Intermolecular Forces and Generalized TDDFT Response Functions in Liouville Space 1 Introduction Consider two interacting systems with nonoverlapping charge distriutions. How can the properties of the comined system e expressed in terms of properties of the individual systems alone? This general prolem appears in a wide variety of physical, chemical and iological contexts [1 3]. In this chapter we will provide a prescription for resolving this issue y the computation of i) response functions and ii) correlation functions of spontaneous fluctuations, of relevant degrees of freedom in the individual systems. The computation of response and correlations is greatly simplified y using the density matrix in Liouville space[4]. Hilert and Liouville space offer very different languages for the description of nonlinear response. Computing dynamical oservales in terms of the wavefunction in Hilert space requires oth forward and ackward propagations in time. In contrast, the density matrix, calculated in Liouville space should only e propagated forward. The choice is etween following the ket only, moving it forward and ackward, or following the joint forward dynamics of the ket and the ra. Artificial time variales (Keldysh loops) commonly used in many-ody theory [5] are connected with the wavefunction. The density matrix which uses the real laoratory timescale throughout the calculation offers a more intuitive picture. Wavefunction-ased theories calculate transition amplitudes, which y themselves are not oservale. The density matrix on the other hand calculates physical oservales. Moreover, dephasing processes (damping of off-diagonal elements of density operator resulting from phase fluctuations) can only e descried in Liouville space. In this chapter we present a method for expressing the joint response of two interacting systems in terms of the correlations and response functions of the individual systems. This factorization appears quite naturally in Liouville space. The p th order response of the individual systems is a linear comination of 2 p distinct (p + 1)-point correlation functions known as Liouville space Contriuted y Shaul Mukamel 1, Adam E. Cohen 2 and Upendra Harola 1 1 Department of Chemistry, University of California, Irvine. 2 Department of Physics, Stanford University. To e pulished in TDDFT ook, Edited y M. Marques, F. Nogueira, A. Ruio and E. K. U. Gross (Springer-Verlag, 2005)
2 pathway [4], which differ y whether the interaction at each time is with the ra or the ket. The p th order contriution to the intermolecular interaction requires a different linear comination of these same Liouville space pathways of oth molecules. The 2 p Liouville space pathways are conveniently comined into p+1 generalized response functions (GRFs)[6,7]. One of the GRFs is the ordinary (causal) response function. The other GRFs represent spontaneous fluctuations, and the response of these fluctuations to a perturation, and are therefore non-causal. The complete set of GRFs is calculated using generalized TDDFT equations in Liouville space. A direct DFT simulation of molecular complexes y treating them as supermolecules is complicated ecause it requires nonlocal energy functionals[8 10]. The response approach makes good use of the perturative nature of the coupling and recasts the energies in terms of properties of individual molecules which, in turn, may e calculated using local functionals[11,12]. 2 Quantum Dynamics in Liouville Space; Superoperators In this section we introduce the notion of Liouville space superoperators and review some of their useful properties. A detailed discussion of superoperators is given in [14]. The elements of an N N density matrix in Hilert space are arranged as a vector of length N 2 in Liouville space. An operator in Liouville space is then a matrix of dimension N 2 N 2, and is called a superoperator. Two special superoperators, A L and A R, are associated with every Hilert space operator, A, and implement left and right multiplication on another operator X: A L X AX, A R X XA. These relations are not written as equalities ecause X is a vector in Liouville space and a matrix in Hilert space. It will e useful to define the symmetric A + 1 2 (A L + A R ) and antisymmetric A A L A R cominations. Hereafter we shall use Greek indices to denote superoperators A ν with ν = L, R or ν = +,. Recasting these definitions in Hilert space using ordinary operators we get A + X 1 2 (AX + XA); A X AX XA. (1) A product of ± superoperators constitutes a series of nested commutators and anticommutators in Hilert space. It is easy to verify that (AB) = A + B + A B +, (AB) + = A + B + + 1 4 A B. (2) We now consider products of superoperators that depend parametrically on time. We introduce a time ordering operator T in Liouville space, which orders all superoperators to its right such that time decreases from left to right. This is the natural time-ordering which follows chronologically the
