Density matrix functional theory. Rob Klooster

Transcription

1 Density matrix functional theory Rob Klooster March 26, 2007

2 Summary In this report, we will give an overview of Density Matrix Functional Theory (DMFT). In the first part we will discuss the extended Hohenberg-Kohn theorem for non-local potentials, which claims a one-one mapping between the groundstate wavefunction and the reduced density one-matrix. The eigenequations for the natural orbitals are derived, which provide a means to apply DMFT in practice. The main part of the report consists of the discussion of several functionals which have been proposed to include electron correlation. Of the 5 discussed functionals (BB, GU, BBC1, BBC2 and BBC3), BBC3 performs best for the potential energy curves of small molecules. It is however not applicable to infinite systems, such as the homogeneous electron gas. A universal functional remains to be found. 1

3 Contents 1 Introduction 3 2 The formalism of DMFT The Hohenberg-Kohn theorem Generalization of the Hohenberg-Kohn theorem to nonlocal potentials Minimization of the energy functional Eigenproblem for the natural orbitals Density matrix functionals The Buijse-Baerends functional The Goedecker-Umrigar self-interaction correction The corrected Buijse-Baerends functionals DMFT calculations on the potential energy curves of diatomics Conclusions A Properties of the one-matrix 28 B Perturbation of the one-matrix eigenequation 32 2

4 Chapter 1 Introduction The description of electron correlation is one of the central problems in quantum chemistry. In Density Functional Theory (DFT), correlation effects are described as a functional of the one-particle electron density ρ(r). One of the drawbacks is that the kinetic energy is not known as a functional of the density. A way to circumvent this is to introduce the auxiliary Kohn-Sham system of non-interacting particles. In Density Matrix Functional Theory (DMFT) the main variable is the one-particle density matrix γ(r, r ). 1 4 An advantage of using the density matrix instead of the density is that the kinetic energy as a functional of the density matrix is known. Therefore there is no need to introduce an auxiliary system. Moreover, the unknown exchange-correlation part of the energy only has to describe electron-electron interactions, whereas in DFT also a kinetic part is included. In the first part of this report, we will provide the theory behind DMFT. The Hohenberg-Kohn theorem for non-local potentials is derived, 3 which proofs a one-one mapping between the groundstate wavefunction and the density matrix. This is the formal justification of the theory. The electronelectron interaction is a functional of the one-matrix, but in most approximations it has been given in terms of the eigenfunctions and eigenvalues of the one-matrix; the natural orbitals and occupation numbers respectively. The effective potential for the natural orbitals is then derived, 5 which provides a means to implement DMFT in a practical way. We will not consider the application to the homogeneous electron gas, 6, 7 the extension to open shells, 8 response properties 9 or time-dependent density matrix functional theory. 10 In the second part, we will discuss some of the functionals which have 3

5 been suggested to describe the electron-electron interaction. The functionals we discuss are all based on the Buijse-Baerends functional (BB). 11, 12 A self-interaction correction has been suggested by Goedecker and Umrigar (GU), 13, 14 while some other corrections have been suggested in the corrected BB functionals (BBC1, BBC2 and BBC3) by Gritsenko, Pernal and Baerends. 15 These are the only functionals which will be discussed in this report, but these are by no means the only functionals available The third part consists of some DMFT calculations on potential energy curves of diatomics. The results of the various functionals are compared to accurate Configuration Interaction (CI) calculations. For most systems, the BBC3 functional provides the best agreement with CI results, but there is still much room for improvement. 4

6 Chapter 2 The formalism of DMFT 2.1 The Hohenberg-Kohn theorem Hohenberg and Kohn (HK) proved that, for a system with a non-degenerate groundstate, there exists a one-one mapping of the groundstate 1-particle density ρ and the local external potential v(r). 27 They also proved that there is a one-one mapping of the external potential to the groundstate wavefunction, so the groundstate wavefunction can be written as a functional of the density. v Ψ ρ (2.1) This means that all groundstate properties, in particular the energy, can be expressed in terms of the density. The HK theorem states that there must exist an energy functional of the density, such that the minimum gives the groundstate energy if and only if the density is the real ground state density. The minimum can be found by utilizing a variational procedure, where the density can be varied under the constraint that it is N- and v-representable. The proof of the one-one correspondence between the groundstate density, the groundstate wavefunction and the local external potential goes as follows. The Hamiltonian of a system with local potential v(r) is Ĥ = ˆT + ˆV + Ŵ, (2.2) where the kinetic energy ˆT, the external potential ˆV and the electronelectron interaction Ŵ have been defined as ˆT = 1 2 i 2 ˆV = i i v(r i ) 5

7 Ŵ = 1. (2.3) r i<j ij The groundstate wavefunction Ψ 0 is the eigenfunction of this Hamiltonian with the lowest eigenvalue E 0. E 0 = Ψ 0 Ĥ Ψ 0 (2.4) We assume that the groundstate is non-degenerate. The electron density is defined as ρ(r) = d2... dn Ψ(rσ, 2,..., N) 2. (2.5) σ We use the compound index i = x i = r i σ i. It is clear that the density is a functional of the wavefunction, which in itself is a functional of the potential. However, at this point we can not say if these mappings are one-one or many-one. The proof for the one-one mapping between the external potential and the groundstate wavefunction is fairly straightforward. Consider two distinct potentials v and v (with associated Hamiltonians Ĥ and Ĥ ) which give the same groundstate wavefunction Ψ Substracting the two equations gives ĤΨ = EΨ Ĥ Ψ = E Ψ (2.6) (Ĥ Ĥ )Ψ = (E E )Ψ ( ˆV ˆV )Ψ = (E E )Ψ (2.7) Since ˆV is a multiplicative operator and Ψ can not be zero on a set of nonzero measure, we can devide by Ψ and we get for the potentials ˆV = ˆV + C (2.8) So, the potentials that give the same groundstate wavefunction can not differ by more than a constant. Now consider two systems with external potentials v(r) and v (r), which differ by more than a constant. And let us assume both systems give the same groundstate density ρ. The groundstate wavefunctions of the two systems, Ψ 0 and Ψ 0, will not be equal. Using the variational principle, we can write Ψ 0 Ĥ Ψ 0 = E < Ψ 0 Ĥ Ψ 0 = Ψ 0 Ĥ + v (r) v(r) Ψ 0 E < E + dr(v (r) v(r))ρ(r). (2.9) 6

8 Exchanging the primed and unprimed quantities, we get E < E + dr(v(r) v (r))ρ(r). (2.10) When we add the last two equations, we get the inequality E + E < E + E, (2.11) which is impossible. This means that the density can not be the same for two different external potentials (or groundstate wavefunctions), which completes the proof. 2.2 Generalization of the Hohenberg-Kohn theorem to nonlocal potentials Now we will extend the Hohenberg-Kohn theorem to nonlocal external potentials (v(x, x )). This was derived by Gilbert. 3 In contrast to local potentials, there is no one-one mapping between the nonlocal external potential and the groundstate wavefunction. The reason is that ˆV is not a simple multiplicative operator any more. However, this is of no consequence. There is still a one-one mapping between the groundstate wavefunction and the groundstate density one-matrix γ γ(1, 1 ) = N d2... dn Ψ(1, 2... N)Ψ (1, 2... N) (2.12) v(x, x ) Ψ γ, (2.13) which means that all groundstate properties are contained in the density one-matrix, even while there is not a distinct potential associated with them. From the expression of the one-matrix, it is clear that it is a functional of the wavefunction. We will now proof that the groundstate wavefunction is also a functional of the one-matrix. Consider two nonlocal external potentials v and v (or, equivalently, Hamiltonians Ĥ and Ĥ ) and assume that they give two distinct nondegenerate groundstate wavefunctions Ψ and Ψ. Using the variational principle, we can state that Ψ Ĥ Ψ Ψ Ĥ Ψ = E Ψ ˆV ˆV Ψ E > 0 Ψ Ĥ Ψ Ψ Ĥ Ψ = E + Ψ ˆV ˆV Ψ E > 0 (2.14) Adding these two equations and defining the difference of the potentials δv(x, x ) = v(x, x ) v (x, x ) gives Ψ ˆV ˆV Ψ Ψ ˆV ˆV Ψ > 0 7

