.2. Matrix Representatives. The matrix representative of the Hamiltonian (eqs 1-3) in the product spectral basis Φ(i;j;k;...) of eq 4 takes the form N
from similar block diagonal matrices by appropriate row and column interchangessdetermined by the relative positions of the R and β indices to N - 1 and N, respectivelysin the forms
(5)
β)1
where R ) (R1, R2, ..., RN) gives the position of the N atoms in the laboratory frame and H(R) and V(R,β)(RRβ) are matrix representatives of the corresponding operators in eqs 1-3. The atomic matrix [H(R)] is a diagonal matrix involving energies of the atomic spectral states Φ(R)(i), whereas the interaction matrix [V(R,β)(RRβ)] is nondiagonal and depends explicitly upon the relative positions (RRβ) of the two indicated atoms. Both types of matrices are of identical dimension, determined by the size of the complete product basis of eq 4, and have individual elements determined by the product ordering convention adopted for the direct product states of eq 4. Equation 5 illustrates the rigorous pairwise-additive nature of the total Hamiltonian matrix in terms of the individual interaction matrices when the direct product basis is employed.14-16 Employing the common ordering convention of letting later indices run to completion prior to earlier ones in the produce spectrum of eq 4, which defines the bracket ordering symbol of eq 4, the last atomic matrix [H(N)] and last pair interaction matrix [V(N-1,N)(RN-1,N)] take block diagonal forms, with repeating matrices constructed in the corresponding atomic and atomic-pair product basis, respectively, appearing on the diagonals. Accordingly, the atomic and atomic-pair interaction matrices of eq 5 for an arbitrary pair of atoms (R, β) are obtained
and
{
0
0 0
E(R) 0 · · ·
v(R,β) 0 V(R,β)(RRβ) ) 0 · · ·
E(R) · · ·
0 (R,β)
v 0 · · ·
· · · · · · · · · ·· ·
0 0 v(R,β) · · ·
}
(6) 0
· · · · · · · · · ·· ·
}
(7) 0
employing the ordering symbol of eq 4. Here,
ˆ (R)(i)|Φ(R)(i)〉 E(R) ) 〈Φ(R)(i)|H
(8)
is the diagonal matrix of energies of the atom R arranged in normal (energy-increasing) order along the diagonal and
v(R,β) ≡ v(R,β)(RRβ) ) 〈Φ(R,β)(i;j)|Vˆ (R,β)(i;j)|Φ(R,β)(i;j)〉 (9) is the pair-interaction matrix in the ordered (R, β) product basis
Φ(R,β)(i;j) ) {Φ(R)(i)XΦ(β)(j)}O2
(10)
In eqs 8 and 9 a vector product convention is adopted in which customary matrices having (row-column) ordering labels are formed from the indicated pairs of row vectors Φ(R)(i) and Φ(R,β)(i;j). The subscript (O2) in eq 10 implies ordering only in the direct product space of vectors Φ(R)(i) and Φ(β)(j) in accordance with their positions (R < β) in the overall direct product of eq 4. It should be noted that the matrix E(R) of eq 8 has a generally different dimension than that of the matrix v(R,β) of eq 9, although the complete matrices H(R) and V(R,β)(RRβ) of eqs 6 and 7, respectively, have the same dimensions. Once the matrices of eqs 8 and 9 are constructed, those of eqs 6 and 7 are assembled by performing the row and column reorderings implied by the indicated ordering symbol (O). The latter operations correspond to placing individual matrix elements in positions appropriate to the orderings of the relevant atomic basis functions in the product basis of eq 4. A simple example employing three identical atoms having two spectral states each clarifies the ordering convention and dimensions of the matrices H(R), E(R), V(R,β)(RRβ), and v(R,β)(RRβ).10 It should be stressed, however, that the development of eqs 1-10 entails construction of infinite-dimensional atomic and interaction matrices in the atomic spectral basis and is, therefore, largely formal in nature in the absence of additional considerations. 3. Implementations of the Formal Development A program of implementation of the development of eqs 1-10 is described here which includes atomic- and pair-based approaches. The former approach employs optimal atomic spectral states in a formulation suitable for ab initio finite-matrix computations and for establishing a viable separation theorem for interaction energies in this representation, whereas the latter approach using diatomic potential curves and related pair
Spectral Theory of Physical and Chemical Binding
J. Phys. Chem., Vol. 100, No. 8, 1996 2977
information is seen to be suitable for both ab initio and semiempirical applications. 3.1. Separation Theorem and Stieltjes Representation. Adopting the familiar Fourier representation for the interaction operator of eq 3,38
Vˆ
(R,β)
(R) 2 (R) F(R) g (ER;k) ) |〈ΦΓR (i)|V1 (i)〉| )
(i;j) ) e2
2π
It will be recalled in the above connection that LanczosKrylov sequences generated from a given test function (eq 14) constitute optimal invariant subspaces for representations of the associated spectral density,
nR
∫ (dk/k2)eik‚R
2 k
nR
nβ
i
j
{ZR - ∑eik‚ri}{Zβ - ∑e-ik‚rj} (11)
Rβ
the pair matrix of eq 9 becomes
v(R,β)(RRβ) )
e2 (dk/k2)eik‚RRβ{v(R)(k)Xv(β)(-k)}O2 (12) 2∫k 2π
where nR
v (k) ) 〈Φ (i)|ZR - ∑eik‚ri|Φ(R)(i)〉 (R)
(R)
(13)
i
is the (infinite-dimensional) “spectral response” matrix for the atom R, and the direct product of matrices in eq 12 refers to an ordered matrix product as in eq 10. The response matrices of eq 13 can be determined for the atoms of interest, independent of any particular atomic arrangement, with RRβ in the exponential factor of eq 12 providing the necessary geometrical information appropriate to the particular aggregate of interest. Equation 12 is seen to be a three-dimensional quadrature over the single indicated wave vector k which does not involve the electronic degrees of freedom. Accordingly, once the quantummechanical calculations required in the construction of the matrices of eq 13 are completed, which calculations can be performed once and for all, the Hamiltonian matrix of eq 5 can be assembled by employing eq 12 and performing the row and column interchanges implicit in the notation of eqs 6 and 7. The development of eqs 1-13 is therefore seen to constitute a separation theorem for the evaluation of adiabatic electronic energy surfaces in terms of properties of the individual atoms or other fragments comprising the aggregate of interest. Applications of the development of eqs 1-13 will entail a program of study to address computational issues, particularly those associated with the (formal) use of a direct product of atomic spectral states in the absence of prior enforcement of antisymmetry constraints. Of particular interest in this connection is representation of the response matrix v(R)(k) of eq 13 and study of its convergence, which will reflect the completeness of the atomic spectral basis employed. A finite, denumerable representation of the infinite set of discrete and continuum atomic spectral states Φ(R)(i) required in eq 13 which is complete over finite spatial regions can be devised for this purpose employing previously described L2 Stieltjes methods.19 Specifically, adopting a test function of the form nR
V(R) 1 (i)
2 |〈ΦΓ(R)R (i)|ZR - ∑eik‚ri|Φ(R) g (i)〉| (15)
) {ZR - ∑eik‚ri}Φ(R) g (i)
(14)
i
where Φ(R) g (i) is the (nondegenerate) ground state of the atom R, a so-called Lanczos-Krylov sequence {V(R) k (i), k ) 1, 2, ...} can be generated by recurrence with the atomic Hamiltonian operator following a methodology which has been described in detail elsewhere.19,39,40 Such sequences can provide explicit representational basis sets in place of the (formally) complete sets of atomic spectral states for computational applications of the spectral method.