Intermolecular forces and Generalized TDDFT... 3 various interactions with the density matrix [13]. We can freely commute operators following a T without worrying aout commutations ecause in the end the order will e fixed y T. The expectation value of any superoperator, A ν, is defined as, A ν (t) = Tr{A ν (t)ρ eq }, (3) where ρ eq is the equilirium density matrix. It is easy to see that for any two operators A and B, T A + (t)b (t ) = 0 if t > t. (4) T A + (t)b (t ) is thus a retarded (i.e. causal) function. Equation (4) follows from the definitions (1): A superoperator corresponds to a commutator in Hilert space, so for t < t, T A + (t)b (t ) ecomes a trace of a commutator which vanishes. Similarly, the trace of two operators vanishes: T A (t)b (t ) = 0 for all t and t. (5) We next introduce the interaction picture for superoperators. We partition the Hamiltonian, H = H 0 + H I, into a reference part, H 0, which can e diagonalized, and the remainder, interaction part, H I. We define a corresponding superoperator, H, as H = H 0 + H I. The time evolution of the density matrix is given y the Liouville equation: ρ t = ī h H ρ. (6) Equation (6) has the formal Green function solution, ρ(t) = G(t, t 0 )ρ(t 0 ). Note that the time evolution operator, G, acts only from the left, implying forward evolution of the density matrix. The total time evolution operator { G(t, t 0 ) = T exp ī t } dτh (τ), (7) h t 0 can e partitioned as: G(t, t 0 ) = G 0 (t, t 0 )G I (t, t 0 ) (8) where G 0 descries the time evolution with respect to H 0 { G 0 (t, t 0 ) = θ(t t 0 ) exp ī } h H 0 (t t 0 ), (9) and G I is the time evolution operator in the interaction picture { G I (t, t 0 ) = T exp ī t } dτ h H I (τ). (10) t 0
4 The time dependence of a superoperator in the interaction picture, denoted y a ( ) is defined as à ν (t) = G 0 (t, t 0)A ν (t 0 )G 0 (t, t 0 ). (11) The equilirium density matrix of the interacting system can e generated from the density matrix of the noninteracting system (ρ 0 ) y an adiaatic switching of the interaction H I, starting at time t = : ρ eq = G I (0, )ρ 0. In the wavefunction (Gell-Mann-Low) formulation of adiaatic switching, the wavefunction acquires a singular phase which must e cancelled y a denominator given y the closed loop S matrix[25]; this unphysical phase never shows up in Liouville space. For a set of operators {A i }, the p th order generalized response functions (GRF) are defined as R νp+1...ν1 i p+1...i 1 (t p+1... t 1 ) = ( ) p i T A ip+1ν h p+1 (t νp+1 )... A i1ν 1 (t 1 ) 0, (12) where... 0 represents a trace with respect to ρ 0, the indices ν n = + or, and p denotes the numer of indices in the set {ν p+1... ν 1 }. There are p + 1, p th order GRFs, having different numer of indices. Each memer of the p th order GRF represents to a different physical process. For example, there are two first order GRFs, R ++ i 2i 1 (t 2, t 1 ) = T A i2+(t 2 )A i1+(t 1 ) 0 R + i 2i 1 (t 2, t 1 ) = i h T A i 2+(t 2 )A i1 (t 1 ) 0. (13) Recasting them in Hilert space we have R + i 2i 1 (t 2, t 1 ) = i [ ] h θ(t 2 t 1 ) Tr{A i2 (t 2 )A i1 (t 1 )ρ 0 } Tr{A i2 (t 2 )A i1 (t 1 )ρ 0 } = h 1 θ(t 2 t 1 )ImJ(t 2, t 1 ) R ++ i 2i 1 (t 2, t 1 ) = Tr{A i1 (t 1 )A i2 (t 2 )ρ 0 } + Tr{A i2 (t 2 )A i1 (t 1 )ρ 0 } = ReJ(t 2, t 1 ) (14) where J(t 2, t 1 ) = Tr{A i2 (t 2 )A i1 (t 1 )ρ 0 }. With the factor h 1 the GRF R + has a well defined classical limit [14]. R + is causal [see (5)] and represents the response of the system at time t 2 to an external perturation acting at an earlier time t 1. On the other hand, R ++ is non-causal and denotes the correlation of A at two times. Each index corresponds to the interaction with an external perturation while a + index denotes an oservation. In general, time-ordered Liouville space correlation functions with one + and several indices, R +, R +, R +, etc. give response functions; all + correlation functions of the form R ++, R +++, R ++++, etc. give ground state fluctuations, wheras R ++, R ++, R +++, etc. represent changes in the fluctuations caused y an external perturation.
