Economical Models for Electron Densities

Abstract

We discuss a new theoretical framework for modeling molecular electron densities. Our approach decomposes the total density into contributions from basis function products and then approximates each product using constrained least–squares approximation in a tailored local basis of functions with adjustable non–linear parameters. We show how to solve directly for the expansion coefficients and Lagrange multipliers and present an iterative method to optimize the non–linear parameters. Example products from the Dunning cc-pVTZ basis set are discussed.

1. Introduction

1.1. Background

The central tenet of density functional theory (DFT) is that, notwithstanding its conceptual simplicity, the electron density D(r) in a system is a molecular property from which all others can be derived.¹ Consequently, its computation, representation, and visualization are among the most fundamental features of all modern electronic structure software packages. Our goal in this work is to develop a systematic scheme for efficiently constructing accurate approximations to D(r).

In terms of a set of orthonormal orbitals ψ_i(r), we can write the electron density as^2,3

1

and, if each of these orbitals is expanded in a one-electron basis set ϕ_μ(r), we obtain

2

where P_μν is a density matrix. If each basis function is a contracted Gaussian

3

where Y_μ(r) is the angular factor, C^k_μ is a contraction coefficient, α^k_μ is an exponent, and A_μ is the Gaussian center, then the function products in (2) are

4

where we have introduced the Gaussian density

5

and its primitive prefactors, exponents and centers are, respectively,

6

7

8

The electrostatic energy between two products is given by the two-electron integral

9

and these integrals yield the Coulomb, exchange, and correlation energies.⁴ The evaluation of the (μν|λσ) is one of the key bottlenecks of quantum chemistry, and research over many decades has led to an impressive array of algorithms for their computation, including the methods of Boys,⁵ Dupuis et al.,⁶ McMurchie and Davidson,⁷ Obara and Saika,⁸ Head-Gordon and Pople,⁹ Hamilton and Schaefer,¹⁰ Lindh et al.,¹¹ Gill and Pople,^12,13 Adams et al.,¹⁴ Ishida,¹⁵ Makowski,¹⁶ and Komornicki and King.¹⁷ Nonetheless, although some of these are very computationally efficient, all become expensive when the degrees of contraction K_μK_ν and K_λK_σ are large. For this reason, it was recognized long ago that it is desirable to construct approximations for products.

In the 20th century, a variety of methods for fitting function products in an auxiliary basis were introduced and we note the seminal contributions by Reeves and Fletcher,¹⁸ O-ohata et al.,¹⁹ Newton et al.,²⁰ Hehre et al.,²¹ Stewart,²² Billingsley and Bloor,²³ Baerends et al.,²⁴ Whitten,²⁵ Beebe and Linderberg,²⁶ Dunlap et al.,²⁷ Fortunelli and Salvetti,²⁸ Feyereisen et al.,²⁹ and Eichkorn et al.³⁰ Helpful discussions of the underlying theory were published subsequently by a number of workers.³¹⁻³⁷

In the present paper, however, we eschew a global auxiliary basis in favor of approximating each product on its own locally generated basis. More specifically, we seek an approximation to the Gaussian density (5) that uses a few Gaussians whose coefficients, exponents and centers are explicitly optimized.

Throughout this paper, we will use italic symbols to denote scalars and bold (or double-struck) symbols to denote vectors or matrices.

1.2. Least-Squares Functional

We seek to approximate a given density ρ(r) by a model χ(r) where

10

11

where d = (d₁, ..., d_n) is a list of positive coefficients, c = (c₁, ..., c_m) is a list of expansion coefficients, a = [a₁(r), ..., a_n(r)] and b = [b₁(r), ..., b_m(r)] are lists of normalized Gaussians

12

13

α = (α₁, ...,α_n) and β = (β₁, ...,β_m) are lists of inverted exponents, and A = (A₁, ..., A_n) and B = (B₁, ..., B_m) are lists of Gaussian centers on the z axis. The assumption that all d_i > 0 sometimes requires that the basis set be reconstructed and this is discussed in Section 2.1. We assume that m < n, and we regard all of the c_j, β_j, and B_j as optimizable parameters.

For simplicity, we recast the problem in Fourier space and model ρ̂(k) by χ̂(k) where

14

15

and the elements of â and b̂ are the Fourier transforms

16

17

Our models seek to minimize by least-squares the residual

18

over all of space and, to achieve this, we introduce a Fourier weight function

19

In general, our models must also satisfy l linear constraints on the expansion coefficients c_i. These are conveniently captured by the matrix equation

20

where is an m × l matrix and is a vector of length l. For example, if we seek a model that conserves charge in an ss density, must contain a column of ones and the corresponding element of must hold ρ̂(0), i.e. the charge of ρ(r). We therefore introduce a vector of Lagrange multipliers and seek a stationary point of the functional

21

1.3. Weight Function

The parameter p in (19) determines the type of model that we form and we will write L_p(m) to denote an m-Gaussian least-squares model with parameter p and charge conservation

22

constrained. If p > 0, the fit emphasizes high-frequency components of the density; if p < 0, it emphasizes low-frequency components. Historically, p = 3/2, 1/2, or −1/2 have been advocated, but these are just three points on a continuum of possibilities.

