Local expansion of N-representable one-particle

JOURNAL OF CHEMICAL PHYSICS

VOLUME 119, NUMBER 16

22 OCTOBER 2003

Local expansion of N-representable one-particle density matrices yielding a prescribed electron density Jańos Pipeka) and Szilvia Nagy Department of Theoretical Physics, Institute of Physics, Budapest University of Technology and Economics, H-1521 Budapest, Hungary

共Received 23 May 2003; accepted 28 July 2003兲 Multiresolution 共or wavelet兲 analysis offers a strictly local basis set for a systematic introduction of new details into Hilbert space operators. Using this tool we have previously developed an expansion method for density matrices. The set of density operators providing a given electron density plays an essential role in density functional theory, in the minimization of energy expectation values with the constraint that the electron density is fixed. In this contribution, using multiresolution analysis, we present an excellent quality density matrix expansion yielding a prescribed electron density, and compare it to other known methods. Due to the strictly local nature of the applied basis functions, our construction has the specific advantage that the resulting density matrix is correlated and N-representable in the infinite resolution limit. As a further consequence of this scheme we can conclude that the deviation of the exact kinetic energy functional from the Weizsa¨cker term is not a necessary consequence of the particle statistics. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1611176兴

I. INTRODUCTION

applied mathematics in the fields of localized Hilbert-space basis sets and frames. This branch of mathematics, commonly called as multiresolution or wavelets’ analysis, is pioneered by Morlet, Grossmann and Meyer. Further fundamental results have been obtained by Mallat and Daubechies. A historical introduction and a good theoretical overview can be found in the popular textbook of Daubechies.3 Wavelets are often used in many fields of science, especially in data compression, storage of pictures and sound, as well as in noise reduction. As an illustration of the above, consider, that in a real atomic system the details of the electronic structure are not ‘‘distributed’’ evenly over different parts of a molecule. Typical examples are the nuclear and electron–electron cusps. In our previous studies we have shown4,5 that using wavelets 共scaling functions兲 the electron–electron cusp structure of the two-electron density operator can be reproduced extremely well. In the present study we wish to concentrate on oneparticle density operators 共1-matrices兲. The primary issue of applications is to find 1-matrices yielding a prescribed electron density. Such a one-particle density operator automatically satisfies the nuclear cusp condition 共provided that the prescribed density does兲. Another motivation is that all flavors of density functional methods 共DFT兲 are based on applying a variational principle to the minimization of the expectation value of the total energy in terms of the oneelectron density %共r兲共see, e.g., Refs. 6, 7兲. The original formulation of DFT by Hohenberg and Kohn8 was modified mainly in the works of Levy9,10 and Lieb.11,12 Lieb’s universal energy functional is defined as

In the theory of many-electron systems, a longstanding experience is that numerous physical and chemical aspects are attached to the localized description of the electron structure. Understanding chemical bonding is based essentially on one-particle localized orbitals, moreover in calculating electron correlation effects localization is a powerful tool, as well. For a review see Ref. 1 and references therein. A commonly used philosophy for finding an appropriate representation of the electron structure is to apply first a ‘‘smooth description’’ of the system and to consider details later as corrections. The success of calculating the properties of interacting electrons by oversimplified models, like the homogeneous electron gas, shows that such a consideration is a reasonable approximation. Introducing additional details 共as in the statistical theory of atoms兲 usually improves the results. Numerous useful methods 共e.g., gradient expansions in density functional theory兲 are based on the above recognition. The opposite approach by averaging local properties over larger blocks of atomic sites 共renormalization group transformation兲 led to a deep understanding of critical phenomena. Some aspect of these questions are discussed in Ref. 2. Localized orbitals, however, do not provide a systematic possibility for introducing new details into the calculations and for extending the precision of the theoretical approaches. Instead of applying post-Hartree–Fock localized orbitals as a one-electron basis, a hierarchical expansion allowing increased precision in the fine-graining limit 共or moderate accuracy in the coarse-grained resolution兲 will be introduced in this contribution. We will rely on the tremendous success, which has been achieved during the past two decades of

ˆ 兲其. F L关 % 兴 ⫽ inf Tr兵 ⌫ˆ N 共 Tˆ ⫹W Electronic mail: [email protected]

0021-9606/2003/119(16)/8257/9/$20.00

共1兲

ˆ⌫ →% N

a兲

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© 2003 American Institute of Physics

Downloaded 10 May 2005 to 152.66.102.96. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

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J. Chem. Phys., Vol. 119, No. 16, 22 October 2003

J. Pipek and S. Nagy

ˆ are the Here, ⌫ˆ N is the N-electron density operator, Tˆ and W kinetic and electron–electron interaction energy operators of the system, respectively. The symbol ⌫ˆ N →% means that the infimum is to be searched over all the density operators, producing the fixed density % given in the argument of F L . It can be shown13,14 that the set of the operators ⌫ˆ N to be searched is never empty, as for any integrable, non-negative function %共r兲 there exists at least one pure N-electron wave function resulting in the given %. Previously, Valone15,16 has also studied functional 共1兲 in other aspects. In the case of one-particle energy expectation values 共e.g., the interaction with the external potential or the kinetic energy兲, the above expression reduces to a trace formula containing the one-particle density operator ⌫ˆ 1 satisfying ⌫ˆ 1 →%. We will show that knowing just a given part of the multiresolution analysis 共MRA兲 expansion of ⌫ˆ 1 leads to a straightforward calculation of the kinetic energy functional. A further advantage of the MRA expansion of the 1-matrix is that the systematic 共dissociation兲 error of the Hartree–Fock approximation can be eliminated and explained in a physically transparent way due to the local nature of the one-particle basis. Consequently, electron correlation, besides the Pauli principle, affects the internal structure of ⌫ˆ 1 .