i
regarded as a function of the energy ER, and, accordingly, are also optimal for corresponding representations of spectral integrals over this measure.19,40 Experience has shown that finite sequences of L2 Lanczos-Krylov functions can span everincreasing spatial regions with increasing order, avoid the near linear dependence associated with Rydberg series limits, give pointwise convergent Stieltjes approximations to low-lying discrete states, provide packet-like representations of high-lying discrete and continuum states over finite but large spatial intervals, and give rapidly convergent one-electron transition densities of the type required in evaluation of eq 15.19,40 The formalism furthermore provides rigorous upper and lower bounds on the (cumulative) distribution derived from the density of eq 15. Issues related specifically to the k dependence of Lanczos-Krylov sequences of functions have also been discussed earlier in the context of closely related generalized oscillator-strength densities.41 Although applications of the Stieltjes methodology in the present context will require some extensions of the previously reported approach,19,40 it is clear the method is generally suitable for implementations of the spectral development for potential energy surfaces. Specifically, an arbitrary element of the matrix of eq 13 for a neutral atom (ZR ) nR) is of the form nR
{v (k)}ΓR,Γ′R ) (R)
〈ΦΓ(R)R (i)|ZR
(R) (i)〉 - ∑eik‚ri|ΦΓ′ R i
) ∫r γΓ(R)R,Γ′R(rR)(1 - eik‚rR) drR R
(16)
where nR
(R) (i)〉 γΓ(R)R,Γ′R(rR) t 〈ΦΓ(R)R (i)|∑δ(rR - ri)|ΦΓ′ R
(17)
i
is the one-electron transition density for the indicated atomic states (ΓR, Γ′R) and rR is the electron coordinate r measured from the atomic position RR. Accordingly, it is seen that the test function of eq 14 is appropriate for developing a (LanczosKrylov) Hilbert space by recurrence which gives optimal convergence for the first row and column (cf. eqs 15 and 16) of the v(R)(k) matrix. It can be further anticipated that other rows and columns of the v(R)(k) matrix will converge satisfactorily in the Stieltjes representation so devised, although optimal convergence is assured only for the elements of eq 15. Of course, the size of the Lanczos-Krylov basis and the ranges of k and ER values required for convergence of the development can only be clarified fully in the course of detailed computational applications. It is clear, however, that, in so far as a finite Stieltjes representation can provide convergent approximations to the spectral response matrices of eq 13, the development of eqs 1-13 should provide related convergent approximations to the associated potential energy surfaces. To further clarify the issue of electron antisymmetry in the absence of prior enforcement, note that the exact ground-state
2978 J. Phys. Chem., Vol. 100, No. 8, 1996
Langhoff
one-electron density, N nR
ρg(r) t 〈Φg(i,j,k,...)| ∑ ∑δ(rR - ri)|Φg(i,j,k,...)〉
(18)
R)1 i
required in evaluation of the one-electron terms in eq 9 can be expressed in terms of the transition densities of eq 17. Specifically, introducing the spectral product representation of Φg(i,j,k,...) in eq 18 gives N
∑ { ∑ CΓ(R),Γ′ γΓ(R),Γ′ (rR)} R)1 Γ ,Γ′
(19)
(R) (R) R) (i′)|Γ(n CΓ(R)R,Γ′R t 〈ΦΓ′ g (i′|i)|ΦΓR (i)〉 R
(20)
ρg(r) )
R R
R R
R R
where
R) In eq 20, Γ(n g (i′|i) is the nR-electron reduced density matrix constructed in the usual way from the ground-state wave function,42 with the integral there performed over all electrons in the sets i and i′. Similarly, the exact ground-state two-electron density required in evaluation of the electron pair-interaction terms in eq 9,
γg(r1,r2) t 〈Φg(i,j,k,...)| × N
nR nβ
N
∑ ∑(β > R)∑i ∑j δ(rR - ri)δ(rβ - rj)|Φg(i,j,k,...)〉 R)1β)1
(21)
another due to antisymmetry exchange in the conventional representation can be described by products of single-center densities in the spectral representation. Although the foregoing observations only hold rigorously when the exactsor an approximate but correctly antisymmetrics ground-state wave function is employed in eqs 18-23, it is clear a complete atomic spectral basis is capable of correctly reproducing the essential aspects of chemical and physical binding or antibinding in the absence of explicitly antisymmetric terms in the spectral basis, with the single-center transition densities of eq 17 providing the mechanism for appropriate redistribution of electronic charge amplitudes upon aggregate formation. This emphasis on transition densities in the spectral method, and the absence of prior antisymmetry in the development, has points of similarity with one-particle many-body Greens function representations of electronic energies,43 in which formalism single-particle excitation amplitudes are determined in the absence of prior antisymmetry,44 but convergence of the energy is nevertheless obtained when appropriate nonperturbative (Dyson) methods are employed in constructing the required Greens function.45 Solutions [ΦE(i,j,k,...)] of the full Schro¨dinger equation,
H ˆ (i,j,k,...)ΦE(i,j,k,...) ) E(R)ΦE(i,j,k,...)