9 dx δv(x, x )γ(x, x) x=x dx δv(x, x )γ (x, x) x=x > 0 dx δv(x, x ) [ γ(x, x) γ (x, x) ] x=x > 0 (2.15) It is immediately clear that the one-matrices of two different wavefunctions can never be the same, which proofs the one-one mapping. 2.3 Minimization of the energy functional The two-particle density matrix (two-matrix) is defined as Γ(1, 2, 1, 2 ) = N(N 1) d3... dn Ψ(1, 2, 3... N)Ψ (1, 2, 3... N). (2.16) The pair density can be obtained from Γ by setting 1 = 1 and 2 = 2. The groundstate energy can be expressed completely in terms of the groundstate density one-matrix and the groundstate pair density. E = d1 ( 1 ) v(1, 1 ) γ(1, 1 ) 1=1 + 1 d1 d2 2 Γ(1, 2) r 12 (2.17) The groundstate energy is a functional of the two-matrix, because both the one-matrix and the pair density can be obtained from this quantity. The reason why we do not use the two-matrix as the main variable, is because the constraints for N-representability are not (yet) known. Varying the density two-matrix without applying sufficient constraints can lead to a lower energy than the groundstate energy. The constraints for N-representability of the one-matrix are however known (see Appendix A). An important conclusion from the previous section is that the groundstate energy is a functional of the one-matrix. The first term of the expression of the energy is indeed clearly a functional of the one-matrix. The second term (the electron-electron interaction) should then also be a functional of the one-matrix. The exact form of the electronelectron interaction in terms of the one-matrix is not known and has to be approximated. This is the subject of the next chapter. 2.4 Eigenproblem for the natural orbitals The one-matrix can be expressed completely in terms of its eigenfunctions and eigenvalues γ(x, x ) = µ i ϕ i (x)ϕ i (x ). (2.18) i 8

10 The eigenfunctions {ϕ i } and eigenvalues {µ i } of the one-matrix are called the natural orbitals and occupation numbers, respectively. Instead of writing the energy as a functional of the one-matrix, we can also write it as a functional of the occupation numbers and natural orbitals. E[γ] = E[{ϕ i }, {µ i }] = i µ i h ii + V ee [{ϕ i }, {µ i }], (2.19) with {h ii } the matrix elements of the one-particle Hamiltonian ĥ = v(x). (2.20) The natural orbitals are functionals of the one-matrix up to a phase factor, which for each orbital we can choose arbitrarily. Therefore V ee should be invariant under changes in phase of ϕ i, i.e. V ee [ϕ i, ϕ i ] = V ee [e iφ i ϕ i, e iφ i ϕ i ] (2.21) 0 = δv ee δφ i φi =0 = dx δv ee δϕ i (x) = i dx δϕ i (x) δφ + dx δv ee δϕ i (x) i φi =0 δϕ i (x) δφ i δv ] ee δϕ i (x)ϕ i (x) [ δvee δϕ i (x) ϕ i(x) φi =0 (2.22) This relation will be useful later on. The energy has to be minimalized under the constraint that the onematrix is N-representable. The N-representability of the one-matrix can be assured when we keep the natural orbitals orthonormal, the occupation numbers sum up to the number of electrons and lie between 0 and 1. Using the Lagrange multiplier technique, we get the following functional ( ) Ω[{ϕ i }, {µ i }] = E[{ϕ i }, {µ i }] ν µ i N λ ij ( ϕ i ϕ j δ ij ). i ij (2.23) The constraint that 0 µ i 1 can be applied by defining µ i = cos 2 θ i and varying θ i freely (in practice, we only have to vary θ i between 0 and 1 2 π to access all possible µ i ). This functional should be stationary under variations 9

11 in {ϕ i }, {ϕ i } and {θ i}. When we vary {ϕ i }, we get the set of variational equations ( µ i ĥ + 1 ϕ i (x) ) δv ee [{ϕ i }, {µ i }] δϕ i (x) ϕ i (x) = j λ ij ϕ j (x). (2.24) We get an analogous set of equations when we vary {ϕ i }. Varying {θ i } gives ( sin(2θ i ) h ii + V ) ee[{ϕ i }, {µ i }] ν = 0. (2.25) µ i There are two solutions to this equation, the first is sin(2θ i ) = 0. (2.26) This is satisfied when θ i = 0 or 1 2 π, which corresponds to µ i = 0 or 1. The second solution is h ii + V ee[{ϕ i }, {µ i }] µ i = ν (2.27) This last constraint only applies to partially occupied orbitals, because for completely occupied or unoccupied orbitals equation (2.25) is already satisfied. 3 It has been claimed by Donnely and Parr 28 that the Lagrangian λ should be diagonal for the optimal set of natural orbitals and occupation numbers. However, Pernal showed that this is only true if V ee is an explicit functional of the one-matrix. 5 When V ee is defined in terms of the natural orbitals and occupation numbers, which is the case in almost all proposed functionals, the Lagrangian will not be diagonal. In this case, direct minimization of the functional Ω will not give satisfactory eigenequations for the natural orbitals. It would be useful to have an eigenequation for the natural orbitals of the form (ĥ + ˆv ee)ϕ i (x) = ɛ i ϕ i (x), (2.28) with some sort of effective potential ˆv ee, which is expected to be Hermitian. In general, this potential will be non-local ˆv ee ϕ i (x) = dx v ee (x, x )ϕ i (x ), (2.29) where the kernel v ee (x, x ) is Hermitian. 10

12 In the paper of Pernal, 5 the optimal potential is obtained in two different ways. It can be found by demanding that the energy functional is stationary with respect to variations in the effective potential. δe[{ϕ i [v ee ]}, {µ i }] δv ee (x, x ) = 0 (2.30) This is equivalent to the optimized effective potential (OEP) method for local potentials. 29 The derivatives of the natural orbitals with respect to v ee can be found using first order perturbation theory applied to equation (2.28). Perturbing v ee with δv ee yields δϕ i (x ) δv ee (x, x ) = ϕ j (x)ϕ i(x )ϕ j (x ). (2.31) ɛ j i i ɛ j We get an equivalent expression for the derivative of ϕ i (x ). Now we apply the chain rule to equation (2.30) at fixed occupation numbers ( δe[γ] dx δϕ i (x ) δϕ i i (x ) δv ee (x, x ) + δe[γ] δϕ ) i (x ) δϕ i (x ) δv ee (x, x = 0 ) ( δe[γ] dx ϕ j (x)ϕ i(x )ϕ j (x ) δϕ i j i i (x + ) ɛ i ɛ j δe[γ] ϕ j (x )ϕ ) i (x)ϕ j (x ) δϕ = 0 (2.32) i (x ) ɛ i ɛ j Combining this equation with equations (2.19) and (2.28) gives ϕ j (x)ϕ i(x [ ) (µ i µ j ) ϕ i v ee ϕ j + ɛ i j i i ɛ j ( dx δvee [γ] δϕ j (x ) ϕ i (x ) δv ) ] ee[γ] δϕ i (x ) ϕ j(x ) = 0 (2.33) Because the products ϕ j (x)ϕ i(x ) are linearly independent, we can obtain a solution for the off-diagonal matrix elements of the effective potential ( 1 δvee [γ] i j ϕ i v ee ϕ j = dx µ i µ j δϕ i (x) ϕ j(x) δv ) ee[γ] δϕ j (x)ϕ i (x) (2.34) This expression gives no information about the diagonal matrix elements of v ee or about the matrix element between orbitals with equal occupancy 11