II. DENSITY OPERATORS AND MATRICES

The pure state density operator of an N-particle electron system with the wave function ⌿(x1 ,...,xN ) is ⌫ˆ N ⫽ 兩 ⌿ 典具 ⌿ 兩 , and its kernel, the density matrix, is given as

␥ N 共 x1 ,...,xN 兩 x⬘1 ,...,xN⬘ 兲 ⫽⌿ 共 x1 ,...,xN 兲 ⌿ * 共 x1⬘ ,...,xN⬘ 兲 .

共2兲

The notation xi stands for the ith particle’s space and spin coordinates. Density matrices are Hermitic and fulfill the Pauli principle both in primed and unprimed variables,

␥ N 共 x1 ,x2 ,...,xN 兩 x1⬘ ,x⬘2 ,...,xN⬘ 兲 ⫽ ␥ N* 共 x⬘1 ,x2⬘ ,...,xN⬘ 兩 x1 ,x2 ,...,xN 兲 ⫽⫺ ␥ N 共 x2 ,x1 ,...,xN 兩 x⬘1 ,x2⬘ ,...,xN⬘ 兲 .

共3兲

For calculating the expectation values of (p⫺1)-particle operators, the information content of ␥ N is unnecessarily high, and reduced density matrices can be introduced by a consecutive application of the following partial trace operation:

␥ p⫺1 共 x1 ,...,xp⫺1 兩 x1⬘ ,...,x⬘p⫺1 兲 ⫽

p N⫺p⫹1

冕␥

p 共 x1 ,...,xp⫺1 ,xp 兩 x1⬘ ,...,x⬘p⫺1 ,xp 兲 dxp .

共4兲 The integration over the variable xp means a summation over the corresponding spin variable and integration over space variables. Here, we have applied Lo¨wdin’s normalization convention Tr ⌫ˆ p ⫽( Np ). The expectation values of spinindependent operators are determined by the spin-traced density matrices, defined as

␥ sp 共 r1 ,...,rp 兩 r1⬘ ,...,r⬘p 兲 ⫽

兺

␴ 1 ,..., ␴ p

␥ p 共 r1 ␴ 1 ,...,rp ␴ p 兩 r1⬘ ␴ 1 ,...,r⬘p ␴ p 兲 .

共5兲

Typical examples for one-particle operators are the kinetic energy and the interaction with the external field v (r). The kinetic energy expectation value can be written as

具T典⫽

冕

t 共 r兲 dr,

共6兲

where an advantageous choice7 for the kinetic energy density in atomic units is t 共 r兲 ⫽ 21 “"“ ⬘ ␥ s1 共 r兩 r⬘ 兲兩 r⫽r⬘ .

共7兲

The expectation value of the external potential depends only on the diagonal element %(r)⫽ ␥ s1 (r兩 r),

具V典⫽

冕

v共 r兲 % 共 r兲 dr.

共8兲

III. SCALING FUNCTION BASED DENSITY MATRICES

Scaling functions are the primary objects of multiresolution analysis. Supposing that for given purposes a rough description of a function ␸ (r) of the Hilbert space L 2 (R) is satisfactory, it is possible to choose a basis set with the following properties. Each element of the basis is generated by translating a well-behaved ‘‘mother’’ function s 0 (r) on a regular grid with a grid distance b. The attribute ‘‘well behaved’’ means usually that all translated 共but otherwise identical兲 basis functions are orthogonal to each other, normalized and have a finite support. This basis set expands the rough subspace V 0 傺L 2 (R). As the simplest example we refer to the so-called Haar scaling functions, which are integer translations of the characteristic function of the 关0,1兲 interval. The expanded subspace V 0 contains step functions of the infinite lattice with grid points at all integer positions. We will give more general examples below. The expansion of an arbitrary ␸ (r) by this basis set leads to just a coarse grained approximation of ␸ (r). If a refinement of the description is necessary, it can be achieved by a suitable dilation a of the scaling function as s 1 (r)⫽a 1/2s 0 (ar) and by shrinking the grid distance from b to a ⫺1 b. The factor a 1/2 is introduced to keep the new basis set normalized. This set spans the subspace V 1 . Further refinements can be achieved by similar scaling procedures leading to a sequence of subspaces V 0 , V 1 , V 2 , V 3 , etc. Subspace V m is spanned by its orthonormal basis set 兵 s m (r⫺a ⫺m bl) 兩 r苸R,l苸Z其 . The dilated scaling function s m (r)⫽a m/2s 0 (a m r) and as a shorthand notation we introduce s ml (r)⫽s m (r⫺a ⫺m bl). As we require that the expansions of a function ␸ (r) in consecutive subspaces should be refinements of the previous ones, a necessary condition is that V m 傺V m⫹1 , i.e., the scaling function at resolution level m should be expressed as a linear combination of the basis functions of the (m⫹1)th subspace,



