Assessment of range-separated time-dependent density-functional theory for calculating C 6 dispersion coefficients

Transcription

1 1/10 Assessment of range-separated time-dependent density-functional theory for calculating C 6 dispersion coefficients Julien Toulouse 1,2, Elisa Rebolini 1, Tim Gould 3, John F. Dobson 3, Prasenjit Seal 4, János G. Ángyán 4 1 Laboratoire de Chimie Théorique, Université Pierre et Marie Curie / CNRS, Paris, France 2 Laboratoire de Chimie et Physique Quantiques, Université de Toulouse / CNRS, Toulouse, France 3 Nanoscale Science and Technology Centre, Griffith University, Nathan, Australia 4 CRM2, Institut Jean Barriol, Nancy University / CNRS, Vandoeuvre-lès-Nancy, France Web page: April 2013

2 Abstract We assess a variant of linear-response range-separated time-dependent density-functional theory (TDDFT), combining a long-range Hartree-Fock (HF) exchange kernel with a short-range adiabatic exchange-correlation kernel in the local-density approximation (LDA) for calculating isotropic C 6 dispersion coefficients of homodimers of a number of closed-shell atoms and small molecules. This range-separated TDDFT tends to give underestimated C 6 coefficients of small molecules with a mean absolute percentage error of about 5%, a slight improvement over standard TDDFT in the adiabatic LDA which tends to overestimate them with a mean absolute percentage error of 8%, but close to time-dependent Hartree-Fock which has a mean absolute percentage error of about 6%. These results thus show that introduction of long-range HF exchange in TDDFT has a small but beneficial impact on the values of C 6 coefficients. It also confirms that the present variant of range-separated TDDFT is a reasonably accurate method even using only a LDA-type density functional and without adding an explicit treatment of long-range correlation. J. Toulouse, E. Rebolini, T. Gould, J. F. Dobson, P. Seal, J. Ángyán, submitted to J. Chem. Phys. 2/10

3 C 6 dispersion coefficients The leading term in the expansion of the London dispersion interaction energy between a pair of atoms or molecules at long distance R is [1] E int = C 6 + (1) R6 where the isotropic C 6 dispersion coefficient between the subsystems A and B is given by the Casimir-Polder formula [2,3] C 6 = 3 π 0 duᾱ A (iu)ᾱ B (iu) (2) where ᾱ A (iu) and ᾱ B (iu) are the spherically averaged imaginary-frequency dynamic dipole polarizabilities of A and B, which can be expressed as ᾱ(iu) = f n ω 2 n n +u 2 (3) where the sum is over all excited states n, and f n and ω n are the dipole oscillator strength and the excitation energy for the transition to the excited state n. For closed-shell systems, only singlet singlet excitations contribute to ᾱ(iu). 3/10

4 TDDFT based on range-separated hybrid (RSH) scheme [4] For closed-shell systems, the singlet singlet excitation energies ω n are calculated in the basis of spatial RSH orbitals {φ k (r)} [5] from the familiar symmetric eigenvalue equation [6] MZ n = ω 2 n Z n (4) where M = (A B) 1/2 (A+B)(A B) 1/2 and Z n are normalized eigenvectors. The elements of the symmetric matrices A and B are A ia,jb = (ε a ε i )δ ij δ ab +2 aj ŵ ee ib aj ŵ lr ee bi +2 aj ˆf sr xc ib (5) B ia,jb = 2 ab ŵ ee ij ab ŵ lr ee ji +2 ab ˆf sr xc ij (6) where i,j and a,b refer to occupied and virtual RSH spatial orbitals, respectively, ε k is the orbital eigenvalue of orbital k, aj ŵ ee ib and aj ŵee lr bi are two-electron integrals associated with the Coulomb interaction w ee (r) = 1/r and the long-range interaction wee(r) lr = erf(µr)/r, respectively, and aj ˆf xc sr ib are the matrix elements of the short-range adiabatic exchange-correlation kernel fxc(r sr 1,r 2 ) = δ 2 Exc[n]/δn(r sr 1 )δn(r 2 ). 4/10

