Schrödinger Equation in Born-Oppenheimer Approximation (1)

Transcription

1 Schrödinger Equation in Born-Oppenheimer Approximation (1) Time-independent Schrödinger equation: Solution: Wave-function: (where x=(r,s)) Energy: E 2

2 Schrödinger Equation in Born-Oppenheimer Approximation (2) Knowledge of the wavefunction Ψ(x 1,x 2,..,x N ) is sufficient to obtain all observable quantities, electron density ρ(r) in particular: ρ(r)= dx 1 dx 2..dx N i N δ(r-r i ) Ψ(x 1,x 2,..,x N ) 2 3

3 Schrödinger Equation in Born-Oppenheimer Approximation (3) Variational principle: E 0 =min Ψ N H Ψ N Ψ N where, E 0 is ground-state energy and Ψ N is a trial N-electron wavefunction Variational principle, is the basis for a large group of methods to obtain an approximate solution of Schrödinger equation (HF, MCSCF, CI, for instance). 3

4 Constrained minimisation of E HK [ρ] for a given ρ B E emb [ρ B ] = min ρdr=n,ρ ρb E HK [ρ] or E emb [ρ B ] = min ρ Adr=NA EHK [ρ A + ρ B ] ρ A ρ B ρ electron density electron density total electron of the QM subsystem of the environment density (can be macroscopic!) (can be macroscopic!) 4

5 Definitions of density functionals through constrained search The minimiser (Ψo) is used to define the following density functionals By a functional, we mean a correspondence which assigns a definite (real) number to each function (or a curve) belonging to some class. [I.M. Gelfand and S.V. Fomin, "Calculus of Variations" Prentice-Hall, Inc. 1963, page 1] 5

6 The non-additive bi-functionals: For a given functional F[ρ], the corresponding non-additive bi-functional is: F nad [ρ A,ρ B ] = F[ρ A +ρ B ]- F[ρ A ] F[ρ B ] A given functional F[ρ] is additive if: F nad [ρ A,ρ B ] = F[ρ A +ρ B ]- F[ρ A ] Fρ B ] =0. Additive functionals: N[ρ], V ext [ρ] Non-additive functionals: J[ρ],T[ρ], E ex [ρ], T s [ρ], E HK [ρ] 6

7 Embedding potentials for variational QC methods Q1: Can conventional methods of Quantum Chemistry be modified by adding some local potential (embedding potential) to perform the constrained minimisation E emb = min ρdr=n,ρ ρb E HK [ρ]? Q2: Can such local potential be obtained as universal functional of ρ A and ρ B? [H 0 + V emb (r)]ψ emb = E emb Ψ emb 7

8 Embedding potentials for variational QC methods Q1: Can conventional methods of Quantum Chemistry be modified by adding some local potential (embedding potential) to perform the constrained minimisation E emb = min ρdr=n,ρ ρb E HK [ρ]? Q2: Can such local potential be obtained as universal functional of ρ A and ρ B? Answers: [H 0 + V emb (r)]ψ emb = E emb Ψ emb α) Φ s emb embedded non-interacting wavefunction (Kohn-Sham system) [Wesolowski & Warshel, J. Phys. Chem. 97 (1993) 8050] Φ s emb, Ψ emb, or Γ emb Auxiliary quantities used to access: β) Ψ s emb embedded interacting wavefunction [Wesolowski, Phys. Rev.A. 77 (2008) ] γ) Γ s emb embedded one-matrix [Pernal & Wesolowski, Intl. J. Quant. Chem.109 (2009) 2520] - the total energy (- energy of the frozen part) - properties directly related to the electronic structure 8

9 Embedding potentials for variational QC methods Embedding potential for methods such as HF, MCSF, CI (embedded «interacting wavefunction») [Wesolowski, Phys. Rev.A. 77 (2008) ] Embedding potential for a non-interacting reference system [Wesolowski & Warshel, J. Phys. Chem. 1993, 97, 8050] or for «one-matrix» [Pernal & Wesolowski, Intl. J. Quant. Chem.109 (2009)2520] 9