One-particle density matrices

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As locality plays an essential role in the description of the electron structure, the application of scaling functions as an expansion basis set offers clear advantages, especially the lack of localization tails due to their compact support. For spin variables scaling function spinors, s mls 共 r, ␴ 兲 ⫽s ml 共 r 兲 ␦ s ␴ ,

共10兲

are introduced, where later the shorthand notations (mls) ⫽ ␮ and (r, ␴ )⫽x will be used. Slater determinants built from scaling function spinors,

␹ ␮ 1 ¯ ␮ N 共 x1 ,...,xN 兲 ⫽ 共 N! 兲 ⫺1/2 det关 s ␮ i 共 x j 兲兴 ,

共11兲

constitute a basis for the N-particle wave functions. The N-particle density matrix constructed according to Eq. 共2兲 can be obviously expressed in terms of bilinear combinations of ␹ ␮ 1 ¯ ␮ N . After straightforward tracing of the space and spin variables, the spin-traced two-particle density matrix can be expanded as5

␥ s2 共 r 1 ,r 2 兩 r 1⬘ ,r 2⬘ 兲 ⫽ FIG. 1. Density expansion 共26兲 of a slowly varying Gaussian function at resolution level m⫽0 and m⫽2, applying Eq. 共27兲. The insets show the A,m scaling functions s 00(r) and s 20(r) used to build the ‘‘basis set’’ ␽ kl (r 兩 r). B,m Note that for real-valued scaling functions ␽ kl (r 兩 r)⫽0. Atomic units were used.

兺

k 1 ,k 2 l 1 ,l 2

␽ A,m 共 r 1 ,r 2 兩 r 1⬘ ,r 2⬘ 兲关 g kA,m 1k2l1l2 k1k2l1l2

⫹g kB,m k l

1 2 1l2

␽ kB,m 共 r 1 ,r 2 兩 r 1⬘ ,r 2⬘ 兲兴 , 1k2l1l2

共12兲

where coefficients g kA,B,m k l l are real numbers. The functions 1 2 1 2

␽ kA,m 共 r 1 ,r 2 兩 r 1⬘ ,r 2⬘ 兲 1k2l1l2 s m共 r 兲 ⫽

兺l h l s m共 ar⫺bl 兲

⫽a ⫺1/2

兺l h l s m⫹1共 r⫺a ⫺1 bl 兲 .

* 共 r 1⬘ 兲 s ml * 共 r 2⬘ 兲 ⫽ 12 关 s mk 1 共 r 1 兲 s mk 2 共 r 2 兲 s ml 1 2 * 共 r 1⬘ 兲 s ml * 共 r 2⬘ 兲 ⫹s mk 2 共 r 1 兲 s mk 1 共 r 2 兲 s ml 2 1

共9兲

Equation 共9兲 is called the refinement equation, and the properties of the scaling functions are coded in the coefficients h l and in constants a and b. In practice, the parameters are commonly chosen as a⫽2 and b⫽1. The Haar scaling functions are defined by h 0 ⫽h 1 ⫽1 and all other coefficients h l ⫽0. Another example is the well-known Daubechies-4 scaling function3 with four nonzero refinement coefficients h 0 ⫽(1⫹ 冑3)/4, h 1 ⫽(3⫹ 冑3)/4, h 2 ⫽(3⫺ 冑3)/4, and h 3 ⫽(1 ⫺ 冑3)/4. Given the coefficients h l , the refinement equation 共9兲 offers the possibility of calculating the numerical value of s 0 (r) at any diadic points of R by an iterative algorithm.17 The Daubechies-4 scaling function calculated using this method is plotted in the insets of Figs. 1 and 2. This function, although it looks quite irregular, is continuous and has a finite support of 关0,3兲. By slightly increasing the number of nonzero coefficients h l , the resulting scaling function becomes differentiable and still has a compact support. In the infinitely fine resolution limit subset V m is dense in the Hilbert space L 2 (R), as m→⬁. The exact mathematical formulation of multiresolution analysis can be found in the textbooks by Daubechies3 and Chui,17,18 as well as in Refs. 19–23. We have also summarized the principles of MRA in our previous publications.4,5 The method given above has a straightforward extension to three-dimensional configuration spaces, but the explanations are simpler and more concise in one dimension.

* 共 r ⬘1 兲 s mk * 共 r 2⬘ 兲 ⫹s ml 1 共 r 1 兲 s ml 2 共 r 2 兲 s mk 1 2 * 共 r 1⬘ 兲 s mk * 共 r 2⬘ 兲兴 ⫹s ml 2 共 r 1 兲 s ml 1 共 r 2 兲 s mk 2 1

共13兲

and

␽ kB,m 共 r 1 ,r 2 兩 r 1⬘ ,r ⬘2 兲 1k2l1l2 i * 共 r ⬘1 兲 s ml * 共 r ⬘2 兲 ⫽ 关 s mk 1 共 r 1 兲 s mk 2 共 r 2 兲 s ml 1 2 2

* 共 r 1⬘ 兲 s ml * 共 r 2⬘ 兲 ⫹s mk 2 共 r 1 兲 s mk 1 共 r 2 兲 s ml 2 1 * 共 r ⬘1 兲 s mk * 共 r 2⬘ 兲 ⫺s ml 1 共 r 1 兲 s ml 2 共 r 2 兲 s mk 1 2 * 共 r ⬘1 兲 s mk * 共 r 2⬘ 兲兴 , ⫺s ml 2 共 r 1 兲 s ml 1 共 r 2 兲 s mk 2 1

