Chapter 9 Long range perturbation theory

Transcription

1 Chapter 9 Long range perturbation theory 9.1 Induction energy Substitution of the expanded interaction operator leads at the first order to the multipole expression of the electrostatic interaction energy. At the second order, we have in the Cartesian tensor formulation E (2) (ind,b) = b 0 1 E B 0b 00 T ˆq Aˆq B + T α (ˆq Aˆµ B α ˆµ A α ˆq B ) + T αβ (ˆq A ˆΘB αβ + ˆΘ A αβ ˆq B ˆµ A α ˆµ B β ) b 0b T ˆq Aˆq B + T α (ˆq Aˆµ B α ˆµA α ˆqB ) + T α β (ˆqA ˆΘB α β + ˆΘ A α β ˆqB ˆµ A α ˆµB β ) (9.1) Performing the integrations on the A subsystem, we obtain the corresponding ground state multipole moments. Grouping the terms according to the multipole rank of the B subsystem, and taking into account that ψ B 0 ˆq B ψ B b = 0, E (2) (ind,b) = (T α q A T αβ µ A α +...) b 0 0 ˆµ B α b b ˆµ B α 0 (T E0b B α q A T α β µa α +...) (T αβ q A + T αβγ µ A γ +...) b 0 0 ˆΘ B αβ b b ˆµB α 0 (T α q A T α β µa β +...) E B 0b (T α q A T αβ µ A α +...) b 0 0 ˆµ B α b b ˆΘ B α β 0 (T α β qa + T α β γ µa γ +...) E B 0b (T αβ q A + T αβγ µ A γ +...) b 0 0 ˆΘ B αβ b b ˆΘ B α β 0 (T α β qa T α β γ µa γ +...)... E B 0b (9.2) 75

2 E (2) (ind,b) = 1 2 F A α α B α,α F A α 1 3 F A α A B α,α β F A α β 1 6 F A αβc B αβ,α β F A α β +... (9.5) Introduce the multipole polarizabilities α αβ = n 0 0 ˆµ α n n ˆµ β ˆµ β n n ˆµ α 0 A α,βγ = n 0 C αβ,γδ = 1 3 n 0 0 ˆµ α n n ˆΘ βγ ˆΘ βγ n n ˆµ α 0 0 ˆΘ αβ n n ˆΘ γδ ˆΘ γδ n n ˆΘ αβ 0 (9.3) and identify the parentheses as the electric field, field gradient, etc. F A α = (T α q A T αβ µ A α +...) F A αβ = (T αβ q A + T αβγ µ A γ +...) (9.4) The induction energy in terms of the field (and its gradients) and the multipolar polarizabilities In the spherical tensor formalism, the general polarizability component α lκ,l κ = n 0 and the induction energy takes the form E (2) (ind,b) = ˆQ lκ n n ˆQ l κ ˆQ l κ n n ˆQ lκ 0 (9.6) lκ l κ α B lκl κ V A lκv A l κ = 1 2 lκ Q B lκlv A lκ (9.7) where Q B lκl are the induced moments of B Q B lκ = l κ α B lκl κ V A l κ (9.8) The multipolar induction energy can be truncated according to the powers of (1/R) N does not ensure that the induction energy is negative according to the maximum rank of the multipole operators (L) always negative 76

3 9.2 Multipole polarizabilities Dipole polarizability is a symmetric tensor α xx α xy α xz α = α yy α yz (9.9) α zz Units are volume (Å 3 or bohr 3 ); the SI units are Fm 2. 1 a.u. = Å 3 = Fm 2. Proportional to the volume, as it can be seen by introducing an average excitation energy and using the resolution of identity α = n ˆr α n n ˆr α 0 2 U ( 0 ˆr 2 α 0 0 ˆr α 0 0 ˆr α 0 ) (9.10) and inversely proportional to the excitation energy. As any second-rank tensor, it can be decomposed into irreducible parts, according to α = α (0) + α (1) + α (2) (9.11) where α (0) αβ = 1 3 Tr(α)δ αβ α (1) αβ = 1 2 (α αβ α βα ) = 0 (9.12) α (2) αβ = 1 2 (α αβ + α βα ) 1 3 Tr(α)δ αβ α (1) vanishes, since α is symmetric. In a principal-axis system we have the decomposition α 0 0 α xx α 0 0 α = 0 α α yy α 0 (9.13) 0 0 α 0 0 α zz α The first trace-invariant quantity that can be formed is the mean polarizability α = 1 3 Trα = 1 3 (α xx + α yy + α zz ) (9.14) and the second trace-invariant is the polarizability anisotropy, γ, defined as the positive root of γ 2 = 1 2 [3Tr(α2 ) Tr(α) 2 ] (9.15) 77

