Exact high-density limit of correlation potential for two-electron density

Size: px

Start display at page:

Download "Exact high-density limit of correlation potential for two-electron density"

Agnes May
5 years ago
Views:

1 JOURAL OF CHEMICAL PHYSICS VOLUME 0, UMBER JUE 999 Exact high-density limit of correlation potential for two-electron density Stanislav Ivanov a) Department of Chemistry and Quantum Theory Group, Tulane University, ew Orleans, Louisiana 708 Kieron Burke Department of Chemistry, Rutgers University, 35 Penn Street, Cadmen, ew Jersey 080 Mel Levy Department of Chemistry and Quantum Theory Group, Tulane University, ew Orleans, Louisiana 708 Received 4 October 998; accepted 8 February 999 Present approximations to the correlation energy, E c n, in density functional theory yield poor results for the corresponding correlation potential, v c (n;r)e c n/n(r). Improvements in v c (n;r), are especially needed for high-quality Kohn Sham calculations. For a two-electron density, the exact form of v c (n;r) in its high-density limit is derived in terms of the density of the system and the first-order wave function from the adiabatic perturbation theory. Our expression leads to a formula for the difference E c nv c (n;r)n(r)dr, valid for any two-electron density in the high-density limit, thus generalizes previous results. umerical results both exact and approximate are presented for both E c n and v c (n;r)n(r)dr in this limit for two electrons in a harmonic oscillator external potential Hooke s atom. 999 American Institute of Physics. S ITRODUCTIO AD DEFIITIOS Density functional theory DFT has provided an effective machinery for quantum chemistry calculations on large molecules. 5 This state-of-the-art theory enables one to replace the complicated conventional wave function approach with the simpler density functional formalism. In density functional calculations an energy functional must be employed. The form of the exact exchange-correlation component, E xc n, of this functional must be approximated. 8 For convenience, E xc n is further partitioned into exchange E x n and correlation E c n contributions, i.e., E xc n E x ne c n. 8 The most widely used implementation of DFT is the Kohn Sham KS theory. 5 In this approach, the interacting system of interest is replaced by a model noninteracting system with the same ground-state density in a new effective multiplicative potential the KS potential incorporating all effects associated with the electron electron interactions. The ground-state properties of the system under investigation are then obtained by means of the density and the noninteracting ground-state wave function that yields this density. The KS potential v s (n;r) is a unique functional of the density n(r), and is usually written as v s n;rvrun;rv x n;rv c n;r, where v(r) is the physical external potential of the system. The Hartree potential u(n;r) is un;r nr rr dr, a Present address: Quantum Theory Project, University of Florida, Gainesville, Florida 36. which is the functional derivative of the Hartree electron electron repulsion energy Un given explicitly in terms of the density by Un nrnr drdr. rr In Eq., the exchange potential v x (n;r) is the functional derivative of E x n, i.e., v x (n;r)e x n/n(r). The exchange energy E x n is known exactly in terms of the noninteracting KS wave function, and is defined as E x n 0 nvˆ ee 0 nun, where Vˆ ee is the operator for the electron electron repulsion, and 0 n is the KS wave function, i.e., the wave function that minimizes the expectation value of the kinetic energy operator only, and yields the same density n(r). Except for certain degenerate cases, the KS wave function is a single determinant. 9 In practice, E x n is usually approximated as an explicit functional of the density. The exact density dependence of E x n is known only for two-electron diamagnetic densities because E x n Un. The last term on the right-hand side of Eq. is the correlation potential v c (n;r), which is the functional derivative of E c n, i.e., v c (n;r)e c n/n(r). The correlation energy functional, E c n, is formally defined as E c nntˆ Vˆ ee n 0 ntˆvˆ ee 0 n. In Eq. 5, Tˆ is the kinetic energy operator and n is that antisymmetric wave function that minimizes Tˆ Vˆ ee and yields the density n(r) /99/0()/06/7/$ American Institute of Physics