共14兲

are suitable to expand any density matrix ␥ s2 . Although density matrices do not constitute a Hilbert space, the set of 兵 ␽ kA,B,m 其 acts similarly to an expansion basis. For this rea1k2l1l2 son we will call them loosely a ‘‘basis set’’ for ␥ s2 . Tracing one of the space variables, the one-particle spintraced density matrix can be written as A,m B,m B,m ␥ s1 共 r 兩 r ⬘ 兲 ⫽ 兺关 g A,m kl ␽ kl 共 r 兩 r ⬘ 兲 ⫹g kl ␽ kl 共 r 兩 r ⬘ 兲兴 , k,l

共15兲 where the basis functions are


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FIG. 2. Density expansion 共26兲 of a model density of the carbon monoxide molecule 共dotted line兲 at resolution levels m⫽0, m⫽3, and m⫽6. For the expansion coefficients we have applied Eq. 共27兲. The insets illustrate the scaling functions s 00(r), s 30(r), and s 60(r), respectively. Note that the scales of the insets and the molecular density are not comparable for technical reasons. Also, the proportions of the vertical and horizontal dimensions of the scaling functions are not exactly realistic. Atomic units were used.

1 * 共 r ⬘ 兲 ⫹s ml 共 r 兲 s mk * 共 r ⬘ 兲兴 , ␽ A,m kl 共 r 兩 r ⬘ 兲 ⫽ 2 关 s mk 共 r 兲 s ml

共16兲

IV. APPROXIMATION OF THE ELECTRON DENSITY AT RESOLUTION LEVEL m

and i * 共 r ⬘ 兲 ⫺s ml 共 r 兲 s mk * 共 r ⬘ 兲兴 . ␽ B,m kl 共 r 兩 r ⬘ 兲 ⫽ 关 s mk 共 r 兲 s ml 2

共17兲

The mth level decomposition of the electron density arises by taking the diagonal element of 共15兲, % m共 r 兲 ⫽

A,m B,m B,m 关 g A,m 兺 kl ␽ kl 共 r 兩 r 兲 ⫹g kl ␽ kl 共 r 兩 r 兲兴 . k,l

共18兲

The expectation value of one-particle operators can be rewritten by substituting ␥ s1 for its scaling function expansion 共15兲 in Eq. 共6兲 using definition 共7兲 and in Eq. 共8兲. The kinetic energy expectation value reads as A,m B,m B,m 具 T 典 ⫽ 兺共 g A,m kl T kl ⫹g kl T kl 兲 , k,l

共19兲

where the quantities ⫽ T A,B,m kl

1 2

冕

d d A,B,m ␽ 共 r 兩 r ⬘ 兲兩 r⫽r ⬘ dr dr dr ⬘ kl

共20兲

are universal for any systems and depend only on the value of 兩 k⫺l 兩 . Since s ml are compactly supported, even for moderate distances 兩 k⫺l 兩 the scaling functions as well as their derivatives do not overlap, causing that only a few nonzero T A,B,m appear in expansion 共19兲. The expectation value of kl the external potential can clearly be decomposed using 共18兲 as A,m B,m B,m 具 V 典 ⫽ 兺共 g A,m kl V kl ⫹g kl V kl 兲 , k,l

共21兲

with ⫽ V A,B,m kl

冕

v共 r 兲 ␽ A,B,m 共 r 兩 r 兲 dr. kl

共22兲

Again, for larger 兩 k⫺l 兩 differences, V A,B,m disappear. More kl precisely, if s 0 (r) is supported on the interval 关 0,L), all ⫽0 and V A,B,m ⫽0 for 兩 k⫺l 兩 ⭓L. quantities T A,B,m kl kl

According to DFT philosophy, the primary quantity %(r) determines the expectation values 具 T 典 and 具 V 典 by finding the minimum of expressions 共6兲 and 共8兲 among those one-particle density matrices ␥ s1 that are compatible with the preselected density. This statement can be traced at the scaling function expansions 共15兲, 共18兲, 共19兲, and 共21兲 in the following way. At first, we note that all considered expressions contain the same expansion coefficients g A,B,m . However, in kl % m (r) those terms do not contribute to the summation, for which 兩 k⫺l 兩 ⭓L, as s mk (r) and s ml (r) do not overlap in these cases. As we have mentioned previously, a similar argumentation is valid for the expectation values 共19兲 and 共21兲, as well. Recognizing this fact, we might conclude that if we were able to calculate the expansion coefficients g A,B,m with kl 兩 k⫺l 兩 ⬍L from %(r), the electron density would directly determine 具 T 典 and 具 V 典 . In the remaining part of this publication we will investigate the question how and to what extent the above project can be realized. Two remarks are appropriate here. As the set 兵 s mk (r) 其 is a basis in the subspace V m , the product functions s mk (r)s ml (r ⬘ ) form a basis set in the direct product space, consequently, given a ␥ s1 (r 兩 r ⬘ ) the expansion coefficients can be, in principle, determined unambiguously for all g A,B,m kl values of k and l. A similar statement, however, is not valid for %(r), since the functions s mk (r)s ml (r) do not form a basis set in a Hilbert-space sense. We conclude that a presewith 兩 k⫺l 兩 ⬍L lected density %(r) determines only g A,B,m kl 共which would not be a problem in calculating 具 T 典 and 具 V 典 ), but these values are probably not unique. We will show below a method for constructing one possible set of expansion coefficients for a given density %(r). As the scaling functions known and widely used in the literature are real, in the following consideration we will asA,m * , yielding all ␽ B,m sume s mk ⫽s mk kl (r 兩 r)⫽0 and ␽ kl (r 兩 r) ⫽s mk (r)s ml (r). We will also assume that the scaling functions are compactly supported on the interval 关 0,L). As %(r)⭓0 and 兰 %(r)dr⫽N, the function % 1/2(r) is square integrable and can be approximated at resolution level m in subspace V m as



