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. 2024 Jul 10;128(29):6019–6025. doi: 10.1021/acs.jpca.4c02712

Discontinuities of Kinetic Energy Densities within Finite and Complete Basis Sets

Conrad C Moore 1, Viktor N Staroverov 1,*
PMCID: PMC11285366  PMID: 38985544

Abstract

graphic file with name jp4c02712_0007.jpg

Although electron densities are always continuous, other ingredients of density-functional approximations can be sharply discontinuous at isolated points. In particular, the positive-definite, Weizsäcker, and Pauli kinetic energy densities expressed in terms of Slater-type orbitals all have discontinuities at the positions of the atomic nuclei in molecules. The first two of those quantities are similarly discontinuous even in the basis-set limit. These striking features are not as widely recognized as they deserve to be. We show in detail how discontinuities of kinetic energy densities arise from asymmetric electron–nucleus cusps of molecular wave functions and point out instances of their significance in electronic structure theory.

1. Introduction

Most quantum chemistry calculations use one-electron basis sets consisting either of Gaussian-type functions (GTF) or of Slater-type orbitals (STO).1 A fundamental difference between GTFs and STOs is that the former are smooth everywhere, whereas the latter may have cusps at the positions of the atomic nuclei or wherever the orbital is centered. Since exact electronic wave functions are also known to have electron–nucleus cusps,2−4 STOs are in principle better suited than GTFs for approximating electronic wave functions near atomic nuclei.5 Nuclear cusps of various density functional ingredients is a topic of sustained interest in fundamental density-functional theory6−12 and in kinetic energy potential methods.13,14

In molecules, nuclear cusps of exact electronic wave functions are not spherically symmetric with respect to the nucleus.15−17 The same is true about nuclear cusps of approximate wave functions expanded in STOs. This means that, although electronic wave functions and densities themselves are continuous, quantities that depend on their derivatives may have sharp jump discontinuities at the positions of atomic nuclei. Examples of such quantities include the total “positive-definite” kinetic energy density, τ(r), and the gradient of the electron density, |∇ρ(r)|. Bader and co-workers18−20 were apparently the first to draw attention to these features of τ(r) and |∇ρ(r)|.

Gradients and kinetic energy densities are widely used as ingredients of density-functional approximations for the exchange-correlation energy.21−23 They also appear in exact expressions for exchange-correlation potentials.24−31 This suggests that jump discontinuities of kinetic energy densities and |∇ρ(r)| may be significant both in fundamental and applied density-functional theory. There are in fact explicit indications of their relevance in the development of kinetic energy functionals,20 in the theory of exact Kohn–Sham potentials,24 and in the method of local hybrid functionals,32 but so far the whole matter has been dealt with only in passing.

In this work, we focus on and investigate in detail the discontinuous behavior of τ(r) and related quantities in diatomic molecules. We begin by deriving jump discontinuities of kinetic energy densities from model wave functions within minimal STO basis sets. Then we extend our analysis to Hartree–Fock and correlated wave functions within general finite STO basis sets as they approach the basis-set limit. Finally, we discuss the conceptual and practical implications of these findings for electronic structure theory.

2. Methods

2.1. Definitions

Using the nucleus-centered spherical polar coordinates and atomic units, we write a normalized STO as

2.1. 1

where n (n = 1, 2, ...), l, and m are integers, ζ is the orbital exponent, and Zlm(θ, ϕ) are real spherical harmonics. A molecular orbital (whether Hartree–Fock, Kohn–Sham, or natural) expanded in M STOs is

2.1. 2

and the corresponding electron density is then

2.1. 3

where ni (0 ≤ ni ≤ 2) are orbital occupation numbers.

The Weizsäcker kinetic energy density is given by

2.1. 4

Since ρ(r) itself is continuous, τW(r) can be viewed as a proxy for |∇ρ(r)|2. The “positive-definite” total kinetic energy density is

2.1. 5

and the Pauli kinetic energy density is defined as33

2.1. 6

Using the Lagrange identity, once can show that20,28

2.1. 7

There are of course other forms34 of the total kinetic energy density that differ from τ(r) by addition of some fraction of ∇2ρ(r). However, because ∇2ρ(r) diverges at the nucleus, those other forms are singular (infinite) rather than finitely discontinuous at each nucleus, and as such they are not relevant to this work. The “positive-definite” kinetic energy density is special in that it is everywhere finite.