L_3/2(m) models minimize the square of the residual density R(r) itself. Such models were used by Reeves and Fletcher¹⁸ to perform approximate Slater orbital calculations, O-ohata et al.¹⁹ to construct expansions of Slater functions, Newton et al.²⁰ in the PDDO approximation, Hehre et al.^21,22 to construct the STO-nG basis sets, Billingsley and Bloor²³ in the LEDO approximation and Baerends et al.²⁴ in their Hartree–Fock–Slater algorithm. Most of these authors used unconstrained fits, but Baerends et al. constrained charge conservation.

L_1/2(m) models minimize the squared norm of the electric field of R(r). Such models were advocated by Whitten²⁵ for modeling densities without charge conservation, used by Dunlap et al.²⁷ in his Coulomb fitting approach, developed further by Feyereisen et al.²⁹ and Eichkorn et al.³⁰ and analyzed in detail by Hall and Martin.³¹

L_–1/2(m) models minimize the square of the potential value of R(r). They were introduced by Fortunelli and Salvetti²⁸ and defined rigorously by Gill et al.³²

The constraint (22) ensures that the integral in (21) exists when weight functions with −2 < p ≤ 0 are used but, for p ≤ −2, additional constraints must be applied. For certain isolated p values, for example p = 0, the weight function (19) is not positive definite and the functional (21) does not have a well-defined stationary point. However, if they are needed, such models can be obtained by taking limits in the neighborhood of p.

To illustrate this, we modeled a single Slater function by a single Gaussian function. In Fourier space, this corresponds to modeling ρ̂(k) = (1 + k²)⁻² by χ̂(k) = exp(−βk²). By minimizing (21), one finds that the optimal β satisfies

23

where U is the Tricomi hypergeometric function.³⁸ Analysis of (23) reveals that

24

and Figure 1 shows that the optimal β varies smoothly and almost linearly over a wide domain.

Figure 1.

Optimal β for modeling a Slater function as the weight function ω_p(k) varies.

1.4. Cost-Benefit Analysis

In a spatially extended system, most pairs of basis functions ϕ_μ and ϕ_ν are well separated, and the product ϕ_μ(r)ϕ_ν(r) can therefore be neglected. It is easy to show that the number of significant ϕ_μ(r)ϕ_ν(r) products grows only linearly with N and, thus, because the shell pair economizations are independent, the cost of economizing all significant shell pairs is O(N). However, it is reasonable to ask whether this effort is worthwhile. There are two types of calculations where the cost is easy to justify.

First, calculations in which the electron density D(r) will be evaluated at a large number K of points r as, for example, on a van der Waals surface. The cost of evaluating (2) at a point r is O(N) and the total cost for all points is therefore O(KN). In such situations, the economization cost will be more than compensated by the subsequent savings in density evaluations.

Second, calculations in which the shell pairs will be used to calculate two-electron integrals (9) as, for example, in a hybrid density functional calculation. The number of significant shell quartets, and therefore the number of significant ERIs, is O(N²). Thus, whereas the cost of economization scales linearly with N, the savings that accrue from the use of economized shell pairs to form ERIs scale quadratically with N and will more than compensate for the economization cost if N is sufficiently large.

2. Methods

2.1. Basis Set Reconstruction

To facilitate the modeling process, we required all contraction coefficients to be positive. This is true of some basis functions (for example, the 5-fold contracted function on H atoms in the Dunning cc-pVTZ basis,³⁹ see Table 1) but it is not true of many basis functions (for example, the 10-fold contracted functions on C atoms in the same basis, see Table 2). However, it is often possible (at least, in shared-exponent basis sets such as those of Dunning³⁹ and Jensen⁴⁰) to reconstruct the basis on each atom, creating all-positive basis functions by linearly combining the originals. Because the reconstructed basis set necessarily has the same span as the original, it is equivalent for quantum chemical purposes.

Table 1. cc-pVTZ Exponents and Coefficients^39,41 of an s Function on H.

Primitive	Exponent	Coefficient
1	33.87	+0.006068
2	5.095	+0.045308
3	1.159	+0.202822
4	0.3258	+0.503903
5	0.1027	+0.383421

Table 2. cc-pVTZ Exponents and Coefficients^39,41 of Two s Functions on C.

