LoProp: local property calculations with quantum chemical methods

Transcription

1 LoProp: local property calculations with quantum chemical methods Laura Gagliardi Dipartimento di Chimica Fisica F. Accascina Università degli Studi di Palermo Italy Gunnar Karlström, Jesper Krogh, and Roland Lindh Deptartment of Theoretical Chemistry University of Lund Sweden 1

2 Outline Background Method Results 2

3 Premises and Motivations A quantum chemical calculation gives an energy and a wave function: transform this information into a form that allows for comparison between different systems. Small systems: enough to calculate molecular properties like the dipole or quadrupole moment. Response properties characterized with a polarizability or hyperpolarizability. Large systems: a partitioning of the molecular charge distribution or the response property into local contributions is needed. 3

4 From Quantum Chemistry to Molecular Simulations Empirical potentials are used in most of existing molecular simulation packages. The accuracy is good for simulations on systems similar to those for which the potential parameters were fitted. Semiempirical potentials: the electrostatics from the molecular wf and the dispersion and repulsion parameters from empirical parameters. Quantum chemical methods can be used to calculate the intermolecular forces for the entire system at each step of the simulation. 4

5 The NEMO potential E = E el.stat. + E ind. + E disp. + E exc.rep. E el.stat. from a multi-center multipole expansion of the density of each monomer. E ind. by computing distributed polarizabilities in each molecule. E disp. from the interaction between the distributed polarizabilities (London force). E exc.rep. proportional to the square of the overlap; exponentially decaying function of the atom-atom distance. Fit the parameters of E exc.rep. to quantum chemical energies minus E el.stat., E ind. and E disp.. 5

6 Localization schemes Should converge with basis set saturation. Should be computational simple. Should give physically meaningful and transferable local properties. Should exactly represent molecular properties. Should allow comparision. Available methods: Mulliken, NAO, ESP, AIM, GAPT. 6

7 The new method LoProp Meets all 5 requirements Only constraint: the ground state valence orbitals of the atoms in the studied system are well described with the used basis set. The only parameter which effects the LoProp scheme is the classification of the basis functions into occupied and virtual. 7

8 Localization scheme T1 Gram-Schmidt orthonormalization of the atomic basis. T2 One Löwdin orthonormalization in the occupied and one in the virtual subspace. T3 Gram-Schmidt orthonormalization to project the occupied out of the virtual subspace. T4 Löwdin orthonormalization in the virtual subspace. Total transformation matrix T = T1 T2 T3 T4 8

9 9

10 Localized Properties < O >= T r(do) = µν D µν < µ O ν > (1) becomes < O >= T r(t 1 DT)(T 1 OT) (2) transform the integrals and 1-electron density matrix to the new basis and restrict the trace to the subspace of functions of a single center or the combination of two centers. 10

11 Localized Polarizabilities The localized polarizability is computed from the change of the localized dipole moments and the fluctuating charges due to the applied electric field. α (ij) κλ = µ(ij) κ (F + δ λ ) µ (ij) κ (F δ λ ) 2δ λ + ( Q(ij) (F + δ λ ) Q (ij) (F δ λ ))(R (i) R (j) ) κ 2δ λ 11

12 To proceed in our analysis and partition of the molecular polarizability we define Q (ij) s as the charge transfer from center j to center i as induced by the change of the electric field (note the permutational symmetry: Q (ij) = Q (ji) ). The charge transfer from center j to i as a function of the change of the applied electric field is not uniquely defined. The sum of the Q (ij) s for a fixed i gives q (ii) (F) = j i Q (ij) (F) where q (i) (F) = q (i) (F) q (i) (0) 12

13 A reasonable requirement is that the Q (ij) s are designed to be as small as possible and that the charge transfer is short range. For this purpose we construct the Lagrangian L = ij ( Q (ij) ) 2 f(r ij ) + i λ i ( j i Q (ij) q (ii) ) where the r ij, being the distance between center i and j, and f(r ij ) being a penalty function which will give the desired localized charge transfer. 13

14 In our investigations we have worked with two penalty functions, where n is an integer, and f(r ij ) = r n ij (3) f(r ij ) = Exp(α(r ij /(r BS i + r BS j )) 2 ) (4) where α is a constant and r BS is the Bragg-Slater radius of the respective atoms. 14

15 The Lagrangian leads to the equation system Aλ = q where the matrix elements of A are computed as A ij = 1 2f(r ij ) + C and A ii = j i 1 2f(r ij ) + C C is an arbitrary constant which is added to shift the eigen vector corresponding to i q(i) = 0. The equation system is now trivially solved by computing the inverse of the A matrix and the 15

16 Q (ij) are computed as Q (ij) = λ i λ j 2f(r ij ) 16

17 Water: Oxygen total charge LoProp 631g LoProp cc-pvnz LoProp ANO Mulliken 631g Mulliken cc-pvnz Mulliken ANO NBO 631g NBO cc-pvnz B1 B2 B3 B4 17

18 Formaldehyde: SCF Dipole Moment (a.u.) Table 1: ANOS=3s2p1d on C, O and 2s1p on H; ANOL=4s3p2d1f on C, O and 3s2p1d on H ANOS ANOL µ x µ y µ z µ x µ y µ z H ± ± C O HH HC ± ± HO ± ± CO SCF Total CASSCF Total CASPT2 Total Experiment: µ=

19 Formaldehyde: Polarizability (a.u.) Table 2: ANOS:=3s2p1d on C, O and 2s1p on H α xx α yy α zz α xy α yz α zx H ± C O HH HC ± HO ± CO Total SCF Total CASSCF Total CASPT ᾱ: Exper. 16.5; CASPT2 ANOS 14.41; ANOL

20 Eclipsed and Staggered Ethane Eclip. Stag. Eclip. Stag. Exp. Basis set ANO-small ANO-large q C q H µ C µ C1 C µ H µ C1 H ᾱ C ᾱ C1 C ᾱ H ᾱ C1 H ᾱ

21 Propane versus Butane Prop. But. Prop. But. µ C µ C1 C µ H µ C1 H µ H µ C1 H ᾱ C ᾱ C1 C ᾱ H ᾱ C1 H ᾱ H ᾱ C1 H ᾱ ᾱ exp

22 Benzene vs. Naphtalene 22

23 23

24 (% A 7 B> 1 24 %

25 <! > 1 = 25

26 A 7B>1 < 26 8

27 - A 7 B>1 27

28 LoProp In Summary Easy implementation and not CPU intensive. Correct basis set convergence, transferable properties. Useful to generate force fields for molecular simulations. 28