are constructed in the Stieltjes representation of the spectral product basis of eq 4 employing a linear vibrational development. Letting the column vector ΦE represent the linear variational coefficients in the spectral basis for a particular eigenstate ΦE(i,j,...) of energy E(R)
takes the form
ΦE(i,j,...) ) Φ(i;j;...)‚ΦE
γg(r1,r2) ) N
(25)
gives the familiar matrix equation
N
∑ ∑ (β > R) Γ∑,Γ′ Γ∑,Γ′CΓ(R,β),Γ′ ;Γ ,Γ′γΓ(R),Γ′ (rR)γΓ(β),Γ′(rβ)
R)1 β)1
(24)
R R
R R
β β
R R
β β
H(R)‚ΦE ) EΦE
(22)
β β
in the spectral product basis. Here, (R,β) R+nβ) t 〈ΦΓ′ (i′;j′)|Γ(n (i′,j′|i,j)|ΦΓ(R,β) (i;j)〉 (23) CΓ(R,β) g R,Γ′R;Γβ,Γ′β R,Γ′β R,Γβ
(i;j) refers to the pair-product functions of eq 10, where ΦΓ(R,β) RΓβ R+nβ) Γ(n (i′,j′|i,j) is the (nR + nβ)-electron ground-state reduced g density matrix,42 and rR, rβ are the electron coordinates r1, r2 measured from the atomic positions RR and Rβ, respectively. Equation 19 shows the one-electron density in the spectral product basis to be comprised of atomic-centered densities [(γΓ(R)R,ΓR(rR)] and overlap or transition densities [γΓ(R)R,Γ′R(rR), ΓR * Γ′R]. The former terms allow for expansion or contraction of atomic electronic distributions in the presence of physical or chemical interactions, whereas the latter allow for “exchange” density addition or depletion in binding or antibinding regions, respectively, in accordance with Hellmann-Feynman considerations.8,9 Similarly, eq 22 shows that the two-electrton density required to evaluate the electron pair-interactions terms of eq 9 is comprised in the spectral product representation of pairwise products of dispersion-like single-center atomic transition densities. Comparisons of these spectral-product representations of Fg(r) and γg(r1,r2) with corresponding expressions obtained from contemporary ab initio methods employing antisymmetric configurational state functions constructed in molecular orbital basis sets11 indicate the one-center transition densities of the former representation must fulfill the function of multicenter overlap densities in the latter representation. Such comparisons further indicate that terms in Fg(r) and γg(r1,r2) corresponding to delocalization of two or more electrons from one center to
(26)
for the eigenvalues and vectors (E(R),ΦE). It should be noted, prior antisymmetry constraints in integral forms and associated Lagrangian multipliers can be incorporated in the development to help eliminate the unphysical solutions spanned by the spectral product basis of eq 4 in the absence of antisymmetrized basis states.10 Alternatively, the correct physical states and energies can be identified from the solutions of the unconstrained development on the basis of their electron permutation symmetries or their long-range perturbation theory limits. Since the Stieltjes development can provide accurate approximations to low-lying discrete states but treats higher-lying and continuum states as spectral packets,19,40 it is convenient to partition the Hamiltonian matrix problem of eq 26 in the form31
(
)( )
( )
HQQ(R) HQP(R) ΦE(Q) Φ(Q) ‚ (P) ) E(R)‚ E(P) HPQ(R) HPP(R) ΦE ΦE
(27)
where the Q space refers to the low-lying direct product space and P (≡1 - Q) is its closure. This partitioning implies a particular ordering rule (eq 4) be adopted to insure that the enumeration of the Q space is complete before the P space is included. Eliminating the P-space component of the solution vector of eq 27 in the usual way gives
{HQQ(R) + HQP(R)‚{E(R)IPP - HPP(R)}-1‚HPQ(R)}‚ ΦE(Q) ) E(R)‚ΦE(Q) (28) Here, HQQ(R) contains the zeroth-order energies and first-order electrostatic repulsions and attractions in the reference space
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J. Phys. Chem., Vol. 100, No. 8, 1996 2979
Q, whereas the second or optical potential term in eq 28 contains the “dispersive” physical or chemical interaction terms associated with higher spectral excitations. The Stieltjes methodology,19,40 when employed in conjunction with the matrix ordering conventions of eqs 4-13, is particularly suitable for constructing approximations to the optical potential matrix of eq 28, and can provide either a discrete or explicit integral representation of the latter.40,46 Since interest centers in the present development on the ground or low-lying potential energy surfaces of an aggregate of atoms, the correct analytical behavior of this resolvent matrix in the neighborhood of the higher-lying roots of the HPP matrix, which are not individually reproduced by the Stieltjes development, is largely irrelevant to the development, and either discrete or continuous representations of the optical potential term in eq 28 should be suitable. 3.2. Pair-Interaction Representation. The spectral method can also be implemented employing molecular electronic diatomic pair eigenstates and corresponding interaction energy matrices in a two-step transformation from diatomic-pair to atomic-product basis, in the spirit of the so-called diatomicsin-molecules methods.15,16 The Stieltjes representation of the atomic-product states in this development plays the role of a convenient scaffold for the pair interaction, rather than a computational basis. That is, the required diatomic wave functions can be obtained in this approach employing conventional methods11 which do not directly involve the atomicproduct spectral states. Since an explicitly antisymmetric set of (diatomic) functions is employed as a projector on the spectral product states, it can be anticipated that unphysical solutions spanned by the latter states are avoided in this version of the development, provided that the spectral product basis of eq 10 spans completely the spatial domain of the contributing diatomic states. Letting the row vector Φ(R,β)(ir;jr) represent the (R,β) product states of eq 10 in a coordinate system in which colinear zr axes oriented along the RRβ direction in the laboratory frame are employed at the two atomic sites R and β, the product states in the two coordinate systemsslaboratory and rotatedsare connected by the transformation