	Coefficient
	cc-pVTZ		rec-cc-pVTZ
Primitive	Exponent	v1	v2	u1	u2
1	8236.	+0.000531	–0.000113	+0.000529	+0.000133
2	1235.	+0.004108	–0.000878	+0.004094	+0.001026
3	280.8	+0.021087	–0.004540	+0.021012	+0.005238
4	79.27	+0.081853	–0.018133	+0.081555	+0.019875
5	25.59	+0.234817	–0.055760	+0.233901	+0.053661
6	8.997	+0.434401	–0.126895	+0.432330	+0.077970
7	3.319	+0.346129	–0.170352	+0.343379	0
8	0.9059	+0.039378	+0.140382	+0.041603	+0.143342
9	0.3643	–0.008983	+0.598684	+0.000529	+0.533186
10	0.1285	+0.002385	+0.395389	+0.008666	+0.355805

We formulate our problem as follows. Given a set V of linearly independent vectors , we seek a set U of all-positive vectors {u₁, ..., u_n} with span(U) = span(V).

We define v^min_i as the minimum (i.e., most negative) component of v_i and we then reorder the v_i so that v^min₁ > v^min₂ >··· > v^min_n. We then try to construct a single all-positive vector u₁ from V by writing

25

and seeking the γ_i that maximize u^min₁. This maps onto the linear programming problem

26

with the auxiliary variable t, and solved, for example, by converting to canonical form and applying the Simplex Method.⁴² The optimized t is maximized u^min₁.

If the resulting u^min₁ value is positive, the original vectors are reconstructible. We then form

27

where δ_i is the smallest value that yields u^min_i = 0, and normalize the resulting basis functions. This has the additional benefit of reducing the order of contraction by one even before modeling. We add the prefix “rec-” to the name of a reconstructed basis set.

It is important to note that the reconstructed basis functions are not orthogonal and, in some cases, can be almost parallel. As a consequence, we hope that, in the future, (a) improved reconstruction algorithms will be devised and (b) basis set developers will produce basis sets with all-positive coefficients, so that reconstruction is unnecessary.

If this method is applied to the 10-fold contracted functions on C in the cc-pVTZ basis set (Table 2), we find that u^min₁ is a concave piecewise linear function of γ₂ given by

28

which reaches its maximum value (+0.000529) at γ₂ = 0.0158885. We then form

29

and renormalize both u₁ and u₂. The rec-cc-pVTZ basis functions are shown in Table 2 and we note that their overlap integral is S₁₂ = 0.456.

2.2. Modeling Theory

2.2.1. Linear Parameters

The basic units of the optimization are the scalar

30

the vector

31

and the matrix

32

From these, we can assemble the augmented vector

33

and the augmented matrix

34

and then write (21) as

35

where the coefficients and Lagrange multipliers have been conflated into the vector

36

Requiring that the gradient of (35) with respect to x vanish yields the optimal coefficients and Lagrange multipliers

37

and the resulting minimal value

38

For greatest numerical stability, (37) should be computed by solving a linear system.

Having thus solved for the linear parameters c_i and , we now have a functional (38) that depends only on the nonlinear parameters β_i and B_i. We now discuss the minimization of (38) with respect to those parameters.

2.2.2. Initial Guesses: Spherical Density

If ρ(r) has spherical (K_h) symmetry, we choose all B_i = 0 and eschew further optimization. The exponents β are more challenging, and to construct an initial guess for these, we convert the modeling problem into a quadrature problem. First, we form the integral representation

39

where β₀ > 0 is a scale factor and an Inverse Mellin Transform³⁸ provides the weight function

40

We then approximate (39) by the m-point Gauss–Christoffel quadrature formula⁴³

41

where the weights c_i > 0 and roots 0 < u_i < 1 are related to polynomials that are orthogonal on [0, 1] with respect to w(u). In the Golub-Welsch method⁴⁴ (see Supporting Information), the c_i and u_i are found from the moments

42

and, in order to sample ρ̂(k) well, we choose the arbitrary scale factor to be

43

Finally, by substituting u_i = exp(−β_i/β₀) into (41), we obtain an approximation that we call the Q(m) model

44

Although this model is less accurate than least-squares fitting, it may sometimes be useful in its own right. However, its low cost and reasonable accuracy make it a useful initial guess.

There is an important caveat to the use of a quadrature scheme to generate an initial guess. There is no guarantee⁴⁵ that the Gauss–Christoffel roots u_i and weights c_i will exist or be satisfactory unless the weight function w(u) in (39) is a non-negative function on [0, 1]. It was to ensure this that we assumed that all of the coefficients d_i in (10) are positive.

2.2.3. Initial Guesses: Cylindrical Density

Generating guesses for the exponents β_i and centers B_i when ρ(r) has cylindrical (C_∞v) symmetry is not difficult. In general, because of the primitive prefactors (6), the coefficients d_i of the n primitives in ρ(r) range over many orders of magnitude and ρ(r) is therefore dominated by the primitives with the largest absolute coefficients |d_i|. We exploit this by using the α_i and A_i of the m primitives with the largest |d_i| as our guesses.

We start by ordering the α_i and A_i by |d_i| value and take the first m primitives that are not already “covered” by a larger primitive. A primitive i is considered “covered” if its A_i value is within 0.25 au of the position A_j of a larger primitive j (|d_j| > |d_i|). This is because primitives that are close to each other tend to be represented by a single Gaussian in models, even if they both have large coefficients.