2 J. Chem. Phys., Vol. 0, o., June 999 Ivanov, Burke, and Levy 063 In order to arrive at very the best approximations to E c n and E x n, knowledge of their exact properties is needed. While many exact conditions on E c n are known to guide the construction of approximations; relatively few exist for v c (n;r). The purpose of this work is to present new constraints for v c (n;r) in its high-density scaling limit for two-electron diamagnetic densities. The correlation potential v c (n;r) is very difficult to mimic because of its pointwise nature in contrast to E c n, which is just a number. Many accurate approximations to E c n yield poor approximations to v c (n;r). In addition, properties like ionization and excitation energies, polarizabilities, and hyperpolarizabilities, are very sensitive to the quality of the approximations used to generate the KS potential. Designing new functionals with improved properties of their functional derivatives is crucial for high-quality KS calculations. An important technique for deriving the properties of the unknown density functionals is the scaling of the electron density n(r). Coordinate scaling provides key requirements for the exact dimensional properties of E x n and E c n. When the density is scaled uniformly along each direction in space, i.e., n x,y,z 3 nx,y,z, the following simple scaling occurs with Un and E x n: 8 and Un Un E x n E x n. Upon uniform scaling of the electron density, E c n does not scale homogeneously. Instead, E c n has the following expansion for large : E c n E c n E c 3 n E c 4 n j j E c j n. In other words, E c n is bounded, as, and is equal to a second-order energy, 0 4 i.e., lim E c n E c n The quantity E c () n is particularly important because it is the initial slope in the adiabatic connection method coupling-constant formula for E c n. 0,,5 9 It is also believed that E c n is relatively insensitive to coordinate scaling, i.e., E c () ne c n for small atoms. As a result, knowledge of E c () n can be used for constructing accurate approximations to E c n. The link between uniform scaling of the electron density and scaling of the electron electron interaction has been developed by Görling and Levy in a series of papers.,0 The effective potential and the electron electron interaction along the coupling constant path, which connects a noninteracting and a fully interacting system with the same electron density, have been used by them to construct a DFT perturbation theory, yielding with E c n 0 nĥ n n nĥ n 0 n, Ĥ nvˆ ee un;r i v x n;r i i Vˆ ee v xu n;r i, i and () n being the solution to Ĥ 0 ne 0 ne Ĥ n 0 n. 3 In Eq. 3, E () is the first-order correction to the energy due to Ĥ () n from the standard Rayleigh Schrödinger perturbation theory. In the above equation, u(n;r) and v x (n;r) are the functional derivatives of Un and E x n with respect to the density, and v xu (n;r)v x (n;r) u(n;r). In Eq., the KS wave function 0 is the ground-state solution to the noninteracting Schrödinger equation, Ĥ 0 n k Tˆ v s n;r i k E 0 k k ; i E 0 E 0 E k 0, 4 where we shall assume that E 0 (E 0 0 ) is nondegenerate. The energies E 0 and E k 0 are the eigenvalues corresponding to 0 and k, respectively. Since most of the present approximations to E c n give relatively good results for E c n, but the shapes of the corresponding correlation potentials are not quite satisfactory, we shall derive an expression for the exact form of v c () (n;r) for two-electron diamagnetic densities. This expression should prove useful in the process of obtaining new improved approximations to v c (n;r). By taking the functional derivative of E c () n, we arrive at a formal expression for v c () (n;r) featuring the density of the system of interest, n(r), and the first-order wave function from Görling Levy GL adiabatic perturbation theory, associated with n(r). Our pointwise identity for v c () (n;r) leads to a generalization of a previous result obtained by Görling and Levy 0 with the caveat that all components of the KS potential vanish at infinity. DIFFERET EXPRESSIOS FOR E c n Before approaching the subtle subject of taking the functional derivative of E c () n, we introduce two different forms of the GL expression for the high-density scaling limit of E c n. First, as has been pointed out by Görling and Levy,,0 nˆ 0 n 0 nˆ n0, 5 because the density is held fixed along the adiabatic path. In Eq. 5, ˆ is the density operator. Formula 5 implies that