Φ(R,β)(i;j) ) Φ(R,β)(ir;jr)‚R(R,β)(φRβ,θRβ)
(29)
The row vector of diatomic pair states Ψ(R,β)(ir,jr), defined as solutions of the Schro¨dinger equation for the pair Hamiltonian in the rotated coordinate system r,
(H ˆ (R)(ir) + H ˆ (β)(jr) + Vˆ (R,β)(ir;jr))Ψ(R,β)(ir,jr) ) E(R,β)(RRβ)‚Ψ(R,β)(ir,jr) (33) transforms irreducibly under the operations of the cylindrical group C∞V, is antisymmetric with respect to interchange of any two of the electrons in the combined group i and j, and is energy ordered in a conventional manner. The transformation matrix U(R,β)(RRβ) between this diatomic-pair basis and the atomicproduct basis states in the rotated coordinate system is given by the expression
Φ(R,β)(ir;jr) ) Ψ(R,β)(ir,jr)‚U(R,β)(RRβ)
(34)
U(R,β)(RRβ) ) 〈Ψ(R,β)(ir,jr)|Ψ(R,β)(ir;jr)〉
(35)
where
describes the extent of mixing of the atomic product states Φ(R,β)(ir;jr) in the antisymmetric eigenstates Ψ(R,β)(ir,jr). Explicit construction of the transformation matrix of eq 35 and aspects of its unitarity are discussed below. Employing the two foregoing transformations, the matrix v(R,β)(RRβ) of eq 9 is obtained from the diagonal matrix of potential energy curves for the (R,β) pair (eq 33),
E(R,β)(RRβ) ) 〈Ψ(R,β)(ir,jr)|H ˆ (R)(ir) + H ˆ (β)(jr) + Vˆ (R,β)(ir;jr)|Ψ(R,β)(ir,jr)〉 (36) in the form
v(R,β)(RRβ) ) H(R,β)(RRβ) - E(R,β)
(37)
where
H(R,β)(RRβ) ) R(R,β)(φRβ,θRβ)†‚U(R,β)(RRβ)†‚E(R,β)(RRβ)‚U(R,β)(RRβ)‚
where
R
(R,β)
(φRβ,θRβ) ) 〈Φ
(R,β)
(R,β)
(ir;jr)|Φ
R(R,β)(φRβ,θRβ) (38) (i;j)〉
) {D (φRβ,θRβ,0)XD (φR,β,θRβ,0)}O2 (R)
(β)
is the transform of E(R,β)(RRβ), and
(30)
is the required transformation matrix, employing an ordered direct product of rotation matrices for the sites R and β. The individual matrices D(R)(φRβ,θRβ,0) and D(β)(φRβ,θRβ,0) are block diagonal, having rotation matrices of appropriate angular momentum along their diagonals.20 The so-called “passive” rotation convention is employed in the transformation of eqs 29 and 30, in which the atomic spectral states transform according to the expression (Γ) (Γ) (Γ) (i) ) ∑ΦM′ (ir) DM′,M (φRβ,θRβ,0) ΦM
(31)
(Γ) (Γ) (Γ) (φRβ,θRβ,0) ) 〈ΦM′ (ir)|ΦM (i)〉 DM′,M
(32)
M′
where
is the rotation matrix of angular momentum value specified by the label Γ.20
{ } { }
(R) 0 E(R,β) ≡ E(R,β)(RRβ f ∞) ) E 0 0
+
O2
0 0 0 E(β)
(39) O2
is the separated-pair limit of eq 36, with E(R) and E(β) the atomic eigenvalue matrices of eq 8. Equations 36-39 insure that the interaction energy matrix, rather than the total pair potential energy matrix for the (R,β) pair, is employed in the development to avoid overcounting the atomic energies E(R) and E(β). Equation 38 evidently constitutes two sequential transformations of the pair-interaction potential energy matrix from the diatomic-pair-state representation (eq 33) to the atomic-product representation in the rotated (colinear) coordinate pair frame (eqs 34 and 35) and finally to the required laboratory-frame spectral basis (eqs 29 and 30). Computational applications of this formal transformation, and of the pair approach of eqs 2939 more generally, require finite representations of both the atomic-product and pair states. In a purely ab initio version of the development, both sets of states can be described in
2980 J. Phys. Chem., Vol. 100, No. 8, 1996 Stieltjes forms, in which case the unitarity of the resulting transformation matrix of eq 35 provides a test of closure. In this approach convergence is determined by the number of pair states (eq 33) required and the number of spectral product states (eq 10) needed to represent them. It can be anticipated on the basis of the discussion of section 3.1 that the pair-energy matrix will generally converge more rapidly than the wave functions of eq 34 in such calculations. As an alternative to an ab initio approach, semiempirical or completely empirical perspectives can be adopted in which finite-dimensional transformation matrices U(R,β)(RRβ) which satisfy ad hoc unitary constraints are constructed employing available pair information, in the spirit of the diatomics-in-molecules methods.15,16 Of course, justification of the approximations employed in the latter case must be provided on the basis of the underlying development of eqs 2939 in order to have confidence in the predictions made in this way. 4. Selected Illustrative Applications Although detailed computational applications of the spectral method are reported separately elsewhere,10,32 illustrative results are presented here to clarify connections with related formalisms.14-24 Specifically, pairwise-additive and long-range limits of the development are described, and both analytical and numerical results are reported for alkali-metal-seeded (Na) cryogenic rare-gas (ArN) aggregates.27 4.1. Pairwise-Additive and Long-Range Limits. Solutions of the Schro¨dinger equation in the spectral product basis (eqs 24-28) obtained from a conventional (Rayleigh-Schro¨dinger) perturbation theory development are easily seen to incorporate the pairwise additive approximation to the ground-state potential energy surface of an arbitrary aggregate when interaction terms diagonal in pair labels (R,β) are retained to all orders and threebody and higher pair interactions are neglected in the development. Introduction of the Fourier representation of eq 11 for the interaction operators gives in this way a generalization5 of the familiar Casimir-Polder separation theorem2 for pair interaction energies in lowest (second) order, whereas nonadditive (three-body) interaction terms are seen to arise in thirdorder and to reduce to the familiar Axilrod-Teller form3 when a conventional long-range (dipole) expansion is employed for the interaction Hamiltonian.10 Although these results are not entirely unexpected, it is satisfying that the spectral development contains these familiar approximations in the correct perturbation theory and long-range