In the case of a tie for the mth largest “uncovered” primitive, there are two options. If it would not break another tie, the (m – 1)th largest primitive is excluded, and the α_i and A_i of the mth and (m + 1)th largest primitives are used. Otherwise, the average of the mth and (m + 1)th largest primitives’ quantities are taken.

Finally, if the guess formed this way has k < m primitives, the m – k largest previously excluded “covered” primitives are included as well.

2.2.4. Matrix Elements and Their Derivatives

For the basis functions (17), one can show from eq (6.631.1) of Gradshteyn and Ryzhik⁴⁶ that both the matrix (32) and its derivatives with respect to the nonlinear parameters (β_i and B_i) can be conveniently expressed in terms of the function

45

where Γ is the gamma function³⁸ and M is the Kummer hypergeometric function.³⁸ The matrix itself is given by

46

and, using properties of the hypergeometric function, one finds the first derivatives

47

48

and the second derivatives

49

50

51

For half-integer values of p, all six of these can be expressed in terms of Gaussians and the error function.³⁸

To construct the gradient of Z, we require the first derivative vectors and matrices

52

53

and, to construct its Hessian, we need the second derivative vectors and matrices

54

55

56

Differentiating (38) shows the gradient and Hessian of Z with respect to β and B are

57

58

where

59

60

61

62

63

and X = diag(x). The square brackets in (57) and (58) indicate that only the first m elements of the enclosed vector, or the first m-by-m block of the enclosed matrix, is retained.

It is well-known, however, that the optimal exponents of Gaussian expansions of exponential densities form a roughly geometrical sequence^35,47 and that the exponents typically span several orders of magnitude. We therefore choose to work with the log-exponents

64

which are roughly evenly spaced and typically of the order of unity. By the chain rule, the elements of the gradient and Hessian of Z with respect to the λ_i are

65

66

67

68

2.2.5. Optimization Step

Given the gradient (g) and Hessian (H) of (38), we are in a position to find the optimal β and B via an iterative scheme. Although the Newton–Raphson method could be used, it is effective only if current β and B are near-optimal and can be disastrous otherwise. We prefer the slower (but more robust) approach taken by the Levenberg–Marquardt method,^48,49 wherein a small fraction σ > 0 of the identity matrix is added to the Hessian to form the shifted-Newton step

69

The shift σ should decay as the optimization proceeds and we have found that σ = 10|g| is effective for all the densities discussed below and hundreds of others that we have explored.

When optimizing β and B, our goal is to reduce Z to within 1% of its minimum value. This is usually achieved when H is positive definite and |Δ| < 10^–4.

2.2.6. Algorithm

The inputs to our algorithm are the density ρ(r), the number m of Gaussians in the desired model, the weight function parameter p, the constraint matrix and the constraint vector . The way in which we generate initial guesses for the nonlinear (β and B) parameters depends on whether the Gaussian density is spherical or cylindrical, but after that, the algorithm becomes the same. The overall algorithm, together with links to key equations, is shown in Table 3.

Table 3. Construction of an m-Gaussian Model for a Gaussian Density.

Step	Description	Equations	Cost
1	Input		O(1)
	If ρ is spherical
2a	Compute ⟨uj⟩ and β0	(42), (43)	O(m)
3a	Compute initial guess for β	(44)	O(m2)
4a	Set B = 0		O(m)
	else if ρ is cylindrical
2b	Sort primitives by |di| value		O(n)
3b	Compute initial guess for β and B		O(m)
	end if
5	Compute f00, f10, f01, f20, f11, f02	(33), (52)–(56)	O(m)
6	Compute F00, F10, F01, F20, F11, F02	(34), (52)–(56)	O(m2)
7	Compute E10, E01, E20, E11, E02	(59) – (63)	O(m)
8	Compute U10, U01, H20, H11, H02	(59) – (63)	O(m3)
9	Compute x	(37)	O(m3)
10	Compute g	(57)	O(m2)
11	Compute H	(58)	O(m3)
12	Compute (β, B) ← (β, B) – Δ	(69)	O(m3)
13	If |Δ| > 10–4 or H is not positive definite, go to Step 5		O(m3)

3. Results and Discussion

3.1. Spherical Densities

3.1.1. Preamble

Armed with the necessary theory, we now consider how it can be applied in calculations using the Dunning cc-pVTZ basis set.³⁹ If the basis functions are concentric, their density is spherical and has the Fourier transform (14)

70

and the associated weight function (40) is

71

where δ is the Dirac delta. If all d_i > 0, then w(u) is non-negative and, from (43), we have

72

To find the Q(m), we need the moments

73

and, to find the L_p(m), we need the integrals

74

75

The Q(m) and L_p(m) models are constructed to mimic the original density ρ(r) as well as possible over all space. It is also important, however, to assess the resulting models χ(r) by measuring the largest pointwise difference between ρ(r) and χ(r). For a spherical density, it is natural to measure this through the maximum Jacobian-weighted error

76

and we report this for each of the models described below.