13 (i.e. µ i = µ j ). Note that a possible degeneracy in {ɛ i } does not pose any problems in equation (2.33). When we look at the eigenequation of the natural orbitals (2.28), it can be seen that a different effective potential, ṽ ee (x, x ) = v ee (x, x ) + i λ i ϕ i (x)ϕ i (x ), (2.35) gives rise to the same natural orbitals for any {λ i }. The eigenvalues change however by ɛ i = ɛ i + λ i. So we can choose the diagonal matrix elements of the effective potential in such a way that any possible degeneracy is avoided. Pernal has shown that we get the same results when we use the relation which was formally shown by Gilbert 3 v ee (x, x ) = δv ee[γ] δγ(x, x). (2.36) If V ee is an explicit functional of the one-matrix, then the functional derivative can be directly calculated. If it is an implicit functional of the onematrix, via the natural orbitals and occupation numbers, the effective potential can still be calculated using the chain rule. The obtained kernel will however not satisfy the formal identity δv ee [γ] δϕ i (x) = µ i dx v ee (x, x )ϕ i (x ), (2.37) contrary to what Gilbert assumed. The derivatives of the natural orbitals and occupation numbers with respect to the one-matrix are obtained using first-order perturbation theory applied to the eigenequation of the onematrix. The derivation is available in Appendix B. We get for the derivative of the natural orbitals and their complex conjugates δϕ i (x) δγ(x, x ) = j i δϕ i (x) δγ(x, x ) = j i ϕ j (x )ϕ i (x )ϕ j (x) µ i µ j (2.38) ϕ j (x )ϕ i (x )ϕ j (x) µ i µ j. (2.39) The derivatives of the occupation numbers are given by δµ i δγ(x, x ) = ϕ i (x )ϕ i (x ) (2.40) 12

14 Using the chain rule, we can now construct an expression for the effective potential. v ee (x, x ) = ϕ i (x)ϕ j (x [ ) dx δvee µ i i µ j δϕ i (x ) ϕ j(x ) j i δv ee δϕ j (x ) ϕ i (x ) ] + i ϕ i (x)ϕ i (x ) δv ee δµ i (2.41) The off-diagonal matrix elements of this potential are identical to the effective potential derived in the OEP-like scheme (2.34), which justifies this method of finding the effective potential. The eigenequation (2.28) and the effective potential (2.41) determine the natural orbitals. The eigenvalues can be found by multiplying the eigenequation by ϕ k (x) and integrating ɛ k = h kk + ϕ k v ee ϕ k = h kk + δv ee δµ k = ν k, µ k 0, 1, (2.42) where we used equation (2.27). This implies that all partially occupied orbitals (in general an infinite number) must be degenerate with eigenvalue ν. This has an important consequence for the effective potential of equation (2.41). We define the (off-diagonal elements of the) matrix ξ by [ 1 δvee ξ ij = dx µ i µ j δϕ i (x) ϕ j(x) δv ] ee δϕ j (x)ϕ i (x) (2.43) ξ ij = ϕ i v ee ϕ j, i j, (2.44) so that we can write the effective potential as v ee (x, x ) = i ϕ i (x)ϕ i (x ) δv ee δµ i + i,j i ϕ i (x)ϕ j(x )ξ ij. (2.45) Assuming that all orbitals are partially occupied, we can use equation (2.27) to rewrite the first term of the previous equation i δv ee δµ i = ν h ii ϕ i (x)ϕ i (x ) δv ee δµ i = νδ(x x ) i ϕ i (x)ϕ i (x )h ii. (2.46) 13

15 The one-particle hamiltonian can be expressed as a sum of diagonal and off-diagonal matrix elements in the following way ĥ = ij = i h ij ϕ i (x)ϕ j(x ) h ii ϕ i (x)ϕ i (x ) + h ij ϕ i (x)ϕ j(x ). (2.47) i,j i Combining the last two equations with equation(2.45) yields v ee (x, x ) = νδ(x x ) ĥ + ϕ i (x)ϕ j(x )(h ij + ξ ij ) (2.48) i,j i From the eigenequation (2.28) and using equation (2.44), it can be seen that the last term is zero ϕ i ĥ + ˆv ee ϕ j = ɛ i ϕ i ϕ j h ij + ξ ij = ɛ i δ ij = 0 i,j i (2.49) The paradoxal result follows from equation (2.48) that the operator in the eigenequation is in fact a delta function ĥ + ˆv ee = νδ(x x ). (2.50) All possible functions are eigenfunctions of this operator with eigenvalue ν. One would expect that this form of the effective potential is of no use, because it can not be used to find the natural orbitals. However, the effective potential only has this delta function form when we use the converged natural orbitals. In the SCF procedure, the natural orbitals will converge to the correct solution and at the same time the potential will converge to the delta function. This means that the potential v ee has a very special dependance on the natural orbitals and occupation numbers, such that v ee (x, x )[{ϕ i }, {µ i }] = νδ(x x ) ĥ (2.51) whenever {ϕ i } and {µ i } are the correct solutions that make the functional (2.19) stationary. 14

16 Chapter 3 Density matrix functionals This chapter will summarize some of the approximations which have been made to describe the electron-electron interaction in terms of the one-matrix. 3.1 The Buijse-Baerends functional In his thesis, 11 Buijse presents an approximation to the pair density in terms of the natural orbitals. The pair density can, in general, be partitioned in one of the following ways. Γ(1, 2) = ρ(1)ρ(2) + Γ XC (1, 2) (3.1a) = ρ(1)ρ cond (2 1) (3.1b) ( ) = ρ(1) ρ(2) + ρ hole (2 1) (3.1c) The exchange-correlation part of the pair density is contained in Γ XC (1, 2). The conditional density ρ cond (2 1) is the density of the (N 1) remaining electrons when electron 1 is located at r 1 with spin σ 1. It depends parametrically on the coordinates of electron 1. The hole density ρ hole (2 1) is defined as the difference between the conditional density and ρ(2). Defined in this way, the hole density integrates to -1 electron. We are only interested in closed-shell systems. In the Hartree-Fock approximation, the hole density only incorporates exchange-type interactions, which are a consequence of the Fermi exclusion principle. It is therefore called the Fermi hole. It can be written as the square of an amplitude. ρ Fermi (r 2 r 1 ) = ϕ Fermi (r 2 r 1 ) 2 (3.2) ϕ Fermi (r 2 r 1 ) = N/2 ϕ i 2 (r 1) i=1 ρ(r1 ) ϕ i(r 2 ) (3.3) 15

17 So the Fermi hole amplitude is a linear combination of occupied orbitals. The coefficients depend only on the orbital value and density in the reference point r 1. The difference between the total (full CI) hole and the Fermi hole is defined as the Coulomb hole ρ hole (r 2 r 1 ). ρ hole (r 2 r 1 ) = ρ hole (r 2 r 1 ) ρ Fermi (r 2 r 1 ) (3.4) The Coulomb hole describes the correlation effects which are not included in the HF approximation. Neglecting this quantity leads for example to the incorrect dissociation limit of H 2 and a number of other problems, especially in highly correlated materials. The Coulomb hole in terms of the XC part of the pair density is ρ hole (r 2 r 1 ) = ΓCI XC (r 1, r 2 ) ρ CI (r 1 ) ΓHF XC (r 1, r 2 ) ρ HF (r 1 ) (3.5) This shows that the definition of the Coulomb hole (3.4) is not quite satisfactory, because the total hole density and the Fermi hole have different densities associated with them. Therefore, the correlation effects it describes are not purely electron-electron interactions; it also includes some one-electron contributions. In the Levy-Lieb partitioning of the pair density, a HF-like pair density is used as a zeroth order approximation. Γ 0 (r 1, r 2 ) = ρ(r 1 )ρ(r 2 ) 1 2 γ(r 2, r 1 )γ(r 1, r 2 ) (3.6) This is essentially the HF pair density, except that we use the correlated density and one-matrix. The total correlated pair density can then be partitioned in the zeroth order approximation and an unknown rest term. The energy can be partitioned as Γ(r 1, r 2 ) = Γ 0 (r 1, r 2 ) + Γ rest (r 1, r 2 ) (3.7) E = E 0 + E rest E 0 = E (1) [γ] + E (2) [Γ 0 ] E rest = 1 Γ rest (r 1, r 2 ) dr 1 dr 2 (3.8) 2 r 12 The one-electron energy E (1) is a functional of the one-matrix and is exact when the one-matrix is the exact correlated one-matrix. Therefore the rest 16