As it turns out, τ(r), τW(r), and τP(r) all have discontinuities at the positions of the atomic nuclei in molecules whenever the molecular orbitals are expanded in STOs. We will see that, depending on the circumstances, the relative magnitudes of these discontinuities can range from barely noticeable to extraordinary.

2.2. The Origin of Kinetic Energy Density Discontinuities

To get to the heart of the matter, we begin by analyzing kinetic energy densities for two simple systems: the H2 and He2 molecules described with ground-state Hartree–Fock (HF) self-consistent field (SCF) wave functions within a minimal STO basis set. A minimal basis set for these systems consists of two Slater 1s (n = 1, l = 0) orbitals, which we denote χA(r) and χB(r), each having the same orbital exponent ζ and centered on the respective nucleus (A or B). The two HF SCF molecular orbitals in this basis are given by

2.2. 8

where the signs “+” and “–” identify the “bonding” and “antibonding” MOs, and N± are the corresponding normalization factors given by

2.2. 9

The quantity S is the overlap integral35

2.2. 10

where R is the internuclear distance.

2.3. Jump Discontinuities in the H2 Molecule

Within the HF SCF method, the total electron density for H2 is just ρ = 2|ϕ+|2. This means that the Weizsäcker and total kinetic energy densities for H2 coincide

2.3. 11

whereas τP = 0. Substitution of eq 8 into eq 11 gives

2.3. 12

Figure 1 shows that τ(r) is sharply discontinuous at each nucleus. The discontinuities arise because ϕ+(r) has different slopes when approaching a nucleus from different directions. The greatest difference between the slopes occurs along the internuclear axis (z axis). It is clear that τ would also have jumps if the exponents of χA and χB were distinct.

Figure 1.

Figure 1

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Kinetic energy density for H2 (R = 1.4a0) calculated from a HF SCF wave function using a minimal STO basis set with ζ = 1, shown along the internuclear axis. The inset shows the term from which the two jump discontinuities arise.

To understand this phenomenon better, let us investigate the directional dependence of τ(r) at each nucleus. To facilitate the task, we employ a local spherical polar coordinate system centered on the nucleus of interest. Diatomic molecules are cylindrically symmetric, so we need only two variables: the distance r from the nucleus (0 ≤ r < ∞) and the angle θ between r and the internuclear axis (0 ≤ θ ≤ π). We define this angle such that θ = 0 in the direction toward the other nucleus.

If we take A to be the nucleus of interest (and thus the origin of the local coordinate system), then the 1s STO centered on A is simply

2.3. 13

In the coordinate system centered on A, the 1s STO centered on B is given by

2.3. 14

To derive the expression for τ(r) in the local coordinate system centered on A, we note that the gradient of a function of r and θ in this system is given by

2.3. 15

where r̂ and θ̂ are orthonormal basis vectors. Using eq 15 we obtain the one-center terms of eq 12

2.3. 16

and

2.3. 17

The two-center term is evaluated to be

2.3. 18

This term retains its angular dependence even at r = 0, whereas |∇χA|2 and |∇χB|2 become θ-independent there. It is this term that gives rise to the discontinuities of τ(r) (Figure 1).

Substituting eqs 16–18 into eq 12 and setting r = 0 we arrive at

2.3. 19

Note that for θ = π/2, the second term in parentheses vanishes, which means that τ(r) is continuous in the direction perpendicular to the internuclear axis, in agreement with the remarks by Bader and Beddall.19

According to eq 19, the minimal-basis-set kinetic energy density of H2 drops at each nucleus along the internuclear axis by the amount

2.3. 20

when passing through a nucleus toward the other nucleus. This Δτ remains negative for every R and ζ > 0. For a fixed ζ, the magnitude of Δτ decays exponentially with R. Since τW(r) = τ(r) for a HF wave function of H2, eqs 19 and 20 also apply to τW(r).

2.4. Jump Discontinuities in He2

In the He dimer, whose HF SCF wave function has two doubly occupied orbitals, the kinetic energy densities τ(r) and τW(r) are distinct. Direct evaluation of the former using the molecular orbitals of eq 8 gives

2.4. 21

Similarly, using eqs 7 and 8 one finds

2.4. 22

The expression for τW(r) is more complicated but can be written down immediately as the difference between τ(r) and τP(r).

Figure 2 shows that all three of these kinetic energy densities are discontinuous at the nuclei. The discontinuities again come from the two-center term ∇χA·∇χB. Observe that τ(r) rises in the internuclear region of He2, whereas in H2 it drops (Figure 1). The reversal occurs because N+ < N– for any finite R, so the effect of the term ∇χA·∇χB for He2 (eq 21) is opposite of what it was for H2 (eq 12). Another essential observation is that the jumps of τ(r) and τW(r) are similar in magnitude, whereas the jump of τP(r) is much smaller. As we will see later, this is a persistent property of τP(r) that is observed at all levels of theory within STO basis sets.