3.1.2. H(1s1s) Density

Our first example comes from the 5-fold contracted basis function on an H atom in the cc-pVTZ basis set (see Table 1). The product of this function with itself yields a density with n = 15 distinct Gaussians and its Q(m) and L_p(m) models are listed in Table 4.

Table 4. Models for the H(1s²) cc-pVTZ Density.

	Q(m)		L–1/2(m)		L+1/2(m)		L+3/2(m)
m	λi	ci	λi	ci	λi	ci	λi	ci
1	+1.032	1.000	+0.876	1.000	+1.011	1.000	+1.159	1.000
	= 1.8 × 10–1		= 1.7 × 10–1		= 1.8 × 10–1		= 2.1 × 10–1
2	+0.421	0.610	+0.315	0.572	+0.497	0.700	+0.719	0.824
	+1.827	0.390	+1.802	0.428	+2.083	0.300	+2.455	0.176
	= 5.1 × 10–2		= 5.5 × 10–2		= 4.2 × 10–2		= 6.7 × 10–2
3	+0.004	0.262	+0.027	0.308	+0.219	0.456	+0.439	0.621
	+0.935	0.533	+1.102	0.563	+1.396	0.476	+1.741	0.348
	+2.250	0.204	+2.583	0.129	+2.974	0.068	+3.430	0.031
	= 2.8 × 10–2		= 1.8 × 10–2		= 1.2 × 10–2		= 2.0 × 10–2
4	–0.174	0.161	–0.139	0.186	–0.036	0.249	+0.316	0.519
	+0.660	0.465	+0.765	0.521	+0.908	0.521	+1.478	0.412
	+1.564	0.295	+1.849	0.260	+2.027	0.208	+2.838	0.066
	+2.776	0.080	+3.361	0.032	+3.581	0.022	+4.621	0.003
	= 1.2 × 10–2		= 5.4 × 10–3		= 4.5 × 10–3		= 1.2 × 10–2
5	–0.196	0.147	–0.165	0.169	–0.130	0.189	–0.057	0.228
	+0.550	0.325	+0.707	0.499	+0.758	0.506	+0.831	0.496
	+1.008	0.254	+1.709	0.278	+1.787	0.261	+1.854	0.238
	+1.833	0.233	+2.917	0.050	+3.074	0.042	+3.148	0.037
	+3.183	0.041	+4.455	0.004	+4.810	0.002	+4.924	0.002
	= 5.7 × 10–3		= 1.6 × 10–3		= 6.6 × 10–4		= 7.1 × 10–4
6	–0.196	0.147	–0.190	0.152	–0.176	0.161	–0.135	0.183
	+0.539	0.305	+0.610	0.412	+0.660	0.459	+0.730	0.490
	+0.961	0.255	+1.225	0.211	+1.436	0.214	+1.684	0.254
	+1.738	0.219	+1.927	0.187	+2.043	0.132	+2.421	0.046
	+2.411	0.049	+3.147	0.036	+3.192	0.033	+3.268	0.026
	+3.456	0.024	+4.874	0.002	+4.936	0.002	+4.975	0.002
	= 4.7 × 10–3		= 2.5 × 10–4		= 1.5 × 10–4		= 1.8 × 10–4

The exponents of the L_p(m) change smoothly and predictably as p and m are varied. The L_–1/2(m) are the most diffuse (i.e., their λ_i are most negative) and the L_+3/2(m) are the most compact (i.e., their λ_i are the most positive). This is reasonable because, whereas the L_–1/2(m) use a Fourier weight 1/(2π k⁴) that emphasizes low-frequency components of the density and therefore its tail, the L_+3/2(m) use a weight 1/(2πk⁰) that places greater emphasis on the high-frequency components and therefore the region near the nuclear cusp. The L_+1/2(m) exponents are remarkably close to the average of the analogous L_–1/2(m) and L_+3/2(m) exponents, thus generalizing the near-linear behavior seen in Figure 1.

The exponents of Q(m) also change smoothly with m. Their most diffuse exponents are similar to those in L_+1/2(m) when m is small and similar to those in L_–1/2(m) when m is larger but the range λ_m – λ₁ of the Q(m) exponents is smaller than in any of the least-squares models. For applications for which a quick and reasonably accurate approximation is required, the Q(m) models appear useful.

3.1.3. C(1s2s) Density

Our second example comes from the product of the two 10-fold contracted rec-cc-pVTZ basis functions on a C atom (see Table 2). Their product yields a density with n = 55 distinct Gaussians and its Q(m) and L_p(m) models are listed in Table 5.

Table 5. Models for the rec-cc-pVTZ C(1s2s) Density (S = 0.456).