limits. Moreover, the necessity of explicitly summing pairwise interactions to all orders in a perturbation theory development to obtain the pairwise-additive approximation from the atomic-product-based development emphasizes the importance of explicit matrix diagonalization in this version of the spectral method. Of course, a perturbation theory development based on a product form of zeroth-order state does not insure antisymmetry even in the limit of spectral closure.17,18 4.2. Analytical ResultssNaArN Aggregates. Potential energy surfaces and related attributes of aggregates comprised of a single Na atom and N Ar atoms (N ) 2-250) are studied employing the development of eqs 29-39 to help place the spectral method in the context of previous approximations employed in this case.21-27 The application is simplified by (i) restricting the (Ar) rare-gas atomic states to the ground 1S0 state,47 (ii) including only the lowest-lying (32S,32P,42S,32D) Na atom states in the calculations,47 and (iii) treating the U(R,β)(RRβ) pair transformation matrix of eq 35 semiempirically, as described below. Since the aggregates of interest are physically
Langhoff bound, the model is not unrealistic and, as will be seen, incorporates and extends commonly employed approximations for potential surface construction in such cases.21-27 It is convenient to first write out the result obtained retaining only the lowest two Na states, prior to presenting numerical results for the complete Na-ArN model system. In this restricted case the product spectral states of eq 10 are given by the four-term row vector
Φ(R,β)(i;j) f Φ(k)(i;j) ) (32S(i),32P+1(i),32P0(i),32P-1(i))NaX(1S0(j))Ark (40) where electrons i are on the Na atom and j are on the kth Ar atom (k ) 1-N). Since there is only one Ar state contributing to the present development, the ordering symbol of eq 10 is not needed. The corresponding order of molecular NaAr pair states of eq 33 is given by the four-term row vector.28-30
Ψ(k)(i,j) ) (XΣ+(i,j),AΠ+1(i,j),BΣ+(i,j),AΠ-1(i,j))NaArk (41) with the (4 × 4) transformation matrix U(R,β)(RR,β) [fU(Rk)] of eq 35,
(
)
U(Rk) )
〈XΣ+|3P0〉 0 〈XΣ+|3S〉 0 〈AΠ+1|3P+1〉 0 0 0 (42) + + 〈BΣ |3P0〉 0 〈BΣ |3S〉 0 〈AΠ-1|3P-1〉 0 0 0 connecting the two sets of states. In eq 42 a simplified notation is employed which suppresses the Ar ground-state eigenfunction for the six indicated matrix elements. The total (4 × 4) Hamiltonian matrix of eq 5 in the atomicproduct basis in this case is N
N
H(R) ) H(A) + ∑ V(k)(Rk) + ∑ k)1
N
(k > k′)V(k,k′)(Rk,k′) ∑ k)1 k′)1 (43)
where H(A) is the atomic energy matrix of eqs 5 and 6, the single sum over k represents the Na-Ar interactions, and the double sum over k,k′ represents the Ar-Ar interactions. Following the development of eqs 29-39 explicit expressions are obtained for the three types of matrices appearing in eq 43 in terms of the Na and Ar atomic state energies,47 the Ar2 and NaAr potential energy curves,27-30 and the six non-zero elements of the transformation matrix of eq 42. By explicitly enforcing unitarity of the matrix U(Rk),
U(Rk)†‚U(Rk) ) U(Rk)‚U(Rk)† ) I
(44)
only a single independent element 〈XΣ+|3P0〉, which gives the contribution of the indicated atomic product state [32P0(i)X1S0(j)] to the NaAr ground state XΣ+(i,j), is required. In an ab initio version of the pair-based development, as indicated above, the transformation matrix U(Rk) is obtained from eq 35 and calculated diatomic and atomic-product wave functions, in which case (not reported here) its unitarity can be established in a convergence study. In order to display analytically the potential surfaces obtained from the present development, it is convenient to represent the eigenvalues in terms of optical potentials using partitioning methods,31 as in eqs 27 and 28, in solution of the Hamiltonian matrix problem of eqs 26 and 43. Retaining terms to second-
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J. Phys. Chem., Vol. 100, No. 8, 1996 2981
order in the Na-Ar interactions gives for the ground-state surface in this way (Ar) Eg(R) ≈ E3(Na) + 2S + NEg N
(Ar) {EX(NaAr)(Rk) - (E(Na) ∑ 3 S + Eg )} + k)1 2
N
N
(k > k′){EX(Ar )(|Rk - Rk′|) - 2E(Ar) ∑ ∑ g }+ k)1 k′)1 2
N
1 (E(Na) 32S
-
N
(Rk) × ∑ ∑ (k * k′)H(NaAr) 12
k)1 k′)1 E3(Na) 2P )
(Rk′) cos ωkk′ (45) H(NaAr) 21 where
(Rk) ) H(NaAr) (Rk) ) H(NaAr) 12 21 〈XΣ+|3P0〉{EX(NaAr)(Rk) - EB(NaAr)(Rk)} (46) and ωkk′ is the angle between the vectors Rk and Rk′ measured from the Na atom location. In eq 45, the first line gives the atomic energies and the pairwise-additive Na-Ar interactions, the second line gives the pairwise-additive Ar-Ar interactions, and the third line gives nonadditive interaction terms involving the Na atom and two distinct Ar atoms. The latter terms evidently give an angular dependence in the ground-state potential surface missing in the pairwise-additive limit. In contrast to the atomic-product-based development of section 4.1, the pair-based development is seen to automatically include the pairwise-additive approximation in lowest-order and to give rise to nonadditive terms in second order, rather than in third order as in the former development. Proceeding in a similar manner, the excited (32P) adiabatic electronic energy surfaces are found in second order to be the eigenvalues of the 3 × 3 matrix (Ar) )Ibb + H32P(R) ≈ (E3(Na) 2P + NEg
Figure 1. Ground (2A1) state energy surface for C2V NaAr2 obtained from the present development. Depicted are the energies as a function of the Ar-Na-Ar apex angle at fixed Ar-Na distances of 4 Å, as obtained from the indicated computational approximations. The ground state energies of the Na and Ar atoms are arbitrarily set to zero, and the Ar-Ar interaction energy has been deleted to better illustrate the angle-dependent non-pairwise-additive Ar-Na-Ar contributions to the energy: (‚‚‚) 1 × 1 pairwise additive approximation; (- - -) 4 × 4 matrix approximation; (-‚-) 5 × 5 matrix approximation; (s) 10 × 10 matrix approximation.