Figure 2.

Figure 2

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Kinetic energy densities for He2 calculated from a HF SCF wave function using a minimal STO basis set with ζ = 1 for R = 2a0, shown along the internuclear axis.

Using the technique of Sec. 2.3, we deduce the following directional dependence of the minimal-basis-set τ(r) for He2 around each atomic nucleus

2.4. 23

The analogous result for the Pauli kinetic energy density is

2.4. 24

The formula for Inline graphic is more complicated, but there is no need to state it explicitly because it follows immediately from eq 6.

The angular dependence of τ(r) in eqs 19 and 23 is described by the function cos θ. It is significant that the same function occurs in the integrated form of the electron–nucleus cusp condition obeyed by exact electronic wave functions in molecules.15−17 This shows that our simple model captures the correct directional behavior of exact wave functions and kinetic energy densities.

In what follows, we quantify the jumps of τ(r), τW(r), and τP(r) along the internuclear axis by their signed magnitudes denoted Δτ, ΔτW, and ΔτP, respectively. We say that a jump is negative if the kinetic energy density drops in the internuclear region (as in Figure 1) and positive if it rises (as in Figure 2).

According to eq 23, when passing through a He nucleus along the internuclear axis in the direction toward the center of the bond, the total kinetic energy density jumps up by

2.4. 25

According to eq 24, the Pauli kinetic energy density jumps along the internuclear axis by

2.4. 26

The Weizsäcker kinetic energy density changes by

2.4. 27

Figure 3 shows that all these jumps remain positive as functions of ζ for a fixed R, which is to be contrasted with the Δτ of eq 20, which is negative.

Figure 3.

Figure 3

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Jump discontinuities of the minimal-basis-set HF kinetic energy densities along the internuclear axis of He2 (R = 2a0) for varying values of the orbital exponent ζ.

Figure 4 complements Figure 3 and further reveals that, for a fixed ζ, the magnitude of the kinetic energy density jumps in He2 increases without bound as R → 0. For Δτ, this result follows from eq 25: in the R → 0 limit the χA and χB orbitals become identical, so their difference tends to zero and the normalization factor N– in eq 25 increases without bound, whereas all other terms remain finite. The ΔτP of eq 26 diverges in the R → 0 limit because of the vanishing denominator.

Figure 4.

Figure 4

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Jump discontinuities of the minimal-basis-set HF kinetic energy densities along the internuclear axis of He2 for various internuclear distances. The jumps tend to infinity in the R → 0 limit.

Thus, kinetic energy density discontinuities can be either positive or negative and can vary in magnitude over a wide range, depending on the nucleus. Another finding of this section is that the jumps of τP(r) are smaller than those of τ(r) and τW(r). We will now show that these conclusions hold in general.

2.5. Kinetic Energy Density Discontinuities in Larger Basis Sets

So far we have seen kinetic energy density jump discontinuities only for minimal-basis-set HF wave functions of H2 and He2. It is natural to ask how such discontinuities vary with the nuclear charge, the type of the wave function, and the size of the basis set. To answer these questions, we evaluated the quantities τ(r), τW(r), and τP(r) for several diatomic molecules (H2, He2, and F2) using HF and configuration interaction (CI) wave functions constructed within various STO basis sets. Different basis sets were employed on purpose to explore the range of discontinuities. Note that, within each basis set, only the 1s and 2s STOs contribute to the jump discontinuities of molecular orbitals.

Table 1 summarizes the results of our calculations. Here, the notation [ms, mp, md, ...] for a basis set means that the basis includes ms STOs of s type (with possibly different values of n) on each nucleus, mp STOs of p type (again with possibly different values of n) on each nucleus, and so on. The names DZ, TZP, QZ4P refer to the basis sets of ref (36), whereas VB1, VB2, and VB3 are the names of the basis sets of ref (37). The precise definitions of these basis sets may be found in the corresponding papers cited. Figures 5 and 6 represent two selected rows of Table 1 visually.