4 or in components which can be written in a principal axis system γ 2 = 1 2 [3α αβα βα α αα α ββ ] (9.16) γ 2 = 1 2 [(α xx α yy ) 2 + (α yy α zz ) 2 + (α zz α xx ) 2 ] (9.17) and makes clear that this quantity vanishes for spherically symmetric systems. For linear molecules α zz = α and α xx = α yy = α and the polarizability anisotropy, γ γ = α α (9.18) For linear molecules the dipole-quadrupole polarizability can be written in terms of the unit vector along the symmetry axis, e = (e x, e y, e z ) as A αβγ = 1 2 A (3e α e β e γ e α δ βγ ) + A (e β δ αγ + e γ δ αβ 2e α e β e γ ) (9.19) where A = A zzz and A = A xxz = A yyz. For tetrahedral molecules the dipole-dipole polarizability is isotropic, the dipolequadrupole polarizability contains only one independent principal axis component, A = A xyz. This is the first anisotropic polarizability for tetrahedral symmetry Translational invariance The dipole-dipole polarizability is always origin-independent. Higher-rank multipole polarizabilities are (always) origin-dependent. For instance, the A z,zz component involves the ˆΘ zz operator. After a shift of the origin by c = (0, 0, c) ˆΘ C zz = ˆΘ O zz 2cˆµ O z (9.20) and the dipole-quadrupole polarizability at the new origin is A C z,zz = n 0 0 ˆµ O z n n ˆΘ O zz 2cˆµ O z 0 = A O z,zz 2cα zz (9.21) This means that the origin (and the dipole polarizability) should always be specified when A α,βγ is given. 78

5 9.2.2 Multipole polarizabilities spherical tensors The general multipole polarizabilities, defined in terms of the (irreducible) multipole moment operators, ˆQlm α l1 m 1,l 2 m 2 = n 0 0 ˆQ l1 m 1 n n ˆQ l2 m ˆQ l2 m 2 n n ˆQ l1 m 1 0 (9.22) is a reducible spherical tensor quantity, which can be decomposed into irreducible parts as α l1 l 2 :kq = m 1 m 2 C(l 1 l 2 k; m 1 m 2 q)α l1 m 1,l 2 m 2 (9.23) where C(l 1 l 2 k; m 1 m 2 q) are the Clebsch-Gordan coefficients. Since C = 0, unless k = l 1 +l 2, l 1 +l 2 1,..., l 1 l 2, the nonzero irreducible parts of the dipole-dipole polarizability can be only with k = 0, 2, for the dipole-quadrupole polarizability, k = 1, 3, etc. For instance, 1 α 11:00 = 3 (α xx + α yy + α zz ) 2 α 11:20 = 3 γ (9.24) 9.3 Distance dependence of the induction energy Interaction of a spherically symmetric polarizability α αβ = αδ αβ with multipoles: charge E(ind, q) = 1 2 qt αα αβ T β q = 1 2 q2 T α T α α = 1 2 q2 α αrα 2 R 3 R = 1 q 2 α 3 2 R 4 (9.25) dipole E(ind, µ) = 1 2 µ αt αβ α ββ T β α µ α = 1 µ 2 α(3 cos 2 θ + 1) (9.26) 2 R 6 quadrupole E(ind, Θ) R 8 (9.27) 79

6 9.4 Convergence of the multipolar induction energy The multipole-expanded interaction Hamiltonian of a H-atom + proton system is V mult = 1 [ ( ) n r 1 P n (cos θ)] (9.28) R R The exact second order multipole energy (Dalgarno & Lynn, 1957) The ratio of successive terms E (2) = n=1 n=0 (2n + 2)!(n + 2) n(n + 1)R (2n+2) (9.29) 2n(n + 3)(2n + 3) lim = (9.30) n (n + 2)R 2 indicates that for any value of R the series is divergent. 9.5 Dispersion energy Dispersion interaction energy in spherical tensor formalism E (2) (disp) = a 0 b 0 l A m A l B m B 00 ˆQ A l A m A T la m A,l B m B ˆQB lb m B ab ab ˆQ A l T A m l A A m ˆQ B A,l B m B l B B 00 m E0a A + E0b B Applying the Casimir-Polder method (9.31) E (2) (disp) = 2 π T l A m A,l B m B T l A m A,l B m B 0 ˆQ A l A m A a a ˆQ A l dω A A 0 ω m 0a 0 ˆQ B l B m B b b ˆQ B l B B 0 ω m 0b (9.32) ω0a 2 + ω 2 ω0b 2 + ω2 a 0 b 0 Dynamic (frequency-dependent) multipole polarizabilities in molecule-fixed frame α lm,l m (ω) = n 0 2ω 0n 0 ˆQ lm a a ˆQ l m 0 ω 2 0a ω 2 (9.33) related to the charge density susceptibility α lm,l m (ω) = drdr R lm (r)α(r, r ω)r l m (r ) (9.34) 80