Specifically, the matrix H(NaAr) (Rk) appearing there is given by ab (NaAr) † [H(NaAr) (R ) ) H (R ) ] ba k ab k
H(NaAr) (Rk) ) D(0)(φk,θk,0)†‚H(NaAr) (Rk)‚D(1)(φk,θk,0) (49) ab 12 explicitly displaying the angular dependence, where H(NaAr) 12 (Rk) is the three-term row vector
(Rk) ) (0, 〈XΣ+|3P0〉{EX(NaAr)(Rk) - EB(NaAr)(Rk)}, 0) H(NaAr) 12 (50)
N
(Ar) {H(NaAr) (Rk) - (E3(Na) ∑ bb P + Eg )Ibb} + k)1 2
N
N
) (k > k′)H(Ar ∑ ∑ bb (|Rk - Rk′|) + k)1 k′)1 2
1
N
{∑
(Na) (E3(Na) 2P - E32S )
N
∑ (k * k′) × k)1 k′)1 (Rk)‚H(NaAr) (Rk′)} (47) H(NaAr) ba ab
(Rk) matrix is Here, Ibb is the (3 × 3) unit matrix, the H(NaAr) bb given by the expression
(
H(NaAr) (Rk) ) D(1)(φk,θk,0)†‚ bb EA(NaAr)(Rk) 0 0
EB(NaAr)(Rk)
0 0
)
‚D(1)(φk,θk,0) (48)
EA(NaAr)(Rk) where D(1)(φk,θk,0) is the L ) 1 rotation matrix20 and the 2) H(Ar bb (|Rk - Rk′|) matrix is comprised of pairwise-additive Ar-Ar interactions, as in the second line of eq 45. The last terms of eq 47 can be shown to give nonadditive angulardependent corrections to the excited potential energy surfaces. 0
0
As for the ground-state surface, the results of eqs 47-50 are seen to automatically include the correct pairwise-additive approximation for degenerate states in all orders23-27 and to give rise to (additional) nonadditive terms in second order, rather than in third order, as in the atomic-product-based development. The matrix H(NaAr) (Rk) of eq 48 is identical with earlier bb perturbation theory results21-27 obtained in a manner entirely different in origin and appearance from the present development. Moreover, the development of eqs 40-50 can be made identical in form to a version of the diatomics-in-molecules method in which basis set overlap is neglected and an appropriate choice of matrix U(Rk) is employed. Accordingly, it is clear the perturbation theory methods used for treatment of state splittings in collision broadening21,22 and trapped radical spectroscopy23-27 have much in common with the latter diatomics approach,16 all placed here in the context of a more general and rigorous theoretical development. 4.3. Computational ResultssNaArN Aggregates. In Figures 1 and 2 are shown potential energy curves (cm-1) obtained from the present development for the four lowest-lying (2A1, 2B , 2B , 2A ) electronic states of a NaAr aggregate in C 1 2 1 2 2V symmetry. Depicted are energies as a function of the angle ω between the directions to the two Ar atoms from the Na atom employing fixed Na-Ar distances of 4 Å. As indicated in the figures, the ground-state Na and Ar atomic energies are
2982 J. Phys. Chem., Vol. 100, No. 8, 1996
Figure 2. As in Figure 1, for the excited 2B1, 2B2, and 2A1 state energy surfaces for C2V NaAr2: (‚‚‚) 3 × 3 matrix approximation; (- - -) 4 × 4 matrix approximation; (-‚-) 5 × 5 matrix approximation; (s) 10 × 10 matrix approximation.
arbitrarily set to zero, and the additive Ar-Ar interaction energy, having a well depth of ≈100 cm-1, has been deleted to better exhibit the non-pairwise-additive Ar-Na-Ar contributions to the potential energy surfaces obtained from the present development. Referring first to the ground-state 2A1 results of Figure 1, the values labeled 1 × 1 refer to the pairwise-additive approximation to the Na-Ar interactions, whereas the 4 × 4, 5 × 5, and 10 × 10 values show the effects of adding additional (3P,4S,3D) atomic Na states to the development. The necessary diatomic potential curves are obtained from the available calculations,28,29 whereas the elements of the transformation matrix of eq 35 are obtained by requiring the atomic product basis to reproduce calculated molecular (NaAr) dipole and transition moments28,29 following procedures generally similar to those of eqs 4044.10,32 The 10 × 10 results shown require use of the L ) 2 rotation matrix,48 whereas the 4 × 4 and 5 × 5 results require only the L ) 0 and 1 matrices.20 These results (4 × 4, 5 × 5, 10 × 10) are seen to be smoothly convergent and to contribute significantly to the total energy, a circumstance which can be attributed to the “near” 3S-3P degeneracy of the Na atom energy levels, which energy difference appears in the denominator of the second-order approximation of eq 45. Of course, the computational results shown in Figures 1 and 2 are obtained from diagonalization of the indicated matrices and do not involve the perturbation analysis of section 4.2. Since all the alkali metals exhibit near S-P degeneracy, the nonadditive “caging” effectshigher energy in the collinear Ar-Na-Ar configurations depicted in Figure 1 may be a common occurrence in alkali metals trapped in or on rare-gas aggregates more generally. Referring to Figure 2, the nonadditive angle-dependent contributions to the 2B1 surface are seen to be zero for the 4 × 4 and 5 × 5 results and very small for the 10 × 10 results. This is a consequence of the largely out-of-plane (3Px) atomic Na nature of the 2B1 state, to which the 3S (4 × 4) and 4S (5 × 5) Na atom states do not contribute, with the potentially contributing 3Dxz (10 × 10) Na atom state evidently giving a very small shift. These results are seen to be in accordance with the C2V symmetry of the aggregate. The energy variations of the 2B2 and excited 2A1 states in Figure 2 are seen to be relatively large even in the lowest (3 × 3) approximation. Of course, these variations refer to the degeneracy lifting and splitting associated with perturbation of the atomic 3P Na states,
Langhoff
Figure 3. Radial distribution functions for NaAr6 aggregate constructed employing the indicated potential energy surfaces and classical Monte Carlo calculations at 10 K:27 (‚‚‚) 3 × 3 matrix approximation; (- - -) 4 × 4 matrix approximation; (-‚-) 5 × 5 matrix approximation; (s) 10 × 10 matrix approximation. Examination of the geometries of these aggregates (not shown) finds the Na atom on the periphery in every case.