兺k c mk s mk共 r 兲 ,

% 1/2共 r 兲 ⬇


8261

共23兲

where cm k⫽

冕

% 1/2共 r 兲 s mk 共 r 兲 dr⫽% 1/2共 ␰ 兲

冕

s mk 共 r 兲 dr.

共24兲

In the second equality we have applied the integral mean value theorem for continuous functions. The position ␰ is in the support of s mk (r), i.e., ␰ 苸2 ⫺m 关 k,k⫹L). Using the definition of s mk (r) and the fact known from MRA studies17 that 兰 s 0 (r)dr⫽1, we have 兰 s mk (r)dr⫽2 ⫺m/2, which leads to ⫺m/2 1/2 ⫺m cm % „2 共 k⫹ ␦ 兲 …. k ⫽2

共25兲

The parameter ␦ should be in support of s 0 , i.e., 0⭐ ␦ ⬍L. In practical calculations, ␦ is used to get the best fit of %(r). Substituting 共25兲 into 共23兲, we arrive at % m共 r 兲 ⫽

兺kl g A,m kl s mk 共 r 兲 s ml 共 r 兲 ,

共26兲

with the expansion coefficients ⫺m 1/2 ⫺m g A,m % „2 共 k⫹ ␦ 兲 …% 1/2„2 ⫺m 共 l⫹ ␦ 兲 …. kl ⫽2

共27兲

We would like to emphasize that, for the expansion 共23兲 of % 1/2, all the coefficients c m k with indices k苸Z are necessary, are relevant for while according to 共26兲 only those g A,m kl which the relation 兩 k⫺l 兩 ⬍L holds. The expansion coefficients with 兩 k⫺l 兩 ⭓L can be chosen arbitrarily. As the set 兵 s mk (r),k苸Z其 is dense in L 2 (R) in the m→⬁ limit, expansion 共26兲 with 共27兲 converges to %(r) in the fine resolution limit. In order to examine the convergence of the above described method, we have carried out numerical calculations both for slowly and rapidly varying densities. In all calculations we have used the scaling function Daubechies-4, which is supported on the interval 关0,3兲; consequently, only the diagonal and the first and second off-diagonal elements of macontribute to the expansion of any density functrix g A,m kl tions. In Fig. 1 we have plotted the zero and second level expansions of a slowly varying Gaussian function exp„⫺r 2 /(2 ␴ 2 )… with ␴⫽3. The zero level approximation turns out to be surprisingly good, considering the fact that the width of the Gaussian is only twice the support of the scaling function s 00(r). The second level expansion shown in the figure is almost perfect. The excellent convergence found has an MRA specific mathematical background. The particular property of scaling functions is that at any resolution level m they satisfy the identity ⬁

1⫽2

⫺m/2

兺

k⫽⫺⬁

s mk 共 r 兲 ,

共28兲

called the partition of unity. This means that all constant functions 关which are not elements of L 2 (R)] can also be expanded by scaling functions in a natural way. Statement 共28兲 is proved, e.g., in Refs. 3 and 17. According to this fact, the density of the homogeneous electron gas %(r)⫽% 0 given ⫺m % 0 is exact at any resolution level m. by 共26兲 with g A,m kl ⫽2

FIG. 3. The error of the density expansion in Fig. 2 at various resolution levels. Atomic units were used.

As a consequence, the error of expansion of slowly varying densities is unexpectedly small, even at low resolutions. A further improvement can be achieved by applying special scaling functions called coiflets,3 which are able to describe higher-order polynomials exactly. Rapidly varying densities occur typically at nuclear cusps. In order to test our approach for real molecular systems, we have approximated the one-dimensional intersection of a model density of the carbon monoxide molecule. The molecular density is estimated as a sum of the atomic densities of the individual oxygen and carbon atoms calculated according to Slater’s rule.24 The results of the calculations in Fig. 2 show that the applied expansion method results in a considerably good approximation, even in the neighborhood of the nuclear cusps. We can also realize that the regions far from the cusps are described extremely well, even at low resolution levels. For studying the errors around the nuclei we have plotted the difference between the approximated model density and the approximating function 共18兲 in Fig. 3.