7 Dispersion energy E (2) (disp) = T la m A,l B m B T l A m A,l B m B XAB l A m A l B m B l A m A,l B m B (9.35) The Casimir-Polder or dispersion integrals are defined as X AB l A m A l B m B l = dωα A A m A,l B l 2π A m A,l A m (iω)αb A l B m B,l (iω) (9.36) B m B The dispersion integrals can be calculated by numerical quadrature as X AB = M w(ω j )α A (iω j )α B (iω j ) (9.37) 2π j=1 With a Gauss-Chebyshev quadrature scheme the grid points are chosen as ( ω j = cot π 2j 1 ) 4M and the weights w(ω j ) = 2π 4M sin 2 (π(2j 1)/4M) (9.38) (9.39) Typically, with 5 and 7 grid points one has an accuracy of 0.1 %. Numerically more efficient procedure (for high-rank calculations) is to contract first the polarizabilities with the interaction tensors (in local frame) and perform the numerical integration afterwards (Hättig, 1996). In this form the interaction tensors and X AB are both reducible: they transform according to the double and quadruple product group of SO(3). In terms of irreducible tensors, the dispersion coefficients are of the form (Wormer) C L AL B L l A l A l Bl B = m A m A m B m B X AB l A m A l B m B l factor (9.40) A m A,l B m B The algebraic factor depends on the Wigner 3j and 9j coefficients. These coefficients are coupled to the following form C L AL B L n (n = l A + l B + l A + l B + 2) For atoms in S-state, for example, E (2) disp = C 2n R 2n n=3 n 2 C 2n = C(l; n k 1) l=1 C(l; l ) = (2l + 2l )! (2l)!(2l )! 1 2π dωα A l (iω)α B l (iω) (9.41) 81

8 The dispersion integral depends on the properties of the individual molecules and independent of the intermolecular geometry. The complete expression can be expressed in terms of irreducible spherical tensor components. dipole-dipole leading term R 3 R 3 R 6 dipole-quadrupole would be R 3 R 4 R 7, after averaging over all orientations, it gives zero quadrupole-quadrupole term R 4 R 4 R 8 dipole-octopole and quadrupole-qudrupole R Dipolar dispersion energy The leading multipolar term in the dispersion interaction energy E (2) (disp) = a 0 00 ˆµ A αt αβ ˆµ B β ab ab ˆµA γ T γδ ˆµ B δ 00 b 0 E A 0a + E B 0b (9.42) Casimir-Polder formula E (2) (disp) = T αβ T γδ 2 π dω a 0 0 ˆµ A α a a ˆµ A γ 0 ω 0a ω 2 0a + ω 2 0 ˆµ B β b b ˆµB δ 0 ω 0b ω0b 2 + ω2 (9.43) b 0 Unsöld approximation E (2) E (disp) = T αβ T 0a E A B 0 ˆµ A 0b α a a ˆµ A γ 0 0 ˆµ B β b b ˆµB δ 0 γδ E A a 0 b 0 0a + E0b B ω0a A ω0b B (9.44) In order to factor this latter expression, we use the identity with E0a E A 0b B E0a A + E0b B = U AU B U A + U B (1 + ab ) (9.45) ab = 1/U A 1/E A 0a + 1/U B 1/E B 0b 1/E A 0a + 1/E B 0b (9.46) that can be made negligibly small by choosing appropriate average excitation energies U A and U B E (2) (disp) U AU B 4(U A + U B ) T αβt γδ ααγα A βδ B (9.47) 82

9 In both expressions we can separate the orientation-independent spherically averaged and various orientation-dependent components, by using the decomposition of the polarizabilities to irreducible parts. The spherically averaged component becomes in both cases leading to the general expression with the C 6 coefficient C 6 = 3 T αβ T γδ α A δ αγ α B δ βδ = α A α B T αβ T αβ = α A α B 6 R 6 (9.48) C 6 E (2) (disp) C 6 R 6 (9.49) dωα A (iω)α B (iω) (Casimir-Polder) 3U AU B 2(U A + U B ) αa α B (London) (9.50) Experimental oscillator strength distributions can be used to determine experimental C 6 coefficients. Roughly proportional to the square of the polarizability/volume. Some typical values system C 6 C 8 C 10 H H He He Ne Ne Ar Ar Kr Kr Xe Xe 286 Combining rules can be deduced from the London-formula C AB 6 C AA 6 C BB 6 (9.51) Approximate dispersion formulae The average excitation energy of the London formula is an empirical parameter, either the first ionization potential (quite bad) or the first excitation energy (somewhat better) is used. 83