	Q(m)		L–1/2(m)		L+1/2(m)		L+3/2(m)
m	λi	ci/S	λi	ci/S	λi	ci/S	λi	ci/S
1	4.031	1.000	3.386	1.000	3.685	1.000	3.976	1.000
	= 6.3 × 10–1		= 4.3 × 10–1		= 4.9 × 10–1		= 6.0 × 10–1
2	3.273	0.725	2.349	0.441	2.835	0.642	3.284	0.796
	5.098	0.275	4.445	0.559	4.954	0.358	5.454	0.204
	= 1.5 × 10–1		= 1.6 × 10–1		= 1.4 × 10–1		= 2.1 × 10–1
3	2.855	0.524	1.423	0.117	2.334	0.382	2.872	0.615
	4.150	0.332	3.159	0.572	3.958	0.484	4.657	0.342
	5.561	0.144	5.036	0.312	5.699	0.134	6.423	0.043
	= 6.1 × 10–2		= 1.1 × 10–1		= 5.1 × 10–2		= 8.7 × 10–2
4	2.625	0.416	0.854	0.046	1.708	0.155	2.574	0.467
	3.703	0.298	2.622	0.396	3.165	0.485	4.104	0.396
	4.715	0.216	4.027	0.419	4.646	0.307	5.504	0.127
	6.015	0.070	5.646	0.139	6.272	0.053	7.218	0.010
	= 3.1 × 10–2		= 4.8 × 10–2		= 1.9 × 10–2		= 4.0 × 10–2
5	2.544	0.370	0.494	0.023	1.195	0.071	2.273	0.321
	3.369	0.167	2.117	0.187	2.750	0.403	3.596	0.412
	3.941	0.220	3.216	0.416	4.018	0.352	4.874	0.221
	4.877	0.190	4.554	0.306	5.279	0.156	6.180	0.044
	6.191	0.053	6.098	0.068	6.871	0.018	7.879	0.003
	= 2.6 × 10–2		= 2.5 × 10–2		= 9.8 × 10–3		= 2.1 × 10–2
6	2.182	0.143	0.217	0.013	0.957	0.049	1.958	0.201
	2.768	0.281	1.602	0.081	2.534	0.318	3.208	0.416
	3.682	0.266	2.775	0.369	3.609	0.341	4.437	0.276
	4.457	0.156	3.924	0.331	4.732	0.225	5.576	0.093
	5.177	0.119	5.109	0.179	5.920	0.062	6.875	0.013
	6.424	0.035	6.624	0.028	7.503	0.006	8.583	0.001
	= 1.7 × 10–2		= 1.1 × 10–2		= 3.7 × 10–3		= 1.2 × 10–2

3.1.4. Other Densities

We have applied our algorithm for spherical densities to all of the concentric ss products arising in the cc-pVDZ and cc-pVTZ basis sets of Dunning and the pc-1 and pc-2 basis set of Jensen, constructing L_–1/2(m), L_1/2(m) and L_3/2(m) models with m = 1,..., 6.

The maximum error does not show a strong dependence on p but we find that the smallest is often obtained by L_3/2(m) if m is small, and by L_1/2(m) if m is larger.

In many cases, the number m of Gaussians required to accurately model a density with a large number n of Gaussians is surprisingly small. The explanation for this encouraging discovery appears to lie in the distribution of exponents α^kl_μν in a typical basis function product. We see from (7) that the α^kl_μν are simple sums of the α^k_μ and α^l_ν of the parent basis functions but, because the latter typically form a roughly geometrical sequence, the distribution of α^kl_μν is clustered. To illustrate this, consider a hypothetical basis function with α^k_μ = {1000, 100, 10, 1}. The product of this with itself generates a Gaussian density with α^kl_μμ = {2000, 1100, 1010, 1001, 200, 110, 101, 20, 11, 2}, and although these 10 exponents are strictly distinct, the second, third and fourth Gaussians are similar (as are the sixth and seventh Gaussians) and can be modeled quite well by a single Gaussian.

3.2. Cylindrical Densities

3.2.1. Preamble

If two basis functions are on different centers, separated by a distance R > 0, their product yields a Gaussian density ρ(r) with cylindrical symmetry and, to find the L_p(m), we need

77

78

The overlap S = ∫ρ(r) dr of the functions decays rapidly as R increases and it is interesting to compare the accuracies of the L_p(m) models as a function of R.

The L_p(m) models are constructed to mimic the original density ρ(r) as well as possible over all space. It is also important, however, to assess the resulting models χ(r) by measuring the largest pointwise difference between ρ(r) and χ(r). For a cylindrical density, it is natural to measure this through the maximum axial error

79

and we report this for each of the models described below.

3.2.2. H(1s)H(1s) Density

Our first example is the product of the 5-fold contracted cc-pVTZ basis functions (Table 1) on two hydrogen atoms centered at (0, 0, ±R/2). The density consists of 25 Gaussians, and L_–1/2(m) models with m = 1 to 6 Gaussians are shown in Table 6.