Table 1. Total Energies, Maximum Values of Kinetic Energy Densities, and Kinetic Energy Density Jumps (all in a.u.) for Various Wave Functions and STO Basis Sets.

      total
 Weizsäcker
 Pauli
wave function basis set Ee (Eh) max τ Δτ max τW ΔτW max τP ΔτP
H2 (R = 1.4a0)
HF DZ [2] –1.127951 0.359 –0.182 0.359 –0.182 0.000 0.000
  TZP [3,1] –1.132627 0.418 –0.278 0.418 –0.278 0.000 0.000
  QZ4P [5,2,2] –1.133608 0.410 –0.319 0.410 –0.319 0.000 0.000
H2 (R = 1.4a0)
CIa ref (38) [3,1,1] –1.169837 0.427 –0.319 0.404 –0.315 0.023 –0.004
He2 (R = 2.0a0)
HF VB1 [3,1] –5.601662 7.591 0.064 7.537 0.019 0.334 0.046
  VB2 [4,2,1] –5.602205 7.299 0.090 7.255 0.059 0.330 0.031
  VB3 [5,3,2,1] –5.602497 7.558 0.167 7.524 0.149 0.329 0.018
He2 (R = 2.0a0)
HF ref (39) [3,1] –5.602330 7.931 0.131 7.906 0.137 0.334 –0.006
F2 (R = 2.68a0)
HF VB1 [5,3,1] –198.758882 18264.8 –143.7 18039.5 –151.9 233.5 8.195
  VB2 [6,4,2,1] –198.770522 18440.9 –161.7 18205.1 –159.6 235.8 –2.080
  VB3 [7,5,3,2,1] –198.772013 18486.0 –163.9 18247.8 –162.0 238.2 –1.903
F2 (R = 2.68a0)
HF ref (40) [4,3,1,1] –198.76825 18530.7 –172.7 18291.9 –169.7 238.8 –2.967
CIa ref (40) [4,3,1,1] –198.83774 18516.3 –150.3 18275.6 –148.0 240.6 –2.340
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a

The CI wave functions are from ref (38).

Figure 5.

Figure 5

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Kinetic energy densities of H2 (R = 1.4a0) calculated from the CI STO wave function of ref (38). The inset shows the small discontinuity of τP(r).

Figure 6.

Figure 6

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Kinetic energy densities of He2 (R = 2a0) calculated from the HF STO wave function of ref (39). All discontinuities are relatively small, but nonzero, as shown in the insets.

According to Table 1, τ(r) and τW(r) have discontinuities in all three molecules, for all wave functions and basis sets. For a given system, the jumps Δτ and ΔτW are similar in magnitude. As the nuclear charge increases, the maximum values of τ(r) and τW(r) also increase, so the magnitudes of Δτ and ΔτW relative to the magnitudes of τ(r) and τW(r) at the nuclei decrease overall. For instance, the magnitude of Δτ is 50–75% of the maximum value of τ(r) for H2, but only 1–2% for He2, and then drops further to less than 1% for F2.

To rationalize why the relative magnitude of jump discontinuities Δτ/(max τ) decreases with increasing nuclear charge Z (H2, He2, F2), we note that only the core STOs (1s and 2s) contribute to jump discontinuities, and the optimal exponents of such core orbitals generally increase with Z. According to the equations of sections 2.3 and 2.4, the magnitude of Δτ/(max τ) must tend to zero in the ζ → ∞ limit. Under the assumption that this trend remains in place for all atoms, we see that Δτ/(max τ) should decrease with Z.

Another noteworthy observation is that the signs of Δτ and ΔτW are negative for H2 and F2, but positive for He2. Thus, the sign of the jump appears to indicate whether the interaction is bonding or not.

The Pauli kinetic energy density, τP(r), has much smaller but still nonvanishing discontinuities for all finite-basis-set wave functions of Table 1 except the HF wave functions for H2, where τP(r) is identically zero for mathematical reasons. Nevertheless, we observe that, when the basis set increases, the jump ΔτP generally decreases and sometimes changes its sign. This suggests that ΔτP may be approaching zero in the basis-set limit.

In summary, the strikingly large discontinuities of kinetic energy densities in H2 observed for a minimal-basis-set HF wave function persist within large STO basis sets and for correlated wave functions. For He2, the jumps of τ(r) drop to 1–10% of the maximum value of τ(r) and become even smaller (1% or less of the maximum τ value) for heavier nuclei. These results suggest that jump discontinuities of τ(r), τW(r), and τP(r) are always present within finite STO basis sets, but are prominent only at small-Z nuclei. Already for He2, the discontinuities of τ(r) and τW(r) are easy to miss, and those of τP(r) are barely noticeable (Figure 6).