3 064 J. Chem. Phys., Vol. 0, o., June 999 Ivanov, Burke, and Levy n i 0 n i Ar i 0 n Ar i n 0, 6 where A(r) is any one-body multiplicative operator. By means of Eq. 6, along with A(r)v xu (n;r), Eq. simplifies to E c n 0 nvˆ ee n nvˆ ee 0 n. 7 Another form of Eq. 7, which can be readily developed, considers Vˆ ee alone as the perturbation. We introduce () n( () n) as the solution to Ĥ 0 ne 0 nẽ Vˆ ee 0 n, 8 where Ẽ () and () n are the first-order energy and wave function respectively, when the perturbation is Vˆ ee only. By making use of Eqs. 3 and 8, Eq. 7 can be equivalently expressed as E c n 0 nĥ n n nĥ n 0 n. 9 A comparison of Eqs. 7 and 9 shows that E c () n can be expressed either in terms of Vˆ ee acting on (), the response to Ĥ () n, or in terms of Ĥ () n acting on () n, the response to Vˆ ee. DERIVATIOS To obtain an expression for v c () (n;r), we start with the second-order energy from GL perturbation theory, Eq. 7, and calculate v c n;r E c. 0 r n In the special case of a spin-unpolarized two-electron density, there is only one quantity in Eq. 7 whose density dependence cannot be expressed in terms of the density explicitly. This quantity is the first-order wave function, () n. ote that Vˆ ee is independent of the density for fixed particle number, and 0 n is given in terms of the density n(r) the spin part is omitted for simplicity of the notation by 0 n;r,r n / r n / r. To find the functional derivative of E c () n, we rewrite Eq. 7 as E c n 0 nvˆ ee Ẽ n nvˆ ee Ẽ 0 n. By using the differential equation 8, we arrive at E c n ne 0 Ĥ 0 n n ne 0 Ĥ 0 n n. 3 We shall take the functional derivative of E c () n given by Eq. 3. ote that in the above formula () n, () n, E 0, and Ĥ 0 n depend on the density. If either () n or () n is known exactly, the other one is uniquely determined as well. In order to obtain the form of v c () (n;r), we will make use of the following expression: Gng 0 dr G gr, rn 4 for all g(r), such that g(r)dr0, keeping the particle number fixed through a small variation of the density. With this in mind, a variation of the density leads to E c ng 0 ne 0 Ĥ 0 n n ne 0 Ĥ 0 n n ne 0 Ĥ 0 n n ne 0 Ĥ 0 n n ne 0 Ĥ 0 n n ne 0 Ĥ 0 n n. 5 Here primes indiciate density variations of the terms in the parentheses. In order to find the effect of changing the density upon the terms involving the first-order wave functions, () n and () n, we use the corresponding differential equations, Eqs. 3 and 8, and their complex conjugate counterparts. In general, the left-hand sides of Eqs. 3 and 8 are very convenient for density variations because everything is expressed in terms of the density n(r) only. Also, note that when one varies the left-hand sides of Eqs. 3 and 8, the first-order energies E () and Ẽ () could be conveniently eliminated. It follows from the fact that the expressions for E c () n, Eqs., 7, and 9 are determined within a constant in the perturbation because of the normalization conditions, i.e., n 0 n 0 n n0. 6 When () n is considered, a more severe normalization constraint for the density, Eq. 6, applies. The last term that deserves special attention is ne 0 Ĥ 0 n n ne 0 Ĥ 0 n n. 7 To find the result coming from varying (E 0 Ĥ 0 n) with respect to the density, we use the exact form of Ĥ 0 n. In terms of n(r), Ĥ 0 n is given by