Intermolecular forces and Generalized TDDFT... 5 3 TDDFT Equations of Motion for the GRFs Time dependent density functional theory is ased on the effective one-ody Kohn-Sham (KS) Hamiltonian [15], H KS (n(r 1, t)) = h2 2 2m + U(r 1) + e 2 dr 2 n(r 2, t) r 1 r 2 + U xc(n(r 1, t)), (15) where the four terms represent the kinetic energy, the nuclear potential, the Hartree, and the exchange correlation potential, respectively. We now introduce the reduced single electron density matrix ˆρ [16 20] whose diagonal elements give the charge distriution, n(r 1, t) = ρ(r 1, r 1, t) and the off-diagonal elements, ρ(r 1, r 2 ), represent electronic coherences. Hereafter we shall denote matrices in real space such as ˆρ y a caret. We further denote the ground state density matrix y ˆρ g. The GRF corresponding to the charge density may e calculated y solving the time dependent generalized KS equation of motion for ˆρ[27,22], i h t δ ˆρ = [ĤKS (n), ˆρ(t)] + Ûket(t)ˆρ(t) ˆρ(t)Ûra(t), (16) where δρ(r 1, r 2, t) ρ(r 1, r 2, t) ρ g (r 1, r 2 ) is the change in the density matrix induced y the external potentials U ket and U ra. Equation (16) differs from the standard TDDFT equations in that the system is coupled to two external potentials, a left one U ket acting on the ket and a right one U ra acting on the ra. We next define linear cominations, U (r, t) 1 2 (U ket(r, t) + U ra (r, t)) U + (r, t) 1 2 (U ra(r, t) U ket (r, t)), (17) and the diagonal matrices U (r 1, r 2 ) = U (r 1 )δ(r 1 r 2 ), U + (r 1, r 2 ) = U + (r 1 )δ(r 1 r 2 ). Note that Û is given y a sum and Û+ y a difference. The reason for this somewhat confusing notation is that Û (Û+) will enter in a commutator (anticommutator) in equation of motion for ˆρ(t). δ ˆρ serves as a generating function for GRFs, which are otained y a perturative solution of (16) in U (r, t) and U + (r, t) using H 0 = H KS, as we shall shortly see. Equation (16) can e recast in terms of superoperators, where i h δ ˆρ = HKS t ˆρ(t) + U (t)ˆρ(t) U + (t)ˆρ(t) (18) U ˆρ [Û, ˆρ], U + ˆρ [Û+, ˆρ] +, H KS ˆρ [ĤKS, ˆρ]. (19)
6 The p th order GRFs χ νp+1...ν1, are computed as the kernels in a perturative expansion of the charge density fluctuation, δn(r 1, t)=δρ(r 1, r 1, t), in the applied potentials, U + and U. Adopting the areviated space-time notation x n (r n, t n ), we get δn + (x p+1 ) (p) dr p dt p... dr 1 dt 1 χ νp+1νp...ν1 (x p+1, x p,..., x 1 ) It follows from (16) and (20) that χ νp+1...ν1 (x p+1... x 1 ) = U νp (x p )U νp 1 (x p 1 )...U ν1 (x 1 ). (20) ( ) p i T δn νp+1 (x p+1 )... δn ν1 (x 1 ) (21) h where p denotes the numer of minus indices in the set {ν p+1... ν 1 }. To first order (p=1), we have χ ++ (x 1, x 2 ) = T δn + (x 1 )δn + (x 2 ) = θ(t 1 t 2 ) δn + (x 1 )δn + (x 2 ) + θ(t 2 t 1 ) δn + (x 2 )δn + (x 1 ) χ + (x 1, x 2 ) = i h T δn+ (x 1 )δn (x 2 ) = i h θ(t 1 t 2 ) δn + (x 1 )δn (x 2 ). (22) The standard