3. Discussion

Kinetic energy densities are commonly used as ingredients of density-functional approximations. Since they are discontinuous within STO basis sets, one may wonder if there is a cause for concern from a numerical point of view. In calculations of exchange-correlation energies using numerical quadratures, jump discontinuities of τ(r) and |∇ρ(r)|2 at the nuclei are of little if any consequence because they can be eliminated (along with Coulomb singularities of other quantities) by partitioning the space into atomic cells and using a local spherical polar coordinate system around each nucleus; any irregularity of the integrand at r = 0 disappears when the integrand is multiplied with the Jacobian factor.41

In other situations, however, discontinuities of kinetic energy densities may have tangible consequences. For instance, when imposing nuclear cusp conditions on molecular orbitals42−50 it might be beneficial to go beyond the standard equations for spherically averaged quantities and try to impose cusp conditions with explicit directional dependence. The fact that, in H2, discontinuities of τ(r) can be as large as 75% of the value of τ(r) at the nucleus (Figure 5) may explain why the spherically averaged nuclear cusp condition is harder to impose for hydrogens than for heavier nuclei.47

Another context where kinetic energy density discontinuities can show up is the theory of exact Kohn–Sham effective potentials. Within a complete basis set, the exact exchange-correlation potential corresponding to an interacting wave function Ψ can be written exactly in terms of a few quantities derived from Ψ and from the corresponding noninteracting (Kohn–Sham) wave function.24−27 One such expression is25

3. 28

where τWF(r) and τKS(r) are the wave function-based (interacting) and Kohn–Sham (noninteracting) kinetic energy densities, respectively. The precise definitions of vholeXC(r), ϵ̅KS(r) and ϵ̅WF(r) are irrelevant for our purposes. What is relevant is that the first three terms of eq 28 are continuous at each nucleus in any finite or complete basis set, whereas τWF(r) and τKS(r) are not. If vXC(r) is assumed to be continuous at each nucleus, then relations such as eq 28 imply that the nuclear discontinuities of τWF(r) and τKS(r) should be identical. This conclusion, however, is almost paradoxical, because in general τWF(r) and τKS(r) are very different functions of r. Buijse et al.24 implied that, for systems with only one occupied Kohn–Sham orbital, vXC(r) is continuous at a nucleus in a complete basis set. We will show elsewhere that this is true for exchange-correlation potentials in general. Here, we bring up this question as an example of why the discontinuities of kinetic energy densities matter.

4. Conclusions

Mathematical studies of cusp conditions for electronic wave functions, asymptotic properties of electron densities, discontinuities of density-functional ingredients, etc. are usually concerned with exact solutions of the Schrödinger equation. Since exact solutions are not available except for simple systems, the required proofs often have to be indirect, intricate, and are sometimes open to question. We have demonstrated here and elsewhere51,52 that the mathematical analysis of approximate electronic wave functions expressed in terms of finite basis sets is much simpler but often leads to equally valuable insights.

Specifically, we showed here how and why the kinetic energy densities τ(r), τW(r), and τP(r) evaluated using finite Slater-type basis sets become discontinuous at the positions of the atomic nuclei in molecules. These discontinuities are direct manifestations of the nonspherical symmetry of the electron–nucleus cusps of molecular electronic wave functions, and their relative magnitude is especially large for H atoms. It is noteworthy that the true (cos θ) directional dependence of the electron–nucleus cusps derived by Bingel and other workers15−17 is correctly reproduced by the simple minimal-basis-set model of Sec. 2.2.

For a given direction in a molecule, the discontinuities of τ(r), τW(r), and τP(r) are more dependent on the basis set than on the type of the wave function. In a diatomic molecule, they attain their greatest value along the internuclear axis, where the trend is

4. 29

for a given level of theory.

The observed discontinuities of τP(r) generally decrease with the increasing STO basis set size and are expected to vanish altogether in the basis-set limit. The property ΔτP = 0 (in every direction), which we hypothesize to hold in the basis-set limit, is in fact necessary for molecular exact exchange-correlation potentials to be continuous at atomic nuclei. Discontinuities of τ(r) and τW(r), on the other hand, survive even in a complete basis set. We intend to explore the implications of these findings in the upcoming reports.

Acknowledgments

The authors thank Paul Zimmerman and Philip Hoggan for helpful comments on molecular calculations using STOs. The work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) through the Discovery Grants Program (Application RGPIN-2020-06420) and the Canada Graduate Scholarships Program (C.C.M.)

The authors declare no competing financial interest.

Special Issue

Published as part of The Journal of Physical Chemistry Avirtual special issue “Gustavo Scuseria Festschrift”.

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