Table 6. L_–1/2(m) Models for cc-pVTZ H(1s)H(1s) Densities with Various R.

	R = 4.928, S = 0.100			R = 7.725, S = 0.010			R = 9.995, S = 0.001
m	Bi	λi	ci/S	Bi	λi	ci/S	Bi	λi	ci/S
1	+0.000	+0.015	1.000	+0.000	–0.239	1.000	+0.000	–0.262	1.000
	= 3.6 × 10–3			= 2.2 × 10–4			= 1.0 × 10–5
2	–1.142	+0.282	0.500	–1.287	+0.019	0.500	–0.990	–0.125	0.500
	+1.142	+0.282	0.500	+1.287	+0.019	0.500	+0.990	–0.125	0.500
	= 1.8 × 10–3			= 9.1 × 10–5			= 3.9 × 10–6
3	–1.382	+0.709	0.274	–2.320	+0.657	0.133	–2.743	+0.561	0.062
	+0.000	–0.182	0.452	+0.000	–0.165	0.734	+0.000	–0.193	0.875
	+1.382	+0.709	0.274	+2.320	+0.657	0.133	+2.743	+0.561	0.062
	= 1.1 × 10–3			= 3.6 × 10–5			= 9.6 × 10–7
4	–1.763	+0.913	0.168	–2.323	+0.670	0.132	–2.744	+0.561	0.062
	+0.000	–0.163	0.510	+0.000	–0.184	0.717	+0.000	+3.312	–0.0001
	+0.000	+0.662	0.155	+0.000	+0.583	0.018	+0.000	–0.192	0.875
	+1.763	+0.913	0.168	+2.323	+0.670	0.132	+2.744	+0.561	0.062
	= 8.0 × 10–4			= 3.6 × 10–5			= 9.6 × 10–7
5	–2.229	+1.761	0.030	–3.276	+1.640	0.013	–4.252	+1.696	0.002
	–1.137	+0.554	0.290	–1.982	+0.531	0.146	–2.607	+0.541	0.062
	+0.000	–0.273	0.361	+0.000	–0.198	0.683	+0.000	–0.196	0.870
	+1.137	+0.554	0.290	+1.982	+0.531	0.146	+2.607	+0.541	0.062
	+2.229	+1.761	0.030	+3.276	+1.640	0.013	+4.252	+1.696	0.002
	= 3.1 × 10–4			= 1.1 × 10–5			= 1.8 × 10–7
6	–2.138	+1.915	0.025	–3.327	+1.785	0.010
	–1.368	+0.596	0.220	–2.037	+0.550	0.144
	+0.000	–0.186	0.445	+0.000	–0.195	0.689
	+0.000	+0.872	0.064	+0.000	+0.886	0.002
	+1.368	+0.596	0.220	+2.037	+0.550	0.144
	+2.138	+1.915	0.025	+3.327	+1.785	0.010
	= 2.7 × 10–4			= 9.0 × 10–6

We have considered three values of R, chosen so that the resulting overlap S is close to 0.1, 0.01, or 0.001. These correspond roughly to the most distant H atoms in the equilibrium structures of ethane, propane and butane, respectively.

In the least-squares approximation of smooth functions by other smooth functions, pointwise convergence is expected to be exponential⁵⁰ and this seems to be observed here. As the number m of Gaussians in the model grows, the maximum error decays rapidly. In all cases, the optimal centers B_i are symmetrically and more or less uniformly distributed on the z axis, the central Gaussian is the most diffuse (its λ_i is the most negative) and the Gaussians furthest from the origin are the least diffuse and have the smallest coefficients c_i.

It is particularly encouraging to note that, for R ≈ 10, the 25-Gaussian density can be approximated very accurately by a model with just three Gaussians, or accurately () by a model with just a single Gaussian!

The reported m = 4 model for R ≈ 10 is a local minimum, with a Z value 0.5% larger than the global minimum. The values are the same. It seems clear that adding a fourth Gaussian to the model provides negligible improvement, so its precise λ_i and c_i have little bearing on the overally accuracy. The fifth Gaussian is needed to noticeably lower both Z and , but this is unimportant since this density can be modeled so accurately with even a single Gaussian.

3.2.3. C(2s)H(1s) Density

Our second example is the product of the 10-fold contracted rec-cc-pVTZ 2s basis function on a C atom at (0, 0, −R/2) with the 5-fold contracted cc-pVTZ basis function on a H atom at (0, 0, +R/2) (see Tables 1 and 2). The product consists of 50 Gaussians, and L_–1/2(m) models with m = 1 to 6 Gaussians are shown in Table 7.

Table 7. L_–1/2(m) Models for rec-cc-pVTZ C(2s)H(1s) Densities with Various R.