TDDFT equations which only yield ordinary response functions are otained y setting U ket = U ra so that U + = 0 in (18). By allowing U ket and U ra to e different we can generate the complete set of GRF. The ordinary response function χ + characterizes the response of the density to an applied potential U [4]. Similarly, χ ++ can e formally otained as the response to the artificial external potential, U +, that couples to the charge density through an anticommutator. χ ++ represents equilirium charge fluctuations and is therefore non-retarded. van Leeuwen s resolution of the causality paradox of TDDFT [15] has een recently formulated [27] in Liouville space. 4 Collective Electronic Oscillator (CEO) Representation of the GRF Since the TDDFT density matrix, ˆρ(t), corresponds to a many-electron wavefunction given y a single Slater determinant at all times, it can e separated into its electron-hole (interand) part ˆξ and the electron-electron and holehole (intraand) components, T (ˆξ)[17,21]. δ ˆρ(t) = ˆξ(t) + ˆT (ˆξ(t)). (23)
Intermolecular forces and Generalized TDDFT... 7 It follows from the idempotent property, ˆρ 2 = ˆρ, that ˆT is uniquely determined y ˆξ so that δ ˆρ can e expressed solely in terms of ˆξ: [18,17] ˆT (ˆξ) = 1 } 2 (2ˆρg {Î Î) Î 4ˆξ ˆξ. (24) The elements of ˆξ (ut not of δ ˆρ) can thus e considered as independent coordinates for descriing the electronic structure. We next expand H KS in powers of δn(r, t): H KS = H0 KS + H1 KS + H2 KS +... H0 KS ( n(r 1 )) = h2 2 2m + U(r 1) + e 2 n(r 2, t) dr 2 r 1 r 2 + U xc( n(r 1, t)) ( H1 KS e 2 ) (δn(r 1 )) = dr 2 r 1 r 2 + f xc(r 1, r 2 ) δn(r 2, t), (25) with f xc the first order exchange correlation kernel in the adiaatic approximation where it is assumed to e frequency independent, f xc (r 1, r 2 ) = δu xc [n(r 1 )]. (26) δn(r 2 ) n The second order term in density fluctuations is, H2 KS (δn(r 1 )) = dr 2 dr 3 g xc (r 1, r 2, r 3 )δn(r 2, t)δn(r 3, t) (27) with the kernel(in adiaatic approximation), g xc (r 1, r 2, r 3 ) = δ 2 U xc [n(r 1 )]. (28) δn(r 2 )δn(r 3 ) n A quasiparticle algera can e developed for ˆξ y expanding it in the asis set of CEO modes, ˆξ α, which are the eigenvectors of the linearized TDDFT eigenvalue equation with eigenvalues Ω α [17,16]. ˆLˆξ α = Ω α ˆξα, (29) The linearized Liouville space operator, ˆL is otained y sustituting (25) into (16), ˆLˆξ α = [ĤKS 0 ( n), ˆξ α ] + [ĤKS 1 (ξ α ), ˆρ g ]. (30) Ĥ KS 0 and ĤKS 1 are diagonal matrices with matrix elements H0 KS ( n)(r 1, r 2 ) = δ(r 1 r 2 )H0 KS ( n)(r 1 ) ( H1 KS e 2 ) (ξ α )(r 1, r 2 ) = δ(r 1 r 2 ) dr 3 r 2 r 3 +f xc(r 2, r 3 ) ξ α (r 3, r 3 )(31)