which actually gives nonadditive effects associated with the diagonalization of the relevant 3 × 3 matrix.21,22 The additional nonadditive angle-dependent contributions obtained from the spectral development in these cases are seen to be small but not negligible. Specifically, the 4 × 4 and 5 × 5 results for the 2B2 state are small and should be identically zero at all anglessfailure to achieve exact unitarity in the U matrix employed accounts for the discrepancy.10 The 10 × 10 results in this case are seen to give an ≈200 cm-1 splitting from the 3 × 3 results that removes the exact degeneracy of the two states at ω ) 90°. The 4 × 4, 5 × 5, and 10 × 10 results for the excited 2A1 state all correctly give additional nonadditive angledependent contributions, with the 10 × 10 results giving a large “caging” effect for ω ≈ 180°, as in the ground state, associated with the contribution from the 3D0 Na atom function. In Figures 3 and 4 are shown Na-Ar radial distribution functions and optical absorption spectra for a NaAr6 aggregate obtained from classical Monte Carlo calculations at 10 K,27,32 constructed employing the indicated potential energy surfaces. Referring first to Figure 3, the 3 × 3 results correspond to use of perturbation theory as discussed in section 4.2 for the excited states and the additive approximation for the ground state,21-27 whereas the three other approximations involve the introduction of additional atomic product states in the development, as in Figures 1 and 2. Examination of the geometries of these aggregates (not shown) finds the Na atom on the periphery of the aggregate in every case, consequent of the more attactive nature of the ground-state Ar-Ar interaction relative to the NaAr interaction.27-29 Evidently, the introduction of additional states in Figure 3 gives rise to a noticeable contraction of the Na-Ar distances, which can be attributed to the attractive nature of the nonadditive contributions to the Na-Ar potential for acute angular Ar-Na-Ar configurations, as indicated in Figure 1. Referring to Figure 4, for which results are obtained from Monte-Carlo-determined dipole autocorrelation functions,27,32 small but discernible shifts are evidently present in the spectra as additional states are added to the development. Of particular interest is the blue shift in the highest absorption feature which can be understood by inspection of the trends shown in the states of the simpler NaAr2 system depicted in Figures 1 and 2. The
Spectral Theory of Physical and Chemical Binding
Figure 4. Optical absorption spectra for NaAr6 aggregate constructed employing the indicated potential energy surfaces and dipole autocorrelation functions obtained from classical Monte Carlo calculations at 10 K:27 (‚‚‚) 3 × 3 matrix approximation; (- - -) 4 × 4 matrix approximation; (- ‚ -) 5 × 5 matrix approximation; (s) 10 × 10 matrix approximation.
results of Figures 3 and 4 are in general accord with earlier predictions for NaAr6 clusters based on one-electron potentials which also take into account the effects of Na atom charge distortion or state mixing upon aggregate formation.49 5. Comparative and Concluding Remarks A formally exact method is described for calculations of N-body adiabatic electronic potential energy surfaces, a program of computational implementation is described, and applications are reported to the optical spectra and related attributes of metal radicals trapped in low-temperature rare-gas matrices. The development employs as a conceptual representational basis a direct product of the atomic spectral eigenstates of all of the atoms in the system. Unlike the early development of Moffit,14 to whom one can perhaps attribute the use of such a basis in the theoretical chemistry literature, prior antisymmetry in the electron coordinates is not employed in the present development, in accord with the early observations of Eisenschitz and London,13 emphasized by Musher and Amos,17,18 who indicate the linear dependence consequent to use of such antisymmetry. The total adiabatic electronic Hamiltonian matrix in the direct product spectral basis is seen to be exactly pairwise additive in the atomic interactions, providing certain advantages in the treatment of many-body aggregates of atoms.14 Pairwise interaction-matrix additivity is also employed to good purpose in the diatomics-in-molecules method introduced by Ellison15 and employed and usefully refined by others, particularly Tully.16 However, the linear dependence consequent to use of an explicitly antisymmetric (overcomplete) set of states renders the convergence of these approaches uncertain as ab initio methods in the absence of modifications. Accordingly, these methods are widely regarded as semiempirical in nature, useful in fitting and interpolating potential energy surfaces, but of largely unknown or unexamined reliability in an ab initio context. Widespread use of explicitly antisymmetric product states in the chemical literature generally, and in the diatomics methods specifically, seems to arise from adoption of the methods of the familar theory of atomic electronic structure, in which case a “symmetrical” representational basis is employed.36,37 The present development, by contrast, does not entail use of a symmetrical representational basis, achieves