5 Oscillator strengths and sum rule In the limit of a complete one-electron basis set, the TRK sum rule is satisfied in TDHF, and in TDDFT with pure density functionals [6] and with hybrid approximations such as RSH. 5/10 The oscillator strengths f n are obtained from the eigenvectors Z n [6] f n = 4 ( ) 2 d T α (A B) 1/2 Z n (7) 3 α=x,y,z where the components of the transition dipole moment vector d α are d α,ia = φ i (r)r α φ a (r)dr. The exact (non-relativistic) oscillator strengths obey the well-known Thomas-Reiche-Kuhn (TRK) sum rule f n = N (8) n where the sum is over all transitions and N is the number of electrons. The TRK sum rule determines the asymptotic behavior of the dynamic polarizability at large imaginary frequency, u, ᾱ(iu) N u 2 (9)

6 Computational details 6/10 The TDRSH method has been implemented in a development version of MOLPRO [7]. We use the short-range LDA exchange-correlation density functional Exc,LDA[n] sr = n(r)ǫ sr xc,unif(n(r))dr (10) where ǫ sr xc,unif (n) is the complement short-range exchange-correlation energy per particle of a uniform electron gas with a modified electron-electron interaction, as parametrized from quantum Monte Carlo calculations by Paziani et al. [8]. We choose a fixed value for the range-separation parameter of µ = 0.5 bohr 1.

7 Dynamic polarizability of the Ne atom 7/10 Dynamic dipole polarizability ᾱ(iu) as a function of the imaginary frequency u, with an uncontracted d-aug-cc-pcv5z basis set: Dipole polarizability (a.u.) TDLDA TDRSHLDA TDHF Reference Imaginary frequency (a.u.)

8 C 6 dispersion coefficients of atoms C 6 coefficients (in a.u.) for homodimers of rare-gas and alkaline-earth-metal atoms, with uncontracted d-aug-cc-pcv5z basis sets: TDLDA TDRSHLDA TDHF Reference a He Ne (6) Ar (6) Kr (1) Be (3) Mg (12) Ca (35) a From Ref.[9], including estimated uncertainties in parentheses. = For rare-gas atoms, TDLDA systematically overestimates the C 6 coefficients, and TDRSHLDA and TDHF perform better. = For alkaline-earth-metal atoms, TDHF largely overestimates the C 6 coefficients, and TDLDA and TDRSHLDA are more accurate. 8/10

9 C 6 dispersion coefficients of molecules Isotropic C 6 coefficients of homodimers (table) and heterodimers (graph) for a subset of 27 organic and inorganic molecules from the database compiled by Tkatchenko and Scheffler [10], with d-aug-cc-vtz basis sets: TDLDA TDRSHLDA TDHF Reference a H HF H2O N CO NH CH HCl CO H2CO N2O C2H HBr H2S CH3OH SO C2H CH3NH SiH C2H Cl CH3CHO COS CH3OCH C3H CS CCl M%E 7.6% -3.8% -4.4% MA%E 8.0% 5.2% 6.3% a From experimental dipole oscillator strength distribution data. Calculated C 6 (a.u.) TDLDA 50 TDRSHLDA 0 TDHF Reference C 6 (a.u.) = TDLDA overestimates the C 6 coefficients by about 8%. = TDRSHLDA and TDHF give smaller C 6 coefficients which tend to be underestimated by about 5%. 9/10

10 References [1] F. London, Z. Physik 63, 245 (1930). [2] H. B. G. Casimir, D. Polder, Phys. Rev. 73, 360 (1948). [3] H. C. Longuet-Higgins, Discuss. Faraday Soc. 40, 7 (1965). [4] E. Rebolini, A. Savin, J. Toulouse, Mol. Phys., to appear. [5] J. G. Ángyán, I. C. Gerber, A. Savin, J. Toulouse, Phys. Rev. A 72, (2005). [6] M. E. Casida, in Recent Advances in Density Functional Methods, Part I, edited by D. P. Chong (World Scientific, Singapore, 1995). [7] MOLPRO, version , a package of ab initio programs, H.-J. Werner, P. J. Knowles, G. Knizia, F. R. Manby, M. Schütz, and others, Cardiff, UK, (2012). [8] S. Paziani, S. Moroni, P. Gori-Giorgi, G. B. Bachelet, Phys. Rev. B 73, , (2006). [9] A. Derevianko, S. G. Porsev, J. F. Babb, At. Data Nucl. Data Tables 96, 323 (2010). [10] A. Tkatchenko, M. Scheffler, Phys. Rev. Lett. 102, (2009). 10/10