8 The eigenmodes ˆξ α come in pairs represented y positive and negative values of α, and we adopt the notation, Ω α = Ω α and ˆξ α = ˆξ α. Each pair of modes represents a collective electronic oscillator (CEO) and the complete set of modes ˆξ α may e used to descrie all response and spontaneous charge fluctuation properties of the system. By expanding ˆξ(t) of the externally driven system in the CEO eigenmodes, ˆξ(t)= α z α(t)ˆξ α, where α runs over all modes (positive and negative) and z α are numerical coefficients, and sustituting in (24) and (23), we otain the following expansion for the density matrix δρ(r 1, r 2, t) = µ α (r 1, r 2 ) z α (t) + 1 µ α,β (r 1, r 2 ) z α (t) z β (t) +..., (32) 2 α where we have introduced the auxiliary quantities, ˆµ α = ˆξ α and ˆµ αβ = (2ˆρ g I)(ˆξ α ˆξβ +ˆξ β ˆξα ). By sustituting (23) and (24) in (16) we can derive equations of motion for the CEO amplitudes z α which can then e solved successively order y order in the external potentials, U ν1. To second order we get, i h d z α(t) dt α,β = Ω α z α (t) + K α (t) + β K αβ (t) z β (t), (33) with the coefficients, K α (t) = ν K αβ (t) = ν dr 1 U ν (r 1, t)µ ν α (r 1) dr 1 U ν (r 1, t)µ ν αβ (r 1). (34) Here µ α (r 1) µ α (r 1, r 1 ), µ αβ (r 1) ˆµ αβ (r 1, r 1 ), µ + α (r 1) µ α (r 1, r 1 ) = 1 2 (2ˆρg Î)ˆξ α (r 1, r 1 ) and µ + αβ (r 1) µ αβ (r 1 ) = 1 2 (2ˆρg Î)(ˆξ α ˆξβ ˆξ β ˆξα )(r 1, r 1 ). We further expand z α = zα ν1 + zα ν1ν2 +.., in powers of the external potentials, where z ν1ν2...νp denotes the n th order term in the potentials, U ν1 U ν2...u νp. By comparing the terms in oth sides, we otain equations of motion for zα ν1...νp for each order in external potential. To first order we get, i h dzν1 α (t) = Ω α zα ν1 dt (t) + K α(t) (35) The solution of (35) gives the generalized linear response functions χ ++ (r 1 t 1, r 2 t 2 ) = θ(t 1 t 2 ) α µ α (r 1 ) µ α (r 2 )e ī h Ωα(t1 t2) + θ(t 2 t 1 ) α µ α (r 2 ) µ α (r 1 )e ī h Ωα(t1 t2) χ + (r 1 t 1, r 2 t 2 ) = ī h θ(t 1 t 2 ) α s α µ α (r 1 )µ α (r 2 )e ī h Ωα(t1 t2) (36)
Intermolecular forces and Generalized TDDFT... 9 with s α = sign(α). Higher order GRF can e otained y repeating this procedure [22]. We further consider generalized susceptiilities defined y the Fourier transform of the response functions to the frequency domain, δn + (r p+1 ω p+1 ) (p) = dr p dω p... where the frequency transform is defined as, χ ν1ν2 (r 1 ω 1, r 2 ω 2 ) = dr 1 dω 1 U νp (r p ω p )...U ν1 (r 1 ω 1 ) χ νp+1...ν1 (r p+1 ω p+1, r p ω p,..., r 1, ω 1 ) (37) dt 1 dt 2 exp{i(ω 1 t 1 + ω 2 t 2 )}χ ν1ν2 (x 1, x 2 ). (38) Equation (38) together with (36) gives, χ ++ (r 1 ω 1, r 2 ω 2 ) = i hδ(ω 1 + ω 2 ) [ µα (r 1 ) µ α (r 2 ) ω α 2 Ω α + iɛ µ ] α(r 2 ) µ α (r 1 ) ω 2 + Ω α iɛ χ + (r 1 ω 1, r 2 ω 2 ) = δ(ω 1 + ω 2 ) s α µ α (r 1 )µ α (r 2 ). (39) ω α 2 + Ω α iɛ The CEO representations of the ordinary response functions to third order were given in [16] and the GRF to second order were given in [22]. The linear GRFs, χ ++ and χ +, are connected y the fluctuation dissipation relation, ( ) β hω χ ++ (r 1, ω; r 2, ω) = coth χ + (r 1, ω; r 2, ω). (40) 2 To linear order, the ordinary response function provides the complete information and GRF are not needed. However such fluctuation dissipation relations are not that simple for the higher order response functions[23] and the complete set of GRF are required to descrie the complete dynamics. 