J. Phys. Chem., Vol. 100, No. 8, 1996 2983 antisymmetry in the limit of closure, and can provide clarification of the circumstances in which pair-based methods can be regarded as potentially convergent ab initio techniques or reliable semiempirical approaches. Use of a Fourier representation of the electronic and nuclear Coulomb interaction operators in the spectral development gives a formal separation theorem for adiabatic electronic potential energy surfaces. Atomic spectral response matrices arise in this way which are generalizations of (polarizability) response functions widely employed in studies of long-range interactions. An analysis shows the pairwise-additive approximation and familiar long-range limits are correctly contained in the formal development. Ab initio computational implementations are suggested on the basis of introduction of L2 Stieltjes representations of the required spectral states.19,40 These methods have been widely and successfully employed to represent attributes of the (discrete and continuum) spectra of quantum systems in finite spatial regions, particularly of transition charge densities of the type required in construction of response matrices, suggesting the spectral development may provide a useful ab initio method in this representation. A pair-based variant of the spectral development suggests the direct-product states also provide a convenient representational basis or scaffold in which to describe diatomic states and potentials obtained from ab initio calculations or experimental information. Transformations from diatomic to atomic product basis are accomplished in forms similar to those derived in Tully’s formulation of the diatomics method.16 Specifically, there is a two-step transformation from diatomic to atomicproduct representation in an intermediate (diatomic) coordinate system, followed by rotation of the atomic representation into the laboratory frame. As understood by Tully, the separation into diatomic to atomic-state projection followed by atomicstate rotation penetrates deeply into the nature of many-body atomic interactions and the formation of molecules and clusters of particular spatial symmetries. The spectral development provides a basis for ab initio construction of the important transformation from diatomic to atomic-product representation, which can be used to study the convergence and unitarity of the pair methods and furthermore demonstrates equivalences in appropriate limits between these latter methods and apparently dissimilar perturbation approaches to collisional-broadening and trapped-radical level-splitting calculations.21-27 Illustrative analytical and computational results useful for comparisons with related formalisms are reported for NaAr2 clusters in C2V symmetry. These results give potential energy surfaces for the four lowest-lying (2A1, 2B1, 2B2, 2A1) electronic states which are seen to include smoothly convergent nonpairwise-additive terms as a significant fraction of the total interaction energy. The nonadditive terms are strongly angledependent and are in addition to those that refer to and arise from the degeneracy lifting and splitting associated with the atomic 3P Na states in a perturbing environment.21-27 The large nonadditive contributions obtained can be attributed to the “near” 3S-3P degeneracy of the Na atom energy levels, which energy difference appears explicitly in a second-order approximation to the full matrix diagonalization. Since all the alkali metals exhibit such near S-P degeneracy, significant nonadditive effects may be a common occurrence in alkali metals trapped in rare-gas matrices more generally. Calculated Na-Ar radical distribution functions and optical absorption spectra for NaAr6 clusters at 10 K clearly exhibit the effects of the additional nonadditive contributions to the potential energy surfaces. Specifically, the Na-Ar radial distribution functions are seen to contract upon addition of nonadditive terms to the ground-
2984 J. Phys. Chem., Vol. 100, No. 8, 1996 state potential surface, whereas the calculated optical absorption spectra show significant spectral shifts having similar origins in the upper and lower potential surfaces. Acknowledgments. I thank Drs. M. E. Fajardo, J. A. Boatz, and J. A. Sheehy of Phillips Laboratory for very helpful assistance, commentary, and collaboration, AFOSR for support provided under the auspices of the Research Associates Program, and Phillips Laboratory, Edwards AFB, for support provided under contract with Hughes STX Corporation. Dr. Boatz kindly provided the computational results reported in Figures 1-4 and assisted in devising and verifying the various expressions employed in the calculations, and Dr. Sheehy assisted in preparation of the figures. References and Notes (1) Karplus, M. Rend. Sc. Int. Fis. “Enrico Fermi” 1970, 44, 320. (2) Casimir, H. B. G.; Polder, D. Phys. ReV. 1948, 73, 360. (3) Axilrod, B. M.; Teller, E. J. Chem. Phys. 1943, 11, 299. (4) Langhoff, P. W.; Karplus, M. Applications of Pade’ Approximants to Dispersion Force and Optical Polarizability Calculations. In The Pade’ Approximant in Theoretical Physics; Baker, G. A., Jr., Gammel, J. L., Ed.; Academic: New York, 1970; pp 41-95. (5) Langhoff, P. W. Chem. Phys. Lett. 1973, 20, 33. (6) Heisenberg, W.; Pauli, W. Z. Phys. 1929, 56, 1. (7) Dzialoshinski, I. E. J. Exp. Theor. Phys. 1956, 3, 977. (8) Berlin, T. J. Chem. Phys. 1951, 19, 208. (9) Slater, J. C. J. Chem. Phys. 1972, 57, 2389. (10) Langhoff, P. W. Spectral Theory of Weakly Bound Atomic Aggregates. Phillips Laboratory Report; Edwards AFB, CA, June 1995. (11) Yankony, D. R., Ed. Modern Electronic Structure Theory; World Scientific: London, 1995; Parts I & II. (12) Senatore, G.; March, N. H. ReV. Mod. Phys. 1994, 66, 445. (13) Eisenschitz, H.; London, F. Z. Phys. 1930, 60, 491. (14) Moffit, W. Proc. R. Soc. London 1951, A210, 245. (15) Ellison, F. O. J. Am. Chem. Soc. 1963, 85, 3540. (16) Tully, J. C. In Modern Theoretical Chemistry; Segal, G. A., Ed.; Plenum: New York, 1977; Vol. 7, pp 173-200. (17) Amos, A. T.; Musher, J. I. Chem. Phys. Lett. 1967, 1, 149. (18) Musher, J. I.; Amos, A. T. Phys. ReV. 1967, 164, 31. (19) Langhoff, P. W. Stieltjes Methods for Schro¨dinger Spectra. In Mathematical Frontiers in Computational Chemical Physics; Truhlar, D. G., Ed.; Springer: Berlin, 1988; pp 85-135. (20) Edmonds, A. R. Angular Momentum in Quantum Mechanics; Princeton University Press, Princeton, NJ, 1957.
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