10 Embedding potentials for variational QC methods Embedding potential for methods such as HF, MCSF, CI (embedded «interacting wavefunction») PHYSICAL REVIEW A 77, Embedding a multideterminantal wave function in an orbital-free environment Tomasz A. Wesołowski Département de Chimie Physique, 30 quai Ernest-Ansermet, Université de Genève, CH-1211 Genève 4, Switzerland Received 3 October 2006; revised manuscript received 15 October 2007; published 11 January 2008 Variational methods to treat a many-electron system embedded in the environment, which is represented by means of only its electron density, are considered. It is shown that the embedding operator is a local potential in the case where the electron-electron repulsion is treated exactly and the trial embedded wave function takes the multideterminantal form with a fixed number of determinants. The local embedding potential is constructed by imposing that it leads to the same electron density as the one which minimizes the Hohenberg-Kohn functional. For the limiting cases of single-determinant and configuration interaction forms of the embedded wave function, the expressions for the local embedding potential using commonly known density functionals are given. The relation between the derived local embedding potential and the effective embedding potential in the case of the embedded Kohn-Sham system T. A. Wesołowski and A. Warshel, J. Phys. Chem. 97, is discussed in detail. Note that we take a particular perspective on the relation between the wave-function-based methods and densityfunctional theory. A multideterminantal wave function is considered in this work as an auxiliary quantity used to obtain the approximate solution of Eq. 1 and the corresponding electron density by means of variational calculations, whereas the relevant density functionals are considered to be exact in the derivation of the basic relation. functionals are considered only in the discussion 10

11 Embedded interacting electrons (1) construction of the embedding operator V emb!! " A WF (r) 11

12 Embedded interacting electrons (2) construction of the embedding operator V emb We are looking for a local potential V emb (r), such that E HK [! A WF [V emb ] +! B ] = min"! A dr=n A E HK [! A +! B ] for some arbitrarily chosen! B! 1) The definition of the functional # EWF [$ A,! B ]!! where $ A is of the multi-determinant form ($ A MD ).! 2) showing that 3)!extracting the form of V emb from the condition:! A MD =! WF [V emb ]! 12

13 Embedded interacting electrons (3) construction of the embedding operator V emb Depending on the number of determinants (MD) the following relation holds: where 13

14 Embedded interacting electrons (4) construction of the embedding operator V emb Only the first term depends explicitly on! A MD.! For! AO MD obtained from Euler-Lagrange equations, the sum of the first and the last term is equal to F HK [" A ]=T s [" A ]+J[" A ]+E xc [" A ].! Therefore: # EWF [! A0 MD," B ]= F HK [" A +" B ]+$v ext AB (r)(" A +" B )dr= E HK [" A +" B ]! 14

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16 Embedded interacting electrons (6) construction of the embedding operator V emb Euler-Lagrange equations to minimize 16

17 Embedded interacting electrons (7)! Euler-Lagrange equations to minimize 17

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19 Numerical examples for the embedded Hartree-Fock case: F. Aquilante & TAW, J. Chem. Phys., 135 (2011)

20 Embedding an «interacting wavefunction» The embedding operator is a local potential V emb (r) For a given external potential V emb (r) is a bi-functional of ρ A and ρ B (it is orbital-free therefore). The embedded wavefunction is an auxiliary quantity. Its relation to the exact wavefunction for the whole wavefunction is not direct. Depending on the form of the embedded wavefunction, the functional dependence of the potential V emb (r) on ρ A and ρ B might be different. V emb (r) and the potential V eff emb(ks) [ρ A,ρ B,r] for embedding a noninteracting system (Wesolowski-Warshel, 1993) are closely related. They are equal if the embedded wavefunction has the full CI form. The difference is bound from below by (-E c [ρ]) and from above (it is nonpositive). 20

21 Variation formulations of quantum many-body problem (SE in BO approximation) E o in N- electron system as a func3onal of: Search for the minimum among: What is approximated Difficulty in prac3ce E[Ψ] Ψ : wavefunc1ons nothing large space to search E[Γ 12 ] E[γ] E[{φ i }] Γ 12 : two- body reduced density matrices γ one- body density matrices {φ i } i=1,n one- electron orbitals nothing representability of Γ 12 E xc [γ] E xc [ρ] representability of γ AND approxima1on for the func1onal E xc [γ] approxima1ng the func1onal E xc [ρ] E[ρ] ρ: Electron density E xc [ρ] AND T s [ρ] approxima1ng the func1onals (especially T s [ρ]!) 1

22 Variation formulations of quantum many-body problem (SE in BO approximation) E o in N- electron system as a func3onal of: Search for the minimum among: What is approximated Difficulty in prac3ce E[Ψ] Ψ : wavefunc1ons nothing large space to search E[Γ 12 ] E[γ] E[{φ i }] Γ 12 : two- body reduced density matrices γ one- body density matrices {φ i } i=1,n one- electron orbitals nothing representability of Γ 12 E xc [γ] E xc [ρ] representability of γ AND approxima1on for the func1onal E xc [γ] approxima1ng the func1onal E xc [ρ] E[{φ i } A,ρ B ] {φ i } A i=1,n A and ρ B (only {φ i } A in embedding case) E xc [ρ A ], E xc nad [ρ A,ρ B ], T s nad [ρ A,ρ B ] approxima1ng the func1onals and bi- func1onals (in embedding case: admissibility of ρ B ) E[ρ] ρ: Electron density E xc [ρ] AND T s [ρ] approxima1ng the func1onals 1