V. N-REPRESENTABLE DENSITY MATRICES YIELDING A PRESCRIBED DENSITY

We return now to the question of how the mth level representation of a density % m (r) determines the corresponding mth level expansion of ␥ s1 . As we have mentioned previously 共for real scaling functions兲, the values of g B,m kl do not with 兩 k⫺l 兩 ⭓L. This affect % m (r), nor do the values of g A,m kl provides considerable freedom in choosing ␥ s1 that are compatible with the given mth level approximation of the density. One possibility is using 共27兲 for an arbitrary index combination and setting g B,m kl ⫽0. It may be possible, however, that such a density operator does not correspond to any real physical system. This question is known as the N-representability problem of density operators, which has a full solution for the one-particle matrix ␥ 1 , known as the generalized Pauli principle.25 Briefly, ␥ 1 is N-representable if all of its eigenvalues 共i.e., the natural occupation numbers兲 satisfy the condition 0⭐ ␯ i ⭐1. For the eigenvalues of ␥ s1 we have 0⭐n i ⭐2. It is clear that the eigenvalue problem,


8262


冕␥

s 1 共 r 兩 r ⬘ 兲 ␸ 共 r ⬘ 兲 dr ⬘ ⫽n ␸ 共 r 兲 ,


共29兲

is equivalent to that of the matrix g A,m kl , m m 兺l g A,m kl a l ⫽na k ,

共30兲

where the natural orbitals ␸ (r)⫽ 兺 l a m l s ml (r). Let us consider the eigenvalue problem, 共31兲

Ga⫽na,

where the matrix elements of G are defined according to 共25兲 m m and 共27兲 as G kl ⫽g A,m kl ⫽c k c l . A straightforward calculation leads to the result that one of the eigenvectors of problem 共31兲 a⫽c with the eigenvalue n⫽Tr G. Here, we have introduced the vector notation c⫽(c m k ). All other eigenvectors are orthogonal to c and belong to the eigenvalue zero. As according to Lo¨wdin’s normalization n⫽Tr G⫽Tr ⌫ˆ 1 ⫽N, it is clear that each particle of the system occupies the natural orbital a⫽c, i.e., the density matrix defined by the matrix elements 共27兲 corresponds to a Bose condensate. Consequently, the definition 共27兲 has to be modified in order to describe a fermionic system properly. Note that the discussed approximation corresponds to the mth level expansion s (r 兩 r ⬘ ) of the Weizsa¨cker-type density operator ␥ 1W 1/2 1/2 ⫽% (r)% (r ⬘ ). An obvious possibility for creating an N-representable fermionic one-particle density matrix is to apply the Macke– Harriman construction13,14 of orthogonal one-particle orbitals, each belonging to the same density function. For any integer K the orbital,

␸ K共 r 兲 ⫽

冉冊 %共 r 兲 N

1/2

e i 关 K f 共 r 兲 ⫹ ␾ 共 r 兲兴 ,

共32兲

results in the one-particle density %(r)/N. For any index pairs K and K ⬘ , 具 ␸ K 兩 ␸ K ⬘ 典 ⫽ ␦ KK ⬘ if f 共 r 兲⫽

2␲ N

冕

r

⫺⬁

% 共 x 兲 dx,

共33兲

and ␾ (r) is an arbitrary phase function. It is evident that a Slater determinant built from N orbitals with different K values gives the requested total density %(r). The same N-particle wave function can be used to calculate the s (r 兩 r ⬘ ). By N-representable one-particle density matrix ␥ 1MH choosing the index set K⫽0,...,N⫺1 and ␾ (r)⫽⫺ 关 (N ⫺1)/2兴 f (r), we arrive at N „f 共 r 兲 ⫺ f 共 r ⬘ 兲 … 2 1 1/2 s 1/2 ␥ 1MH共 r 兩 r ⬘ 兲 ⫽ % 共 r 兲 % 共 r ⬘ 兲 . 1 N sin „f 共 r 兲 ⫺ f 共 r ⬘ 兲 … 2 共34兲 sin

Translating the formula 共34兲 to the scaling function representation, we should select the sampling sites r⫽2 ⫺m (l ⫹ ␦ ) and r ⬘ ⫽2 ⫺m (k⫹ ␦ ), leading to ⫺m s g A,m ␥ 1 „2 ⫺m 共 l⫹ ␦ 兲兩 2 ⫺m 共 k⫹ ␦ 兲 …, kl ⫽2

共35兲

FIG. 4. Expansion 共18兲 of a slowly varying six-particle Gaussian density with ␴⫽3, according to the Fourier-type fermionization 共36兲 at resolution levels m⫽2 and m⫽4. Atomic units were used.

where the prefactor 2 ⫺m is a consequence of the normalization of the scaling functions. For the Macke–Harriman density matrix 共34兲, the expansion coefficients are N 共 f „2 ⫺m 共 l⫹ ␦ 兲 …⫺ f „2 ⫺m 共 k⫹ ␦ 兲 …兲 2 1 A,m ⫽ cm cm . g kl 1 N k l sin 共 f „2 ⫺m 共 l⫹ ␦ 兲 …⫺ f „2 ⫺m 共 k⫹ ␦ 兲 …兲 2 共36兲 sin

This formula reproduces the Weizsa¨cker-type density matrix G if 兩 k⫺l 兩 is small or in the fine resolution limit, as m →⬁. We have calculated the electron density resulting from expansion 共18兲 with the coefficients 共36兲 for the slowly varying Gaussian model density used previously. As we see in Fig. 4, the Fourier-type orbitals 共32兲 lead to a relatively crude approximation 共compared to Fig. 1兲 at low resolution levels, although convergence seems to be attained as m→⬁. This experience shows that the representation of the delocalized Fourier-type orbitals 共32兲 by localized scaling functions is inadequate from the practical point of view. VI. INCLUSION OF ELECTRON CORRELATION