5 GRF Expressions for Intermolecular Interaction Energies We now show how the GRF may e used to compute the energy of two interacting systems a and with nonoverlapping charge distriutions. At time t = we take the density matrix to e a direct-product of the density matrices of the individual molecules. Thus initial density matrix at time t, ˆρ g 0, is given y, ˆρg 0 = ˆρa 0 ˆρ 0, where ˆρa 0 and ˆρ 0 represent the density matrices of the two molecules at t. The Liouville space time-evolution operator transforms this initial state into a correlated state. The GRF allow us to
10 factorize the time-evolution operator into a sum of terms that individually preserve the purity of the direct-product form. We start with the total hamiltonian of two interacting molecules H λ = H a +H +λh a, where H a and H represent the Hamiltonians for individual molecules and their coupling H a is multiplied y the control parameter λ, 0 λ 1, where λ=1 corresponds to the physical Hamiltonian. We shall follow the convention that primed and unprimed indices correspond to molecules a and, respectively. The charge densities of molecules a and at space points r and r will e denoted y n a (r) and n (r ), respectively. H a is the Coulom interaction H a = drdr J(r r )n a (r)n (r ) J(R k R k )Z k (R k )Z k (R k) k,k + P,k dr J(R k r )Z k (R k )n (r ) (41) where J(r r ) 1/ r r and R k (R k ) represents the position of kth (k th) nucleus in molecule a() with charge Z k (Z k ). P represents the sum over single permutation of primed and unprimed quantities together with indices a and. The interaction energy of the coupled system is otained y switching the parameter λ from 0 to 1 [10] W = 1 0 dλ H a λ. (42) Here... λ denotes the expectation value with respect to the λ-dependent ground state many-electron density matrix of the system, ˆρ λ. We next partition the charge densities of oth molecules as, n a (r)= n a (r) +δn a (r), n (r )= n (r ) +δn (r ), where n is the average density, n a (r) = ρ 0 a (r, r), and n (r ) = ρ 0 (r, r ). Thus the total interaction energy can e written as, W = W (0) +W (I) +W (II), where, W (0) = drdr J(r r ) n a (r) n (r ) J(R k R k )Z k (R k )Z k (R k) k,k + P,k dr J(R k r )Z k (R k ) n (r ), (43) is the average electrostatic energy, and the remaining two terms represent the effects of correlated fluctuations. 1 W (I) = dλ drdr J(r r ) [ n a (r) δn (r ) λ + n (r ) δn a (r) λ ], 0 + dr J(R k r )Z k (R k ) δn (r ) λ P,k W (II) = 1 0 dλ drdr J(r r ) δn a (r)δn (r ) λ. (44)