4 J. Chem. Phys., Vol. 0, o., June 999 Ivanov, Burke, and Levy 065 Ĥ 0 n i i v 0 n;r i, 8 where v 0 n;r 0 n / r n /. 9 r The potential, v 0 (n;r), is obtained by inverting the corresponding one-particle equation. In Eq. 9, 0 is the orbital energy of the doubly occupied KS orbital, and E 0 n 0. By integrating by parts, we obtain E c ng gr n / r n / r Vˆ ee n 4gr n / r n / r Ĥ n n gr sr dr r r dr 4gr tr nr t n n r t n n n 3 r t n rdrc.c. In formula 3, s(r) and t(r) are defined as follows: sr 0 r,r* r,rdr and tr r,r* r,rdr In the above equations, () (r,r), () (r,r), and 0 (r,r) are functionals of the density. The form of the potential, v c () (n;r), is easily obtained from Eq. 3, and it is v c n;r r n/ r r,r 4 n/ dr rr 4 n/ r n / r rr v xun;r v xu n;r r;rdr sr rr dr 4 tr nr t n n r t n n 3 r t n n 34 rc.c. Formula 34, our key result, features the density of the system under investigation and the first-order wave functions from adiabatic perturbation theory, () n, and from perturbation theory when electron electron repulsion is the perturbation, () n. Very good approximations to either one can be found by using the variation-perturbation approach of Hylleraas. Once, for example, () n is determined, then () n can be easily obtained because n n f r f r nr dr n / r n / r, where f (r) is the solution to nr fr nrun;r nrun;rdr When an appropriate transformation exists, such that the three-dimensional one-particle problem is reduced to three one-dimensional ones i.e., the separation of variables is possible, then the analytic solution to Eq. 36 exists and can be obtained by two consecutive integrations over each of the coordinates. If we multiply both sides of Eq. 34 by n(r) and integrate over all space, we arrive at an identity previously obtained by Görling and Levy 0 and recently expressed in a closed form by Ivanov et al. amely, E c n v c n;rnrdr frnrun;rdr frnrdr, 37 where f (r) is the solution to Eq. 36, and () is given by 4 nrun;rdr. 4 nr 38 By integrating by parts, it can be shown that tr t n nr n t n r n 3 r t n n r dr0. 39 EXAMPLE: HARMOIC OSCILLATOR EXTERAL POTETIAL To illustrate the fact that our results apply to any external potential, we test them on a simple model system, the Hooke s atom, which consists of two electrons repelling each other via a Coulomb repulsion, but bound to an attractive center by a simple oscillator potential of frequency. This model has been treated by many authors before, 3 often to illustrate concepts in DFT. 4 7 Recently, an analytic solution was discovered at. 8 Later, an infinite set of such discrete values were found, at lower frequencies. 9 Since formula 37 is true for a density associated with any kind of external potential, we calculate the difference E c () nv c () (n;r)n(r)dr for density n(r) (a/) 3/ exp(ar ); a0 by applying some recent