18 energy E rest is only a correction for the electron-electron interaction. Lieb 30 proved the relation E 0 E HF, where the equality only holds if γ = γ HF. This means that minimization of E 0 by varying the orbitals and occupation numbers will lead to the HF result. Therefore, Γ 0 may be regarded as a useful zeroth order, uncorrelated reference pair density. Correlation in a two electron system An approximate expression for the hole density can be derived for twoelectron systems in terms of the natural orbitals and occupation numbers. Consider the closed shell two-electron wavefunction in natural form 31 Ψ(1, 2) = i c i ϕ i (1)ϕ i (2). (3.9) The spin-integrated pair density is Γ(r 1, r 2 ) = 2 ij c i c j ϕ i (r 1 )ϕ j (r 1 )ϕ i (r 2 )ϕ j(r 2 ) (3.10) and the spin-integrated one-matrix is given by γ(r 1, r 1) = i µ i ϕ i (r 1 )ϕ i (r 1) with µ i = 2c 2 i. (3.11) The conditional density (3.1b) can be calculated from the expression for the pair density. ρ cond (r 2 r 1 ) = Γ(r 1, r 2 ) ρ(r 1 ) = 2ci ϕ i (r 1) ϕ i (r 2 ) i ρ(r1 ) j 2cj ϕ j (r 1 ) ϕ j(r 2 ) ρ(r1 ) = ϕ cond (r 2 r 1 ) 2 (3.12) So the conditional density can be written as the square of an amplitude, much in the same way as the Fermi hole. The hole density for the twoelectron system is ρ hole (r 2 r 1 ) = ρ cond (r 2 r 1 ) ρ(r 2 ) = 2c i c j ϕ i (r 1)ϕ j (r 1 ) ϕ i (r 2 )ϕ ρ(r ij 1 ) j(r 2 ) i 2c 2 i ϕ i (r 1 )ϕ i (r 1 ) (3.13) 17

19 It should be noted that ρ hole can not be calculated from one-matrix information alone; the phase of the expansion coefficients c i = ± µ i /2 can not be deduced. The hole density can not be expressed exactly as a square of an amplitude, but it can be approximated by one. We use the approximation that the expansion coefficients for i > 1 are small, so we can neglect products of expansion coefficients c i c j = 0 for i, j > 1. Another approximation is that if we choose the dominant expansion coefficient c 1 to be positive, the small coefficients will be negative, i.e. c i = µ i /2 for i > 1. 32, 33 The final approximation is that we choose c 1 1, so that ϕ 1 (r 1)ϕ 1 (r 1 )/ρ(r 1 ) 1 2. It can then be shown 11 that the hole density can be approximated by the square of the hole amplitude ρ hole (r 2 r 1 ) ϕ hole (r 2 r 1 ) 2 ϕ hole (r 2 r 1 ) = µi ϕ i (r 1) ϕ i (r 2 ) (3.14) i ρ(r1 ) The hole amplitude is equal to the Fermi hole amplitude (3.3) when the first N/2 occupation numbers are equal to 2 and the rest equal to 0. Correlation in many-electron systems The results for the two-electron system can be generalised to many-electron systems. The assumption is made that, also for the many-electron case, the hole density can be approximated by the square of a hole amplitude. When the constraint that the approximate two-matrix should integrate to the correct one-matrix is applied, it follows that the hole amplitude is in fact the same as in the two-electron case (3.14). 12 We then get an expression for the trial pair density in terms of natural orbitals and occupation numbers only. Γ BB (r 1, r 2 ) = ρ(r 1 )ρ(r 2 ) ij µi µ j ϕ i (r 1 )ϕ j (r 1 )ϕ i (r 2 )ϕ j(r 2 ) (3.15) The same result was earlier found by Müller 34 in a different way. He considered an ansatz for Γ XC, Γ XC (1, 2) = ij µ 1/2+p i µ 1/2 p j ϕ i (1)ϕ j (1)ϕ i (2)ϕ j(2). (3.16) He optimized the value of p and found that it must be equal to 0, which corresponds to the Buijse-Baerends result. 18

20 3.2 The Goedecker-Umrigar self-interaction correction Calculations with the Buijse-Baerends functional show that in general it overestimates the absolute value of the energy. Goedecker and Umrigar proposed a functional which is essentially the same as the BB functional, except with a self-interaction correction. 13, 14 Γ GU (r 1, r 2 ) = [ µ i µ j ϕ i (r 1 ) 2 ϕ j (r 2 ) 2 i j i ] µ i µ j ϕ i (r 1 )ϕ j (r 1 )ϕ i (r 2 )ϕ j(r 2 ) (3.17) The orbital self-interaction terms, which lead to a lower energy, are removed by limiting the sums to contain only terms with i j. Notice that, while this functional is orbital self-interaction free, it is not electron self-interaction free, because fractional occupation numbers are allowed. Therefore it still does not give the correct energy for the hydrogen atom, the error being in the order of the LDA error. A consequence of removing self-interactions is that this functional does not satisfy the sum rules any more dr 2 Γ(r 1, r 2 ) = (N 1)ρ(r 1 ) (3.18) dr 1 dr 2 Γ(r 1, r 2 ) = N(N 1) (3.19) The sum rules are violated by terms of the order of µ i (1 µ i ). In most systems, where the occupation numbers are either close to zero or one, these terms are small. 3.3 The corrected Buijse-Baerends functionals The overestimation of the energy with the Buijse-Baerends functional is discussed by Gritsenko et al. 15 and some corrections are proposed. To locate the source of the overestimation, we go back to the two-electron case. The exact pair density of this system (3.10) can be split into diagonal terms, products of the stongly occupied orbital ϕ 1 with weakly occupied orbitals and products of weakly occupied orbitals. [ Γ(r 1, r 2 ) = i µ i ϕ i (r 1 ) 2 ϕ i (r 2 ) 2 + a>1 µ1 µ a ϕ 1(r 1 )ϕ a (r 1 ) 19