Although the Macke–Harriman construction leads to an s (r 兩 r ⬘ ) corresponds to N-representable density operator, ␥ 1MH an independent particle approximation. As the role of electron correlation in large electron systems is widely known, we will study here the question of how the correlation appears in scaling function expansions. The dissociation error of the restricted Hartree–Fock 共RHF兲 method can be illustrated by the model example of two well-separated, noninteracting H atoms. The physically inappropriate RHF wave function describes both electrons evenly distributed among




the two atoms. For a heuristic consideration, let the atoms be situated at the positions R⫽2 ⫺m l and R⫽2 ⫺m k, and have the ‘‘1s’’ orbitals s ml and s mk , thus the RHF MOs are

␸ ↑ 共 x兲 ⫽2 ⫺1/2关 s ml↑ 共 x兲 ⫹s mk↑ 共 x兲兴 , ␸ ↓ 共 x兲 ⫽2 ⫺1/2关 s ml↓ 共 x兲 ⫹s mk↓ 共 x兲兴 .

共37兲

The spin-dependent 1-matrix with occupation numbers 1 and 1 for up and down spins is

␥ 1 共 x兩 x⬘ 兲 ⫽ ␸ ↑ 共 x兲 ␸ * ↑ 共 x⬘ 兲 ⫹ ␸ ↓ 共 x 兲 ␸ * ↓ 共 x⬘ 兲 .

共38兲

8263

and 艛I i ⫽Z, thus sets I i cover the entire scaling function index space. We introduce the truncated matrix G⬘ with the elements

⬘⫽ G kl

再

0,

if k苸I i ,

m cm k cl ,

l苸I j ,

if both,

i⫽ j,

k and l苸I i .

共44兲

⬘ is zero if This definition automatically satisfies 共42兲 as G kl indices k and l are in distant sets. As an illustration the structure of blocks of G⬘ is shown here if Z is separated into two disjoint sets I 1 and I 2 .

Using 共37兲 and after spin tracing we arrive at

␥ s1 共 r 兩 r ⬘ 兲 ⫽s ml 共 r 兲 s ml 共 r ⬘ 兲 ⫹s mk 共 r 兲 s ml 共 r ⬘ 兲 ⫹s ml 共 r 兲 s mk 共 r ⬘ 兲 ⫹s mk 共 r 兲 s mk 共 r ⬘ 兲 ,

共39兲

A,m A,m B,m B,m B,m showing that g A,m kl ⫽g ll ⫽g kk ⫽1 and g kl ⫽g ll ⫽g kk ⫽0. On the other hand, the proper arrangement of electrons corresponds to the unrestricted Hartree–Fock 共UHF兲 solution with two separated electrons on the atoms,

␸ ↑ 共 x兲 ⫽s ml↑ 共 x兲 , ␸ ↓ 共 x兲 ⫽s mk↓ 共 x兲 .

共40兲

Calculating the spin traced 1-matrix as above, we get

␥ s1 共 r 兩 r ⬘ 兲 ⫽s ml 共 r 兲 s ml 共 r ⬘ 兲 ⫹s mk 共 r 兲 s mk 共 r ⬘ 兲 ,

共41兲

A,m A,m or g A,m kl ⫽0 and g ll ⫽g kk ⫽1. Note that the eigenvalues of s the RHF-type matrix ␥ 1 are 2 and 0, whereas the UHF-type eigenvalues are 1 and 1. This model leads to the general assumption that those 1-matrices ␥ s1 , which reflect the electron correlation correctly, should have the property

g A,m kl →0,

for large 兩 k⫺l 兩 .

共42兲

This recognition shows that for correlated systems expression 共27兲 fails for large index differences 兩 k⫺l 兩 . Furthermore, the well-known limiting behavior,26,27 s ␥ s1 共 r 兩 r ⬘ 兲 ——→ ␥ 1W 共 r兩r⬘兲,

共43兲

r,r ⬘ →⬁

is valid only if r and r ⬘ approach infinity simultaneously, i.e., if 兩 r⫺r ⬘ 兩 remains small. The above discussed procedures of constructing oneparticle density matrices for a prescribed electron density are either bosonic or independent particle approximations. We will show here that an approach using local arguments leads to a 1-matrix, which is correlated, i.e., fulfills 共42兲, N-representable in the infinite resolution limit and at the same time conserves the excellent quality density of the bosonic construction. As we have mentioned previously, most elements of g A,m kl are irrelevant with respect to expanding %(r). In the following considerations certain elements of matrix G will be set to zero instead of using expression 共27兲. We will summarize below how the zero elements of matrix G should be selected. Let us divide the space into nonoverlapping regions, using the local nature of the scaling functions. We denote a set of scaling function indices that characterizes a given region of the space by I i 傺Z. We demand that I i 艚I j ⫽⭋ if i⫽ j,

Similarly to the bosonic case, the eigenvectors of G⬘ with nonzero eigenvalues are clearly c(1) and c(2) , where c 共ki 兲 ⫽

再

cm k , 0,

if k苸I i , otherwise.

共45兲

The corresponding eigenvalues are n 共 i 兲⫽

兺

k苸I i

G kk .