5 066 J. Chem. Phys., Vol. 0, o., June 999 Ivanov, Burke, and Levy identities of Ivanov et al. This density corresponds to the external potential v 0 (r)a / r, which approaches infinity as r goes to infinity. The exact values of E c () n and dr v c () (n;r)n(r), corresponding to density n(r) (a 3 /)exp(ar) have been obtained in Ref.. First, note that 4a. Then, the radial component of the operator f (r) reads as with f r I A 4a r er / 4 / er r 4a 4 0 r e r / I A,r 3, r er / 40 3, r,r dr, 4 3,0,r. 4 In Eqs. 4 and 4, (k,z,z ) is the generalized incomplete Gamma function: z k,z,z t k e t dt. 43 z A similar expression for f (r) has been developed by White and Brown. 30 By means of expressions 40 43, we calculate the integrals on the right-hand side of Eq. 37 to obtain E c n v c n;rnrdr To test this result, we employ a numerical method to solve Hooke s atom for any. The Hamiltonian may be written as Ĥ r r r r. 45 This is more conveniently expressed in terms of the center of mass R(r r )/ and difference ur r coordinates: Ĥ 4 R u R u 4 u. 46 The advantage of the harmonic oscillator potential is now clear. The center of mass and difference variables may be separated: 0 r,r 0 R 0 u, 47 yielding a simple three-dimensional oscillator equation in R, with mass and force constant, and ground-state solution: 0 RexpR /R 0 / 3/4 R 0 3/, 48 where R 0 /. The equation in u is especially interesting: TABLE I. First three analytic solutions to Hooke s atom. Solution u 0 / c j /c 0 0 / 0 0,0,... u u 4 u u u u. 49 It looks like a three-dimensional oscillator of mass, except for the Coulomb repulsion term /u. The ground state of the eigenvalue, Eq. 49, is spherically symmetric, and will decay as a Gaussian for large u. We expand the wave function in the basis set, u j0 c j u j expu /u 0, 50 where u 0 / is the noninteracting length scale of the u oscillator. ote that this set is not orthonormal. This expansion becomes exact for those values of at which analytic solutions exist, as given in the examples of Table I. We now use dimensionless functions to capture the scale-independent features of the solution. We write the scaled wave function as u0 ;xu 3/ 0 0 u 0 ;u c j jx, j0 where xu/u 0 and,/, ,/,/0,0,... 5 x j jx 3/4 e x /, 5 so that c j j0 for. For values of other than those at which analytic solutions exist, we truncate the series in Eq. 5 at large but finite, and find the lowest-energy level by diagonalizing the Hamiltonian within the truncated basis set. The overlap matrix elements are S ij 4 0 ix jxx dxfij, 53 where f (k)(k)//( ), 3 and is the Gamma function, while the u-component Hamiltonian matrix elements are H ij fij ij 3 u 0 ij fij. 54 u 0 This procedure is numerically unstable, in that a small denominator occurs in the generalized eigenvalue solution. However, using 3-digit accuracy, we find that inversion works reliably for 6. We find that 6 allows us to solve the problem for all frequencies down to about 0 4. We find that the virial relation for the Hooke s atom, TV ext V ee, 55 is satisfied more and more accurately as is increased, being good to about 30 digits when 6. We also find that, for 3 5 7

6 J. Chem. Phys., Vol. 0, o., June 999 Ivanov, Burke, and Levy 067 FIG.. The difference between a dimensionless wave function and a simple Gaussian for and for. FIG.. The correlation potential for Hooke s atom for (v c () ) and for. large, the energy converges rapidly with, but the much tougher test, satisfaction of the electron electron cusp condition, (c /c 0 ), requires large. For example, for 300 and 6, this condition is only satisfied to the first digits. We use this numerical solution to explore the highdensity noninteracting limit. As, (u) tends to a simple Gaussian. The approach to the noninteracting limit is then characterized by the difference of the wave function from this Gaussian as u 0: u0 ;x u 0 u0 ;x 0;x. 56 The function (u 0 0;x) is plotted in Fig., and is clearly finite and well behaved. We also plot (u 0 ;x). We see that the first analytic solution is close to the high-density limit. For the correlation energy, we find the expansion about u 0 0tobe E c u 0 Ou The exact value at u 0 is a.u. 3 In the highdensity limit, T c E c, and we find E c T c 0.007u u The exact value at u 0 is a.u.. 3 In Fig., we plot the correlation potential in this limit. ote that the virial of the correlation potential is E c T c nrr v c n;rdr, 59 which therefore vanishes for v () c (n;r). The integral involving the correlation potential is found to be v c n;rnrdr u 0 Ou The exact value at u 0 is a.u. Several of these results were later confirmed by Huang and Umrigar. 3 For all these energies, the high-density expansion, extrapolated to realistic densities (u 0 ), yields an excellent estimate to within 0% of the exact values, thereby demonstrating the usefulness of studying these limits. By making use of Eqs. 57 and 60, we obtain an excellent agreement for the difference E () c nv () c (n;r)n(r)dr, with the exact value in Eq. 44. Last but not least, in Table II, we compare the exact values for E () c n and v () c (n;r)n(r)dr against the respective values obtained from three different approximations to E c n, respectively, to E () c n, the one by Lee, Yang, and Parr LYP, 33 the one by Wilson and Levy WL, 34 and the recently derived GGA by Perdew, Burke, and Ernzerhof PBE. 35 In Table II, the density depends on the strength of the harmonic oscillator a, but E () c n and v () c (n;r)n(r)dr are independent of a. COCLUDIG REMARKS Formula 34 is the first known expression for the correlation potential in the high-density limit. As a consequence of this pointwise identity for v c () (n;r), we have shown that the closed-form expression, Eq. 37, is valid for any two-electron density. Our general approach of taking a functional derivative of the energy functional given by perturbation theory can be extended to more than two electrons. For more than two electrons, taking the functional derivative with respect to each occupied orbital rather than to the whole density, leads to an orbital-dependent correlation potential. TABLE II. A comparison of the exact values for E c () n and v c () (n;r)n(r)dr for density n(r)(a/) 3/ e ar, with those obtained from different approximations. E c () n v c () (n;r)n(r)dr Exact value E LYP c n E WL c n E PBE c n