21 ϕ 1 (r 2 )ϕ a(r 2 ) + µa µ b ϕ a(r 1 )ϕ b (r 1 ) a>b>1 ] ϕ a (r 2 )ϕ b(r 2 ) + c.c. (3.20) Remember that the phase of the occupation number of ϕ 1 is chosen to be positive, and that we assume that all other occupation numbers are negative. When we compare this to the BB approximation of the pair density (3.15), it is clear that the BB functional incorporates in fact two approximations. The first sum in the previous expression is approximated by the product of the densities of electron 1 and 2 plus the diagonal terms of the sum in (3.15). µ i ϕ i (r 1 ) 2 ϕ i (r 2 ) 2 ρ(r 1 )ρ(r 2 ) µ i ϕ i (r 1 ) 2 ϕ i (r 2 ) 2 (3.21) i i This is only exactly true in two limiting cases: when µ 1 = 2 and µ i>1 = 0 or when µ 1 = µ 2 = 1 and µ i>2 = 0. The second approximation is that the sign of the products of weakly occupied orbitals should be positive, while it is negative in the BB functional. In the article of Gritsenko, Pernal and Baerends, 15 three repulsive corrections are suggested for the BB functional. The first corrected functional (BBC1) restores the sign for the products of weakly occupied orbitals. Γ BBC1 (r 1, r 2 ) = Γ BB (r 1, r 2 ) + Γ C1 (r 1, r 2 ), (3.22) where the correction C1 is given by Γ C1 (r 1, r 2 ) = 2 µa µ b ϕ a(r 1 )ϕ b (r 1 )ϕ a (r 2 )ϕ b(r 2 ). (3.23) a,b a>n/2 It is not possible to write the hole density of the BBC1 functional as the square of a hole amplitude, but there is no physical reason to stick to that approximation. This functional does have the correct permutation symmetry and integrates to the correct electron density. The second correction is applied on top of the first. It corrects the XC interaction between different strongly occupied orbitals. In the BB functional, this interaction between orbitals i and j is given by the term 2 1 µi µ j dr1 dr 2 ϕ i (r 1)ϕ j (r 1 )ϕ i (r 2 )ϕ j (r 2)/r 12 = 2 1 µi µ j K ij. However, when we look at the Levy-Lieb partitioning of the two-matrix (3.6), the exact exchange part of the energy will be 1 4 µ iµ j dr1 dr 2 ϕ i (r 1)ϕ j (r 1 )ϕ i (r 2 ) ϕ j (r 2)/r 12 = 1 4 µ iµ j K ij. Both expressions are equal to the Hartree-Fock 20

22 exchange interaction K ij in the limit µ i = µ j = 2. For fractional occupation numbers, the exact exchange interaction energy is less negative than the corresponding term in the BB functional. Since the BB functional gives in general too low energies, the XC interaction between strongly occupied orbitals will be replaced by exact exchange. Γ BBC2 (r 1, r 2 ) = Γ BBC1 (r 1, r 2 ) + Γ C2 (r 1, r 2 Γ C2 ( µi (r 1, r 2 ) = µ j 1 ) 2 µ iµ j ϕ i (r 1 )ϕ j (r 1 ) i,j i N/2 ϕ i (r 2 )ϕ j(r 2 ) (3.24) Since the correction only involves cross-products between different orbitals, the normalisation is not affected. The third correction is applied on top of both C1 and C2. It is the same correction as C2, but now involving the interaction between the first antibonding orbital ϕ N/2+1 ϕ r and the strongly occupied orbitals, except the corresponding bonding orbital ϕ N/2. Also the GU correction treated in the previous section is applied, so the diagonal terms are excluded from the summation to correct for self-interaction. The diagonal elements of the bonding-antibonding pair of orbitals (the frontier orbitals) are however very important to describe left-right correlation in a dissociating bond. Therefore these diagonal elements are retained. The BBC3 functional is given by Γ BBC3 (r 1, r 2 ) = Γ BBC2 (r 1, r 2 ) + Γ C3 (r 1, r 2 ) Γ C3 (r 1, r 2 ) = ( µi µ j 1 ) 2 µ iµ j [ϕ i (r 1 )ϕ r (r 1 )ϕ i (r 2 )ϕ r(r 2 ) + c.c.] i<n/2 + i N/2,r ( µ i 1 ) 2 µ2 i ϕ i (r 1 ) 2 ϕ i (r 2 ) 2 (3.25) A consequence of applying the GU correction is that the pair density does not satisfy the sum rules (3.18) and (3.19) anymore. Still this approximation provides the best overall agreement with full CI benchmark calculations of the BB series for most systems. 21

23 Chapter 4 DMFT calculations on the potential energy curves of diatomics In the article by Gritsenko et al., 15 the results of calculations with the BB, GU and BBC1-3 functionals are compared to accurate Configuration Interaction (CI) calculations for the potential energy curves of some simple diatomic molecules. Two kinds of calculations are performed. The first is a series of post-ci calculations, where the exact MRSDCI (full CI in the case of H 2 ) one-matrix is used as input for the various functionals to obtain directly the potential energy curve. A succesful functional should reproduce the CI results, so it is interesting to see how the functionals perform with the exact one-matrix. For H atoms the correlation-consistent polarized valence 5-zeta basis set (cc-pv5z) was used, while for the heavier atoms the correlation-consistent polarized core-valence quadruple-zeta augmented basis set (aug-cc-pcvqz) was used. The second is a series of self-consistent field (SCF) calculations. An exact functional should reproduce the exact one-matrix, and in that case the potential energy curve will not differ from the first calculation. The functionals of interest are of course not exact and therefore the one-matrix will change during the SCF cycles. As a consequence the potential energy curve will lower. Because of the increased computational cost compared to the first calculations, a smaller basis set was chosen. For the molecules consisting of at least one H atom (i.e. H 2, LiH, BH and HF), the correlationconsistent polarized valence triple-zeta basis set (cc-pvtz) was used from which the f orbital was excluded for the Li, B and F atoms. For Li 2 the 22