Intermolecular forces and Generalized TDDFT... 11 The expectation values δn a (r 1 ) λ and δn a (r 1 )δn (r 2) λ can e computed perturatively in λh a in the interaction picture. λh a is switched on adiaatically giving the interacting ground state density matrix in terms of the non-interacting one. Sustituting for H a from (41), and expanding in powers of λ yields a perturation series in terms of the p th order joint response function, using x n = (r n, t n ), R a (p) (x, x p, x p... x 1, x 1) = T δñ + a (x)[ñ a (x p )ñ (x p)]...[ñ a (x 1 )ñ (x 1)] 0. (45) Making use of (2) and the fact that the initial density matrix is a direct product of the density matrices of the individual molecules, R (p) can e factorized in terms of GRFs of the individual molecules. For example, the first order joint response function is: R (1) a (x, x 1, x 1) = T δñ + a (x)[ñ a (x 1 )ñ (x 1)] 0 = Tr { T δñ + a (x)[ñ a (x 1 ñ (x 1)] ˆρ 0 a ˆρ 0 }. (46) Sustituting ñ ν a (x)= nν a (x) +δñν a (x), ñν (x )= n ν (x ) +δñ ν (x ), and using the identities, δñ ν a (x) 0a = 0 and n+ a (x) 0a = n a(x), we otain R (1) a (x, x 1, x 1) = i n (r 1)χ + a (x, x 1 ), (47) where χ + a represents the linear order GRF for molecule a (see (21)). Similarly, second and higher order joint response functions can e expressed in terms of the GRFs of the individuals molecules. In the present work we have ignored the contriutions due to the nuclear interactions in (41). δn a (r 1 ) λ and δn a (r 1 )δn (r 1) λ, and consequently the interaction energies W (I) and W (II) can thus e expanded perturatively in terms of the GRFs of the individual molecules [22]. We shall collect terms in W (I) and W (II) y their order with respect to charge fluctuations. The total energy is then, W = j W (j), where W (j) represents contriution from jth order charge fluctuation. W (0) was given in (43) and W (1) = 0. W (j) to sixth order are given in [22]. W (2) = 1 2 W (3) = 1 6 P P dt 2 dt 2 dr 1 dr 2 n (r 1) n (r 2)χ + a (x 1, x 2 )J(s 1 )J(s 2 ) dt 3 dr 1 dr 2 dr 3 J(s 1 )J(s 2 )J(s 3 ) (48) n (r 1)χ + a (x 1, x 2, x 3 ) n (r 2) n (r 3) (49)
12 W (4) = 1 6 P dt 2 dt 3 [ n a (r 2 ) n (r 3)χ + a (x 1, x 3 )χ + (x 1, x 2) + n (r 1) n a (r 3 )χ + a (x 1, x 2 )χ + (x 2, x 3) ] 1 2 P dt 2 dr 1 dr 2 dr 3 J(s 1 )J(s 2 )J(s 3 ) dr 1 dr 2 J(s 1 )J(s 2 )χ ++ a (x 1, x 2 )χ + (x 1, x 2) (50) W (5) = 1 dt 2 dt 3 dr 1 dr 2 dr 3 J(s 1 )J(s 2 )J(s 3 ) 6 P { χ + (x 1, x 2, x 3) [ n a (r 1 )χ ++ a (x 2, x 3 ) + n a (r 2 )χ ++ a (x 1, x 3 ) + n a (r 3 )χ ++ a (x 1, x 2 ) + n a (r 1 )χ + a (x 1, x 3 ) ] + χ ++ (x 1, x 2, x 3)[ n a (r 2 )χ + a (x 1, x 3 ) + n (r 3)χ + (x 1, x 2)] W (6) = 1 dt 2 dt 3 6 P [ χ +++ a (x 1, x 2, x 3 )χ + (x 1, x 2, x 3) + χ ++ a (x 1, x 2, x 3 )χ ++ (x 1, x 2, x 3) } dr 1 dr 2 dr 3 J(s 1 )J(s 2 )J(s 3 ) (51) ], (52) where for revity we have used the notations, dr n = dr n dr n and J(s n ) = J(r n r n). We have now at hand all the ingredients necessary for computing the intermolecular energies. The TDDFT results for the GRF of