	R = 4.669, S = 0.100			R = 7.305, S = 0.010			R = 9.446, S = 0.001
m	Bi	λi	ci/S	Bi	λi	ci/S	Bi	λi	ci/S
1	–0.285	+0.158	1.000	–0.593	–0.090	1.000	–0.712	–0.134	1.000
	= 3.8 × 10–3			= 3.2 × 10–4			= 1.8 × 10–5
2	–1.179	+0.418	0.561	–1.779	+0.310	0.480	–2.135	+0.266	0.303
	+0.889	+0.378	0.439	+0.567	+0.025	0.520	–0.072	–0.082	0.697
	= 3.1 × 10–3			= 1.2 × 10–4			= 4.6 × 10–6
3	–1.320	+0.761	0.344	–2.306	+0.758	0.196	–2.797	+0.677	0.120
	–0.289	–0.099	0.387	–0.437	–0.041	0.699	–0.537	–0.073	0.838
	+1.079	+0.733	0.270	+1.869	+0.662	0.105	+2.171	+0.595	0.041
	= 2.9 × 10–3			= 1.1 × 10–4			= 3.0 × 10–6
4	–1.691	+0.948	0.197	–3.396	+2.405	0.006	–4.158	+1.924	0.003
	–0.274	–0.039	0.487	–2.197	+0.707	0.203	–2.710	+0.653	0.121
	–0.170	+0.776	0.171	–0.434	–0.052	0.681	–0.533	–0.076	0.834
	+1.529	+0.954	0.145	+1.813	+0.646	0.110	+2.136	+0.594	0.042
	=2.7 × 10–3			=8.6 × 10–5			=2.4 × 10–6
5	–2.280	+3.202	0.008	–3.232	+1.878	0.012	–4.037	+1.686	0.005
	–1.345	+0.719	0.329	–2.068	+0.649	0.222	–2.675	+0.641	0.122
	–0.230	–0.152	0.345	–0.418	–0.080	0.640	–0.530	–0.078	0.831
	+0.652	+0.625	0.274	+1.487	+0.568	0.115	+2.061	+0.595	0.041
	+1.978	+1.429	0.044	+2.849	+1.438	0.011	+3.881	+1.783	0.001
	= 1.6 × 10–3			= 9.4 × 10–5			= 2.5 × 10–6
6	–2.249	+3.354	0.007	–3.573	+3.933	0.002	–4.647	+3.942	0.0004
	–1.460	+0.796	0.257	–2.812	+1.213	0.032	–3.766	+1.398	0.007
	–0.318	–0.053	0.436	–1.959	+0.590	0.211	–2.646	+0.625	0.121
	+0.013	+0.918	0.093	–0.406	–0.083	0.631	–0.527	–0.078	0.829
	+1.131	+0.611	0.184	+1.523	+0.582	0.116	+2.052	+0.598	0.041
	+1.915	+1.868	0.024	+2.929	+1.587	0.009	+3.830	+1.672	0.001
	= 1.5 × 10–3			= 3.5 × 10–5			= 8.7 × 10–7

As before, we have considered three values of R, chosen so that the resulting overlap S is close to 0.1, 0.01, or 0.001. These correspond roughly to the most separated C and H atoms in the equilibrium structures of ethane, propane and butane, respectively.

Although the maximum pointwise error of the models almost always decreases as m grows, its decay is slower than for the H(1s)H(1s) densities above. More detailed investigation revealed that the pointwise error is usually largest (and oscillating) in a small region close to the carbon nucleus, but because of volume effects, this error contributes relatively little to the integral in (21) and is therefore only slowly reduced as more Gaussians are added to the model.

As was observed for the H(1s)H(1s) densities, the models in Table 7 are usually dominated by a diffuse Gaussian near the origin, flanked by tighter Gaussians with smaller coefficients, and the central dominance is even more marked in the large-R densities. It is clear from the values that the 50-Gaussian densities arising from the overlap of well separated C(2s) and H(1s) functions can be very accurately modeled by models with very small numbers of Gaussians.

4. Conclusions

The electron density D(r) is a fundamental quantity in all quantum chemical calculations and, in order that those calculations be efficient, it is desirable that D(r) be represented as compactly as possible. Most of the successful previous work in this area projects D(r) into a global auxiliary basis set but, in the present work, we have sought instead to approximate each basis function product that contributes to D(r) in a small, local basis that is explicitly tailored to that product. This is achieved by performing a nonlinear optimization with a few adjustable parameters for each product, and we have presented expressions for all of the first and second derivatives required to use Newton’s method or a similar second-order scheme.

We have applied our algorithm to many of the products that arise when Dunning’s cc-pVnZ or Jensen’s pc-n basis sets are used, and we find that, in many cases, products consisting of many primitive Gaussians can be accurately approximated by models with only a few, well-optimized Gaussians. We anticipate that when such models are used in quantum chemical calculations, the resulting errors will be small while the computational speed will be significantly enhanced.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpca.3c04363.

• The Golub-Welsch algorithm (PDF)

The authors declare no competing financial interest.

Special Issue

Published as part of The Journal of Physical Chemistry Avirtual special issue “Krishnan Raghavachari Festschrift”.