24 2 C3 r 1,r 2 = i N/2 ni n r 1 2 n in r The corresponding BBC3 energy functional is where E BBC3 ee = E BBC2 ee + E C3 ee, E C3 ee = ni n r i N/2 1 2 n in r i r 1 r r 1 i r 2 r r 2 r 12 dr 1 dr i N/2,r n i 1 2 n i 2 i r 1 2 i r 2 2 r 12 dr 1 dr Equations 3.10 and 3.12 are valid for molecules with a single bond, which are considered in this paper, but their generalization for the case of multiple bonds is straightforward. The selective application of the GU correction in BBC3 as well as other corrections of this paper preserve the important orbital structure 2.10 of the original BB, which is Gritsenko, Pernal, and Baerends J. Chem. Phys. 122, responsible for the proper description of nondynamical electron correlation in the dissociating bond. Unlike other corrections antibonding pair of orbitals with occupation numbers differing strongly from 2 and 0. The resulting BBC3 2 can be expressed as follows: applied in Eqs and 3.12 does not preserve the nor- of this section, the GU correction even if selectively BBC3 2 r 1,r 2 = BBC2 2 r 1,r 2 + C3 2 r 1,r 2, malization condition 2.9. However, as will be shown in Sec. 3.9 V, the potential energy curves obtained with the selfconsistent where DMFT calculations with BBC3 provide the best agreement with the benchmark full CI FCI and MRCI curves. FIG. 1. Energy curves for the H 2 molecule with NOs and NO occupation numbers from full CI calculations in the cc-pv5z basis. The drawn curve is the full CI curve labeled CI, the other labels are defined in the text Improved density matrix functional J. Chem. Phys. 122, also compared with the BB and GU curves the curves labeled GU apply the full GU correction of omitting the samespin diagonal i= j terms in the Hartree energy and the diagonal terms in the xc energy of the BB expression, Eq. 2.5, in both cases for all orbitals. The correlation-consistent ba- FIG. 5. Energy curves for the HF molecule with NOs and NO occupation numbers from MRSDCI calculations with the cc-pv5z basis on H and augcc-pcvqz on F. i r 1 r r 1 i r 2 r r 2 + c.c. IV. POST-CI CALCULATIONS OF POTENTIAL ENERGY + i i N/2,r n 1 2 n i 2 i r 1 2 i r 2 2 CURVES WITH THE BBC FUNCTIONALS Calculations with the NOs i and occupations n i obtained from MRCI calculations are of importance for further The corresponding BBC3 energy functional is E BBC3 ee = E BBC2 ee + E C3 ee, 3.11 development of DMFT. Indeed, a successful DMFT functional should reproduce these quantities, so that an assess- 2 molecule with NOs and NO occupation FIG. 3. Energy curves for the Li numbers from MRSDCI calculations with the aug-cc-pcvqz on Li. where ment of the quality of the proposed functionals with the E C3 ee = ni n r i N/2 1 2 n in r best i and n i is in order. sis sets are employed in these calculations, which are rather Figures 1 5 display the potential energy curves for the more extended and more adapted to treat electron correlation prototype -bonded molecules H 2, LiH, Li 2, BH, and HF than the basis sets commonly used in molecular selfconsistent DMFT calculations. For H and He atoms the cal- i r 1 r r 1 i r 2 r r 2 dr 1 dr obtained in this post-ci way with the BBC functionals proposed in the preceding section. They are compared with the 2 r i 2 i N/2,r n 1 2 n i 2 i r 1 2 i r 2 2 potential FIG. curves 1. Energy Improved obtained curves for density with the H matrix a 2 multireference molecule functional with NOs single and NO occupation FIG. 2. Energy curves for the LiH molecule J. Chem. with Phys. NOs 122, and NO occupation 2005 and numbers from full CI calculations in the cc-pv5z basis. The drawn curve is dr 1 dr 2. double CI r the MRSDCI full curve labeled with the CI, ATMOL the otherpackage labels are 25 numbers from MRSDCI calculations with the cc-pv5z basis on H and augcc-pcvqz on Li. defined and in they text. are 12 culations are performed in the correlation-consistent polarized valence 5-zeta basis sets 26,27 cc-pv5z and for heavier atoms the correlation-consistent polarized core-valence quadruple zeta augmented aug-cc-pcvqz basis sets 28,29 are used. NOs i and their occupations n i obtained with MRSDCI have been used in Eqs to produce the BB, BBCn, and GU electronic energies Downloaded also compared 12 Mar 2007 with to the BB and GU Redistribution curves the curves subject labeled to AIP license or copyright, see Since two-electron systems in general, and stretched H 2 Equations 3.10 and 3.12 are valid for molecules with a single bond, which are considered in this paper, but their generalization for the case of multiple bonds is straightforward. The selective application of the GU correction in BBC3 as well as other corrections of this paper preserve the important orbital structure 2.10 of the original BB, which is responsible for the proper description of nondynamical electron correlation in the dissociating bond. Unlike other corrections of this section, the GU correction even if selectively applied in Eqs and 3.12 does not preserve the normalization condition 2.9. However, as will be shown in Sec. V, the potential energy curves obtained with the selfconsistent GU apply the full GU correction of omitting the same- spin diagonal i= j terms in the Hartree energy and the diagonal terms in the xc energy of the BB expression, Eq. 2.5, in both cases for all orbitals. The correlation-consistent bain particular, served as the DMFT paradigm, we start our discussion with the potential curves of Fig. 1 for the H 2 molecule. With the scale chosen, it clearly displays, first of all, the well-known failure of the GU approximation. The latter consistently underestimates the H 2 energy and the corresponding error increases dramatically with the bond length R H H, so that already at R=2.5 a.u. the GU curve is way off the FCI one. The reason for this is the deficiency of the GU correction for NO occupations n i that deviate appreciably from 2 or 0, which is the case for the singly occupied frontier NOs g and u of the dissociating H 2. In this case the GU modification produces a deficient not attractive DMFT calculations with BBC3 provide the best enough xc hole agreement with the benchmark full CI FCI and MRCI curves. xc r 2 r 1 1 r g r 1 2 g r g r 1 u r 1 g r 2 u r 2 + c.c. IV. POST-CI CALCULATIONS OF POTENTIAL ENERGY CURVES WITH THE BBC FUNCTIONALS u r 1 2 u r 2 2, Calculations Improved with the density NOsmatrix i functional and occupations n i obtained J. Chem. Phys. 122, from MRCI calculations are of importance for further FIG. 3. Energy curves for the Li 2 molecule with NOs and NO occupation which integrates erroneously to 1/2 electron. On the other FIG. 4. Energy curves for the BH molecule with NOs and NO occupation development of DMFT. Indeed, a successful DMFT functional should reproduce these quantities, so that an assesscc-pcvqz on B. cause of the proper form 2.10 of its xc hole, the corre- numbers from MRSDCI calculations with the aug-cc-pcvqz on Li. FIG. 5. Energy curves for the HF molecule with NOs and NO occupation numbers from MRSDCI calculations with the cc-pv5z basis on H and aug- hand, BB consistently overestimates the H 2 energy but, be- numbers from MRSDCI calculations with the cc-pv5z basis on H and augcc-pcvqz on F. ment of the quality of the proposed functionals with the sis sets are employed in these calculations, which are rather best i and n i is in order. more extended and more adapted to treat electron correlation Downloaded 12 Mar 2007 to Redistribution subject to AIP license or copyright, see Figures 1 5 display the potential energy curves for the culations are performed in the correlation-consistent polarized valence 5-zeta basis sets 26,27 cc-pv5z and for heavier than the basis sets commonly used in molecular selfconsistent DMFT calculations. For H and He atoms the cal- prototype -bonded molecules H 2, LiH, Li 2, BH, and HF obtained in this post-ci way with the BBC functionals proposed atoms the correlation-consistent polarized core-valence qua- in the preceding section. They are compared with the druple zeta augmented aug-cc-pcvqz basis sets 28,29 are FIG. 2. Energy curves for the LiH molecule with NOs and NO occupation potential curves obtained with a multireference single and used. NOs i and their occupations n i obtained with double CI MRSDCI with the ATMOL package 25 numbers from MRSDCI calculations with the cc-pv5z basis on H and augcc-pcvqz on Li. MRSDCI have been used in Eqs to produce the and they are BB, BBCn, and GU electronic energies. Downloaded 12 Mar 2007 to Redistribution subject to AIP license or copyright, see Since two-electron systems in general, and stretched H 2 in particular, served as the DMFT paradigm, we start our discussion with the potential curves of Fig. 1 for the H 2 molecule. With the scale chosen, it clearly displays, first of all, the well-known failure of the GU approximation. The latter consistently underestimates the H 2 energy and the corresponding error increases dramatically with the bond length R H H, so that already at R=2.5 a.u. the GU curve is way off the FCI one. The reason for this is the deficiency of the GU correction for NO occupations n i that deviate appreciably from 2 or 0, which is the case for the singly occupied frontier NOs g and u of the dissociating H 2. In this case FIG. 3. Energy curves for the Li 2 molecule with NOs and NO occupation the GU modification produces a deficient not attractive numbers from MRSDCI calculations with the aug-cc-pcvqz on Li. FIG. 5. Energy curves for the HF molecule with NOs and NO occupation numbers from MRSDCI calculations with the cc-pv5z basis on H and augcc-pcvqz on F. enough xc hole sis sets are employed in these calculations, Figure which are rather 4.1: Potential energy curves of the xc diatomics 2 1 r 1 1 H 2 g r 1 2 g r 2 2 more extended and more adapted to treat electron correlation 2, LiH, Li 2, BH and culations are performed in the correlation-consistent polarized valence one-matrix 5-zeta basis sets 26,27 cc-pv5z fromandaformrsdci heavier calculation + g r 1 u r(full 1 g 2 u CI r 2 + c.c. for H than the basis sets commonly used inhf molecular using selfconsistent DMFT calculations. For H and He atoms the cal- atoms the correlation-consistent polarized core-valence qua- 2 ). The the druple zeta augmented aug-cc-pcvqz basis sets 28,29 are basis sets which u r 1 2 u r 2 2, 4.1 used. NOs i were used are and their occupations n i cc-pv5z on H and aug-cc-pcvqz on the obtained with MRSDCI have been used in Eqs to produce the other atoms. which integrates erroneously to 1/2 electron. On the other FIG. BB, 4. BBCn, Energyand curves GUforelectronic the BH molecule energies. with NOs and NO occupation numbers from MRSDCI calculations with the cc-pv5z basis on H and augcc-pcvqz on B. 2 cause of the proper form 2.10 of its xc hole, the corre- hand, BB consistently overestimates the H 2 energy but, be- Since two-electron systems in general, and stretched H in particular, served as the DMFT paradigm, we start our discussion with the potential curves of Fig. 1 for the H 2 molecule. Downloaded With 12 the Mar scale 2007 chosen, to it clearlyredistribution displays, firstsubject of to AIP license or copyright, see all, the well-known failure of the GU approximation. The latter consistently underestimates the H 2 energy and the corresponding error increases dramatically with the bond length R H H, so that already at R=2.5 a.u. the GU curve is way off the FCI one. The reason for this is the deficiency of the GU correction for NO occupations n i that deviate appreciably from 2 or 0, which is the case for the singly occupied frontier NOs g and u of the dissociating H 2. In this case the GU modification produces a deficient not attractive enough xc hole hole GU xc r 2 r 1 1 r g r 1 2 g r g r 1 u r 1 g r 2 u r 2 + c.c. 1 23