individual molecules can e used to compute intermolecular interaction energy. The second term in (50) reproduces McLachlan s expression for the van der Waals intermolecular energy [12,24]. Since χ + and χ ++ are related y the fluctuation-dissipation theorem, the McLachlan expression may e recast solely in terms of the ordinary response of two molecules, χ + a and χ +. W (4) vdw = 1 2 h ( β hω coth 2 dω dr 1 dr 2 J(s 1 )J(s 2 ) ) α + a (r 1, r 2, ω)α + (r 1, r 2, ω) (53) where χ ν2ν1 (r 2 ω 2, r 1 ω 1 ) = ( h 1 ) p α ν2ν1 a (r 1, r 2, ω 1 )δ(ω 1 + ω 2 ). Equation (53) gives W (4) vdw = k BT n=0 dr 1 dr 2 α + a (r 1, r 2, iω n )α + (r 1, r 2, iω n )J(s 1 )J(s 2 ), (54)
Intermolecular forces and Generalized TDDFT... 13 where ω n = (2πnk B T/ h) are the Matsuara frequencies. However, life is not as simple for the higher order responses. The (p + 1), p th order generalized response functions, χ νp+1νp..ν1, cannot all e related to the fully retarded ordinary response, χ +.... The complete set of generalized response functions is thus required to represent the intermolecular forces. By comining (48)-(52) with the CEO expansion, we can finally express the intermolecular energies in terms of CEO modes of the separate molecules. For example, sustituting for χ + and χ ++ from (36) in (50), the fourth order term is otained in terms of CEO modes as W (4) = 1 2 h 1 6 h 2 dr 1 dr 2 r 1 r 1 r 2 r 2 P αα P αα s α s α µ α (r 3 )µ α (r 1)µ α (r 2) Ω α Ω α dr 1 dr 2 dr 3 r 1 r 1 r 2 r 2 r 3 r 3 n (r 3) s α µ α(r 1 )µ α (r 2 )µ α (r 1)µ α (r 2) (Ω α + Ω α ) [ n a (r 1 )µ α (r 2 ) + n a (r 2 )µ α (r 1 )]. (55) Expansion of higher order terms in CEO modes are given in [22]. The GRF therefore provide a compact and complete description of intermolecular interactions. Both response and correlation functions can e descried in terms of Liouville space pathways and can thus e treated along the same footing. Acknowledgment We wish to thank Prof. Vladimir Chernyak and Dr. Sergei Tretiak who were instrumental in developing the CEO approach. The support of the National Science Foundation (Grant No. CHE-9526125 ), NIRT (Grant No. EEC 0303389) and the National Institute of Health Grant N0. R)1GM59230-01A2 is gratefully acknowledged. References 1. D. Joester, E. Walter, M. Losson, R. Pugin, H. P. Merkle and F. Diederich, Angew. Chem. 115, 1524 (2003). 2. The Theory of Intermolecular Forces, A. J. Stone (Clarendon Press, Oxford, 1996) 3. See chapters A1.5, C1.3 and references therein in Encyclopedia of Chemical Physics and Physical Chemistry, Vol. I-III, Edited y: J. H. Moore, N. D. Spencer( 2001, Institute of Physics). 4. S. Mukamel, Principles of Nonlinear Optical Spectroscopy, (Oxford University Press, New York, 1995). 5. H. Haug and A.-P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors,( Springer, Germany 1996.)
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