10 Similar expressions can be derived from the Casimir-Polder expression, by considering some general properties of the average dynamic dipole-dipole polarizabilities α(iω) = 1 k 0 2ω 0k 0 ˆx k 2 ω 2 0k + ω2 (9.52) We are looking for the simplest, one-term approximation in the form α(iω) ω 2 ω 2 + ω 2 a (9.53) Parameters a and ω will be found from the asymptotic behaviour of α(iω). For ω = 0 one gets the static polarizability, a = α(0). For ω one gets the Thomas-Reiche-Kuhn sum rule, the number of electrons α(iω) 1 2 E 2 ω 2 0n x 2 0n = n 0 n 2 ω 2 (9.54) In this limit we have a ω 2 /ω 2 = n 2 ω 2 ω 2 = n 2 α(0) ω = 1 n α(0) (9.55) which leads to the Mavroyannis-Stephen (Slater-Kirkwood) approximation C 6 3α A α B (α A /n A ) 1/2 + (α B /n B ) 1/2 (9.56) Take another, equivalent form of the dynamic polarizability α(ω) = α + (ω) + α + ( ω) = 1 3 n 0 r on 2 ω 0n + ω n 0 r on 2 ω 0n ω (9.57) For ω = 0 α + (ω) = 1 α(0) (9.58) 2 For ω ωα + (ω) 1 r 0n 2 = 1 0 r 3 3[ r 0 2 = 1 3 ( r)2 (9.59) n 0 84

11 Considering a one-term approximation α + (ω) = Taking the limiting cases one obtains a (ω + ω) a = 1 2 α(0) ω = 2 ( r) 2 3 α and we get the Salem-Tang-Karplus approximation C 6 = For dimers it is identical to the Alexander upper bound (9.60) (9.61) α A α B α A /( r A ) 2 + α B ( r B ) 2 (9.62) C S( 2)S( 1) = 1 2 α( r B) 2 (9.63) Solutions of the H + H dispersion problem The multipolar interaction Hamiltonian for two H atoms lying on the z axis, separated by a distance R 0 is ˆV = 2[R 3 0 ξ 1 ξ 2 cos θ 1 cos θ 2 + βξ 1 ξ 2 2 cos θ 1 (3 cos 2 θ 2 1) + γξ 2 1ξ 2 2(3 cos 2 θ 1 1)(3 cos 2 θ 2 1) +... (9.64) with polar coordinates ξ 1,2 = r 1,2, α = ( 6/2)R 3 0, β = ( (30)/4)R 4 0 and γ = ( 70/8)R 5. By direct summation over the excited states, Eisenchitz and London obtained for the dipolar dispersion energy E (2) = 12 z0n E 2 0m z0m 2 ) (9.65) R0 6 n,m (1 )(1 1n2 )(2 1m2 1n2 1m2 It is difficult to make converge this expression, because of the discrete-continuum matrix elements. The best value is C 6 = The difficulties of the sum-over-states solution can be avoided by solving directly for the first-order wave function using the Ansatz proposed by Slater and Kirkwood, ψ(r 1, r 2 ) = ψ 0 (r 1, r 2 )[1 + φ(r 1, r 2 )] (9.66) with the ground state unperturbed wave function of the dimer, ψ 0 (r 1, r 2 ). two-particle correlation function, φ, satisfies the differential equation The φ + ( ln ψ 0 ) φ) v = 0 (9.67) 85

12 where v can be one of the multipolar terms of the interaction Hamiltonian. the correlation function in the form leads to a differential equation for R(ξ, ξ ). Taking φ = vr(ξ, ξ ) E 0 (9.68) Slater and Kirkwood (1931) obtained an approximate solution, leading to C 6 = Pauling and Beach (1935) used special orbitals to construct the Hamiltonian matrix and obtained C 6 = , C 8 = and C 10 = Recent exact solutions obtained by Choy (P.R.A. 62 (2000) ) using orthogonal polynomials, confirm these values. 9.6 Convergence of the 1/R-expansion The 1/R expansion of the total interaction energy converges asymptotically to the ground state interaction energy. For each finite R, however, the 1/R expansion may diverge. It had been proven for H 2 molecule and H + 2 ion. The 1/R expansion of the second-order polarization energy for H + 2 ion (induction energy) diverges for all finite R. The 1/R expansion of the second-order polarization energy for H 2 molecule, i.e. its dispersion energy, diverges for all finite R. Given a finite one-particle GTF or STF basis set, the multipole expansion of each order of the polarization series converges, as long as the smallest spheres around the respective nuclei and basis function centers do not touch. However, the limit of this convergence may be quite different from the exact interaction energy. 86

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