25 smaller correlation-consistent polarized valence double-zeta basis set (ccpvdz) was used. Because in both series of calculations different basis sets are used, the results may not be quantitatively compared with each other. The results of the first series of calculations are in figure 4.1. The H 2 molecule can be regarded as the prototype molecule for these calculations. It is a simple two-electron system, but there is a strong left-right correlation when the bondlength gets larger. As was discussed before, the diagonal elements of the frontier orbitals are very important for this type of correlation. The neglect of these terms is the reason that the GU functional performs badly at large interatomic distances. The BB functional overestimates the energy, the reasons for this have been discussed in section 3.3. The error vanishes however at large bond lenghts. For the two-electron case, the BBC1 and BBC2 functionals are in fact equal. The C1 correction is important at shorter distances and it brings the results very close to the CI result. The selective GU approximation in the BBC3 functional results in a small error in the energy at the equilibrium, but the rest of the CI curve is almost exactly reproduced. For the other molecules, the same behaviour of the GU functional can be seen. This functional consistently underestimates the correlation energy. The error at equilibrium distances is small, but it increases with bond length. Therefore the dissociation energy is incorrectly described and the energy curve is too deep. The BB functional on the other hand consistently overestimates the correlation energy. The error gets larger at increasing bond lengths, especially for the molecules with heavier atoms (BH and HF). The first two corrections C1 and C2 bring the energy curve much closer to the CI curve, where C2 becomes more important with increasing number of electrons. The BBC3 functional however completely underestimates the correlation energy. The shape of the curve is similar to the CI curve though, i.e. the error is almost constant for different bond lenghts. In a practical application of DMFT, the potential energy curves should be calculated self-consistently. The self-consistent one-matrix will in general be different than the CI one-matrix, so during the SCF cycles the energies are expected to drop. In the post-ci calculations the BBC2 functional gave the best agreement with the CI results, but in self-consistent calculations the energies get too low. The BBC1 functional gives even lower energies, so these two functionals were not considered in this series of calculations. The results of the calculations are in figure 4.2. For comparison, the full CI (MRCI for the HF molecule), as well as the restricted Hartree-Fock (HF) potential energy curves are shown. First consider the H 2 molecule again. The GU functional performs much 24