7 068 J. Chem. Phys., Vol. 0, o., June 999 Ivanov, Burke, and Levy ACKOWLEDGMET K.B. was supported by an award from Research Corporation. P. Hohenberg and W. Kohn, Phys. Rev. B 36, W. Kohn and L. J. Sham, Phys. Rev. A 40, R. G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules Oxford University Press, ew York, 989, and references therein. 4 R. M. Dreizler and E. K. U. Gross, Density Functional Theory Springer- Verlag, Berlin, 990, and references therein. 5 R. O. Jones and O. Gunnarsson, Rev. Mod. Phys. 6, M. Levy, Proc. atl. Acad. Sci. USA 76, M. Levy, Bull. Am. Phys. Soc. 4, M. Levy and J. P. Perdew, Phys. Rev. A 3, M. Levy, Phys. Rev. A 6, M. Levy, Phys. Rev. A 43, , and references therein. A. Görling and M. Levy, Phys. Rev. B 47, ; For a recent review of Görling-Levy perturbation theory see M. Levy, S. Ivanov, and A. Görling, in Electronic Density Functional Theory: Recent Progress and ew Directions, edited by J. F. Dobson, G. Vignale, and M. P. Das Plenum, ew York, 998. M. Levy, Int. J. Quantum Chem. Symp. 3, A. Görling and M. Levy, Phys. Rev. A 45, S. Liu and R. G. Parr, Phys. Rev. A 53, D. C. Langreth and J. P. Perdew, Solid State Commun. 7, O. Gunnarsson and B. I. Lundqvist, Phys. Rev. B 3, A. D. Becke, J. Chem. Phys. 98, A. D. Becke, J. Chem. Phys. 98, M. Levy,. A. March, and. C. Handy, J. Chem. Phys. 04, A. Görling and M. Levy, Int. J. Quantum Chem. 9, 93995; M. Levy and A. Görling, Phys. Rev. A 5, E. A. Hylleraas, Z. Phys. 65, S. Ivanov, R. Lopez-Boada, A. Görling, and M. Levy, J. Chem. Phys. 09, R. Kestner and O. Sinanoglou, Phys. Rev. 8, K. Burke, J. C. Angulo, and J. P. Perdew, Phys. Rev. A 50, K. Burke, J. P. Perdew, and D. C. Langreth, Phys. Rev. Lett. 73, M. Ernzerhof, K. Burke, and J. P. Perdew, J. Chem. Phys. 05, Z. Qian and V. Sahni, Phys. Rev. A 57, S. Kais, D. R. Herschbach,. C. Handy, C. W. Murray, and G. J. Laming, J. Chem. Phys. 99, M. Taut, Phys. Rev. A 48, R. J. White and W. B. Brown, J. Chem. Phys. 53, ; R. J. White and W. B. Brown, ibid. 53, C. Filippi, C. J. Umrigar, and M. Taut, J. Chem. Phys. 00, C. J. Huang and C. J. Umrigar, Phys. Rev. A 56, C. Lee, W. Yang, and R. Parr, Phys. Rev. B 37, L. C. Wilson and M. Levy, Phys. Rev. B 4, J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, ; 78,

Adiabatic connection for near degenerate excited states

PHYSICAL REVIEW A 69, 052510 (2004) Adiabatic connection for near degenerate excited states Fan Zhang Department of Physics and Astronomy, Rutgers University, 136 Frelinghuysen Road, Piscataway, New Jersey