共46兲

All the other eigenvectors belong to the eigenvalue zero, and follow the block structure of matrix G⬘ as a⫽(a(1) ,0) or a ⫽(0,a(2) ), where a(1)⬜c(1) and a(2)⬜c(2) . The expression 共46兲 shows that a one-particle density matrix expanded by 共15兲 using the matrix elements of G⬘ as expansion coefficients, satisfies the generalized Pauli principle to a good approximation, if the index sets I i are chosen in such a manner that the values of the summations in 共46兲 fulfill the relations 0⭐n (i) ⭐2. Since the support of the scaling functions s mk decreases as ⬃2 ⫺m , any preselected n (i) can be approximated arbitrarily well by the rhs of Eq. 共46兲 if the resolution is fine enough, i.e., the value of m is sufficiently large. Note that the sum of the occupation numbers n (i) of G⬘ gives the total particle number

兺i n 共 i 兲⫽Tr G⬘ ⫽Tr G⫽N.

共47兲

For illustrative calculations we have chosen a sixparticle Gaussian density and divided our scaling function index space to six nonoverlapping intervals, with the requirement n (i) ⫽1, (i⫽1,...,6). The constants c m k are determined according to 共25兲, and we have calculated the eigenvalues of the resulting matrix G⬘ numerically. As it is expected, the error of the nonzero eigenvalues disappears exponentially as m→⬁. We have plotted the density that arises according to 共18兲 in Fig. 5, using the expansion coefficients 共44兲. Although the range of considerable deviations decreases with


8264



FIG. 6. The deviation from the limits of the eigenvalues of the density matrix originated from the two-particle Gaussian density with ␴⫽3, calculated according to G⬙, as a function of the resolution level m. The symbols 䊊 and ⫹ denote the deviation of the eigenvalues tending to 1, the other symbols 共⫻, 〫, *, and 䊐兲 show the magnitude of eigenvalues tending to zero. Atomic units were used.

FIG. 5. Expansion 共18兲 of the slowly varying six-particle Gaussian density with ␴⫽3, using the matrix elements of G⬘ as expansion coefficients at resolution levels m⫽2 and m⫽4. Atomic units were used.

increasing resolution, unfortunately, the amplitude of the error compared to the required %(r) remains constant. In order to avoid this unsatisfactory behavior we will slightly modify the definition of G⬘ in the following way:

By the construction of G⬙ it is clear that the diagonal, first, and second off-diagonal elements of the matrix are the same as those of the Weizsa¨cker-type density operator. According to our previous note, these are the terms that determine the kinetic energy expectation value; thus, we conclude that G⬙ corresponds to a density matrix that is correlated, satisfies the Pauli-principle in m→⬁ limit, but at the same time yields the Weizsa¨cker kinetic energy term. This result shows that presence of the ‘‘statistical’’ correction ˜␥ in the factorization28 s ␥ s1 共 r 兩 r ⬘ 兲 ⫽ ␥ 1W 共 r 兩 r ⬘ 兲 ˜␥ 共 r 兩 r ⬘ 兲

共48兲

cannot be attributed purely to the exchange effects.

VII. SUMMARY

The illustration is for N⫽2. We recall, that for the Daubechies-4 scaling function, only the first and second offdiagonal elements of the matrix g A,m affect the value of the kl density. This leads to the natural assumption that completing the zero blocks of G⬘ by the necessary first and second offdiagonal elements according to 共27兲 results in an excellent %(r) approximation, and at the same time N-representability is retained in the m→⬁ limit. Numerical observations have shown that introducing 2⫻2 blocks according to the diagram defining G⬙ leads also to the above expectations with faster convergence. The density produced by G⬙ is the same as in the bosonic case 共see, e.g., Fig. 1 or 2兲. Regarding N-representability, we consider the case N ⫽2. It can be proved that all but six eigenvalues of G⬙ are exactly zero. As Fig. 6 shows, two of the nonzero eigenvalues converge exponentially to unity, and the remaining four tend to zero at the fine resolution limit.

In density functional approaches the basic interest is to create density matrices resulting in a given density function %(r). We have shown that the tool of multiresolution analysis using localized scaling functions leads to a natural representation of density operators and electron densities at any resolution level. We have developed a method for finding the expansion coefficients of the one-particle density matrix using density data only. In the simplest case the resulting density operator describes a Bose condensate, however, by a proper modification, the method yields an N-representable fermionic one-particle density operator in the fine resolution limit, and at the same time an excellent quality density function, even at low resolutions. We have compared the results of our method to the Macke–Harriman construction based on delocalized Fouriertype orbitals. It turned out that the localized nature of the scaling functions describes the details of the electron density better than the delocalized representation. This observation is similar to the fact found earlier in the field of compression and storage of picture and sound, which made the wavelet based description preferable to the Fourier expansion methods.



Due to the localized representation, we were able to include the electron correlation in the sense that the constructed density operator goes beyond the restricted Hartree– Fock level. We have also shown that the deviation of the exact kinetic energy term from the Weizsa¨cker expression is not a necessary consequence of the Pauli principle since by a direct construction we have found a density operator that is correlated, N-representable in the fine resolution limit, although it results in the Weizsa¨cker kinetic energy term.

ACKNOWLEDGMENTS

This work was supported by the Orsza´gos Tudomańyos Kutata´si Alap 共OTKA兲, Grants No. T032116 and No. T042981. 1

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