26 minimum of an approximate DMFT functional is lower than its energy calculated with any nonvariational NOs and their occupancies, even with the best MRCI i and n i. In particular, bital optimization, the basis sets are smaller than in the preculations while the BBC2 energies obtained in the post-ci calceding section. The absolute energies of Figs are are rather close to the MRCI ones see the preceding therefore not directly comparable to those of Figs For section, the self-consistent BBC2 energies appear to be atoms the cc-pvqz Ref. 26 basis set was employed with consistently lower than the corresponding MRCI or fullconfiguration the exclusion of the g and one of the f orbitals for Be and interaction FCI where it was possible to cal- Ne. For molecules composed of at least one light atom, i.e., culate them values. Since BBC2 is obtained from BBC1 for H 2, LiH, BH, and HF, the correlation-consistent polarized with the repulsive correction C2, the self-consistent BBC1 valence triple-zeta basis set cc-pvtz was used from which energies are lower than the already too low BBC2 energies. the f orbital for heavy atoms was excluded, whereas for the On the other hand, the C3 repulsive correction, which makes Li 2 molecule a smaller cc-pvdz basis set was used. It the post-ci BBC3 energies too high, produces self-consistent should be emphasized that the employed basis sets are still BBC3 energies that appear to be rather close and in some relatively large compared to previous calculations 16,21 and oved density matrix functional cases very close to the MRCI ones. ThisJ. discrepancy Chem. Phys. means, we did not freeze any orbitals throughout the optimization 122, of course, that the self-consistent DMFT NOs and their occupations calculations. Fully variational DMFT calculations involve differ from the corresponding MRCI close to ex- he best performance among the functionals act the same relatively small average absolute i and n i. The ultimate goal, of course, is to formulate NO dependent xc functionals that yield self-consistently for both functionals. optimized orbitals that are close to the exact NOs, so the FIG. 8. Energy curves for the Li the BBC corrections greatly improve the 2 molecule with self-consistent determination of the NOs and NO occupation numbers for each NO functional. For discrepancy between energies based on exact NOs and on potential energy curves obtained with non- basis see text. optimized orbitals would disappear. We note that there is no MFT calculations with MRCI i and n i. proof for variational stability of the functionals we are investigating. For the BB CH functional the possibility of varia- calculation the BBC2 functional shows the ormance. In the following section it will be tional collapse has been investigated for the H energy lowering due to the self-consistent 2 molecule, as a function of basis set and of the number of active orbitals nificantly improves the atomic and molecuined with the BBC3 functional, so that the virtual orbitals that are allowed to acquire occupations. 16 the best self-consistent functional. It was concluded that variational collapse does not show up. This has been confirmed by Herbert and Harriman with larger basis sets and for other systems. 17 ISTENT CALCULATIONS OF THE ERGY CURVES WITH THE Figures 6 10 display the potential energy curves for the NALS prototype molecules obtained with self-consistent calculations with the BB, GU, and BBC3 functionals. They are post-ci calculations presented in the precedseful for the assessment and furtherand devel- BH and He, Be in Table II or MRSDCI results in the were parametrized according to the orthonormality- compared with the FCI results in the case of H 2, LiH, Li 2, DMFT methods, in a full-fledged DMFT Gritsenko, Pernal, and Baerends J. Chem. Phys. 122, hole 4.1, it produces a large positive error case of HF and Ne. As a natural reference, the restricted FIG. 7. Energy curves for the LiH molecule with self-consistent determination of the NOs and NO occupation numbers for each NO functional. For lengths. On the other hand, unlike in the post-c ities should be calculated in a self-consistent Hartree Fock FIG. HF 6. Energy curves curves infor the thesame molecule bases H 2 with areself-consistent also included. Because of the computationally more demanding or- basis see text. determination of the NOs and NO occupation numbers for each NO functional. For and in agreement with the previous s sly, a variational, self-consistent energy basis see text. calculations, 12,16 BB consistently overestimate approximate DMFT functional is lower than ergy not only near equilibrium, but at all R H ated with any nonvariational NOs and their Downloaded 12 Mar 2007 to Redistribution subject to AIP license or copyright, see n with the best MRCI i and n i. In par- bital optimization, the basis sets are smaller than in the preceding section. The absolute energies of Figs are estimation is the difference between the FCI compare Figs. 1 and 6. The apparent reason BBC2 energies obtained in the post-ci calher close to the MRCI ones see the precedself-consistent BBC2 energies appear to be atoms the cc-pvqz Ref. 26 basis set was employed with over-attractive BB functional. But then, the rep therefore not directly comparable to those of Figs For corresponding quantities produced self-consiste er than the corresponding MRCI or fulleraction FCI where it was possible to cal- Ne. For molecules composed of at least one light atom, i.e., the exclusion of the g and one of the f orbitals for Be and tions of BBC3 remove this error. As a result, lently reproduces the FCI potential curve lues. Since BBC2 is obtained from BBC1 for H 2, LiH, BH, and HF, the correlation-consistent polarized R H H see Fig. 6, the largest BBC3 e ve correction C2, the self-consistent BBC1 valence triple-zeta basis set cc-pvtz was used from which 3 kcal/mol at R H H =4 a.u. Similar trends in the self-consistent perfor er than the already too low BBC2 energies. the f orbital for heavy atoms was excluded, whereas for the DMFT functionals follow also for other molecu d, the C3 repulsive correction, which makes Li 2 molecule a smaller cc-pvdz basis set was used. It the post-ci calculations of the preceding sect 3 energies too high, produces self-consistent should be emphasized that the employed basis sets are still that appear to be rather close and in some relatively large compared to previous calculations 16,21 consistent BB greatly overestimates the molec and especially in the dissociation region, so the to the MRCI ones. This discrepancy means, we did not freeze any orbitals throughout the optimization curves are much too shallow see Figs. 7 1 he self-consistent DMFT NOs and their ocfrom thepernal, corresponding and Baerends MRCI close to ex- J. Chem. Phys. 122, the case of H 2, but much less so than the BB calculations. Fully variational DMFT calculations involve consistent GU curves are also too deep around ritsenko,. The ultimate goal, of course, is to formunt xc functionals that yield self-consistently 2 molecule with self-consistent determina- FIG. 9. Energy FIG. curves 10. Energy for the curves BH molecule for the HFwith molecule self-consistent with self-consistent determina- determi- self-consistent GU curves dissociate improper FIG. 8. Energy curves for the Li tion of the NOs and NO occupation numbers for each NO functional. tion For of thenation NOs and of the NONOs occupation and NO occupation numbers for numbers each NO for functional. each NO functional. For LiH, For rising too high as in the post-ci case, th ls that are close to the exact NOs, so the basis see text. basis see text. basis see text. responding error is relatively not as large as tha een energies based on exact NOs and on ls would disappear. We note that there is no minimizing the total energy with respect to both natural Downloaded orbitals and natural occupation numbers. For singlet and triplet Newton Boyden-Fletcher-Goldfarb-Shannon BFGS Ref. 31 preserving 12 Mar scheme 2007 to presented inredistribution Ref. 30 andsubject the quasi- to AIP license or copyright, see nal stability of the functionals we are inves- BB CH functional the possibility of variaas been investigated for the H 2 molecule, as developed and applied some time ago by Kutzelnigg. 24 To variational procedure described above. To assure that the ground states of two-electron systems, such a method was optimization method was implemented in the first step of the is set and of the number of active orbitals s that are allowed to acquire occupations. 16 assure convergence to the minimum and to make the calculations more efficient, we have implemented a two-step pro- sum up to the number of electrons N, we imposed that n i natural occupancies stay between zero and two and that they that variational collapse does not show up. cedure. It consists of minimization of the energy with respect =2 cos 2 p i and i n i =N. For the optimization of the free confirmed by Herbert and Harriman with to the natural orbitals for fixed occupation numbers and a parameters p i we employed the conjugate gradient Polak and for other systems. 17 subsequent variation of the occupancies for the new set of Ribière algorithm display the potential energy curves for the natural orbitals obtained in the first step. This procedure is We start our discussion with the potential curves for the ules obtained with self-consistent calcula- repeated until convergence is achieved. The natural orbitals H 2 molecule in Fig. 6. Optimization improves GU close to B, GU, and BBC3 functionals. They are were parametrized according to the orthonormality- the equilibrium geometry, but again, due to its deficient xc hole 4.1, it produces a large positive error at large bond he FCI results in the case of H 2, LiH, Li 2, lengths. On the other hand, unlike in the post-ci calculations, Be in Table II or MRSDCI results in the and in agreement with the previous self-consistent Ne. As a natural reference, the restricted FIG. 7. Energy curves for the LiH molecule with self-consistent determination of the NOs and NO occupation numbers for each NO functional. For ergy not only near equilibrium, but at all R H H distances calculations, 12,16 BB consistently overestimates the H F curves in the same bases are also inof the computationally more demanding or- basis see text. 2 enurves for the Li 2 molecule with self-consistent determinand NO occupation numbers for each NO functional. For nation of the NOs and NO occupation numbers for each NO functional. For estimation is the difference between the FCI i, n i the FIG. 10. Energy curves for the HF molecule with self-consistent determi- compare Figs. 1 and 6. The apparent reason for this over- Mar 2007 to Redistribution subject Figure to AIP basis license see4.2: text. or copyright, Potential see energy curves of the diatomics H 2, LiH, Li 2, BH and HF using a self-consistent calculation. e total energy with respect to both natural orral occupation numbers. For singlet and triplet of two-electron systems, such a method was applied some time ago by Kutzelnigg. 24 To ence to the minimum and to make the calcufficient, we have implemented a two-step proists of minimization of the energy with respect orbitals for fixed occupation numbers and a riation of the occupancies for the new set of s obtained in the first step. This procedure is convergence is achieved. The natural orbitals trized according to the orthonormalitypreserving scheme presented in Ref. 30 and the quasi- Newton Boyden-Fletcher-Goldfarb-Shannon BFGS Ref. 31 optimization method was implemented in the first step of the variational procedure described above. To assure that the natural occupancies stay between zero and two and that they 25 sum up to the number of electrons N, we imposed that n i =2 cos 2 p i and i n i =N. For the optimization of the free parameters p i we employed the conjugate gradient Polak Ribière algorithm. 31 We start our discussion with the potential curves for the H 2 molecule in Fig. 6. Optimization improves GU close to the equilibrium geometry, but again, due to its deficient xc hole 4.1, it produces a large positive error at large bond FIG. 9. Energy curves for the BH molecule with self-consistent determination of the OnNOs theand other NO hand, occupation unlike numbers in the forpost-ci each NO calculations functional. For lengths. and basis see in text. agreement with the previous self-consistent calculations, 12,16 BB consistently overestimates the H en- minimizing the total energy with respect to both natural orbitals and natural occupation numbers. For singlet and triplet ground states of two-electron systems, such a method was developed and applied some time ago by Kutzelnigg. 24 To assure convergence to the minimum and to make the calculations more efficient, we have implemented a two-step procedure. It consists of minimization of the energy with respect to the natural orbitals for fixed occupation numbers and a subsequent variation of the occupancies for the new set of natural orbitals obtained in the first step. This procedure is repeated until convergence is achieved. The natural orbitals corresponding quantities produced self-consistently with the over-attractive BB functional. But then, the repulsive corrections of BBC3 remove this error. As a result, BBC3 excellently reproduces the FCI potential curve of H 2 at all R H H see Fig. 6, the largest BBC3 error is only 3 kcal/mol at R H H =4 a.u. Similar trends in the self-consistent performance of the DMFT functionals follow also for other molecules. Just as in the post-ci calculations of the preceding section, the selfconsistent BB greatly overestimates the molecular energies, especially in the dissociation region, so the BB potential curves are much too shallow see Figs The selfconsistent GU curves are also too deep around R e except in the case of H 2, but much less so than the BB energies. The self-consistent GU curves dissociate improperly for H 2 and LiH, rising too high as in the post-ci case, though the corresponding error is relatively not as large as that case. In the FIG. 10. Energy curves for the HF molecule with self-co nation of the NOs and NO occupation numbers for each N basis see text. preserving scheme presented in Ref. 30 an Newton Boyden-Fletcher-Goldfarb-Shannon B optimization method was implemented in the fi variational procedure described above. To as natural occupancies stay between zero and two sum up to the number of electrons N, we im =2 cos 2 p i and i n i =N. For the optimizatio parameters p i we employed the conjugate gr Ribière algorithm. 31 We start our discussion with the potential H 2 molecule in Fig. 6. Optimization improves the equilibrium geometry, but again, due to it