• This chapter is, verbatim, the paper: Shubin Liu, Robert G. Parr, and Agnes Nagy, "Cusp relations for local strongly decaying properties in electronic systems", Phys. Rev. A, 52, 2645(1995).

  

where r and r' are density and its derivative, respectively, and Z is the nuclear charge, is usually employed in testing an approximate wave function or density in calculation to see if it behaves properly near and at a nucleus. Another example, showing the importance of a quantity at a nucleus, are the Thomas-Fermi [4-5] and Thomas-Fermi-Dirac [6] models for neutral atoms. It is well-known that these models give an electron density that, although normalizable, diverges at an atomic nucleus. But when one corporate one additional constraint [7] namely,

, (2)

to force densities to satisfy the cusp condition, electron densities and total energies are vastly improved. We will show in later sections that the constraint form (2) is similar to part of an expression for the density at a nucleus.
Importance of a value at a nucleus is not restricted only to the density. In dft, the exchange-correlation potential Vxc has special interest, since in the Kohn-Sham method, the only unknown is this term. Much progress of dft in recent years is related to this quantity [8], for which various approximate forms beyond LDA or LSDA have been obtained and have been found to give an overall improvement to the exchange-correlation energies. Unfortunately, however, in a very recent paper by Umrigar et al. [9], it is shown that although the generalized gradient approximation (GGA) yield improved energies compared to the local density approximation, the exchange-correlation potentials obtained from the GGA and other familiar approximations are in poor agreement with exact potentials, especially near a nucleus. We shall give in the present paper accurate values of Vxc at the nucleus for the neutral atoms He, Be, Ne, Mg, and Ar.
Although the expression to give the value of a quantity may be singular at the nucleus, in most of the cases we consider here its value at this point is definitely finite. In Section II, proof of a number of formulations for this purpose are given based on Green's theorem. One finds that the exact value of any quantity can be obtained at a nuclear cusp in terms of its gradients and Laplacians over the whole range of integration. In section III, application of the formulas are made to obtain densities and their first derivatives. Excellent agreement with Kato's cusp condition is observed. In Section IV, we apply the formulas to the exchange-only and exchange-correlation potentials, and determine the first data for Vx(0) and Vxc(0) for atoms. Discussions are presented in Section V to model the data for Vx(0) and Vxc(0). Final remarks are included in Section VI. Atomic units are used throughout.

7.2 The Basic Cusp Relations
Theorem: For any quantity F(r) in an electronic system such that F(r) falls off faster than or equal to 1/r and its gradient falls off faster than or equal to 1/r2, the value at a nucleus (cusp) can be evaluated from any one of the following expressions:

, (3)

, (4)

, (5)
and

, (6)

where a and b are arbitrary constants. Fall off of F faster than or equal to 1/r means that for large enough r.

Proof of Eqs: We use the two identities of Green's theorem [10]

, (7)

and

(8)

(i) Choose f = F(r) and = 1/|r| and plug them into Eq.(7). Because F(r) decays faster than 1/r, the right hand side of Eq. (7) is zero:

, (9)

In the left side of Eq. (7) we use

, (10)

and

. (11)

One then obtains Eq.(4).
(ii) Choosing f = F(r) and Ñy = e-r"Symbol"1/|r|, where a is an arbitrary parameter, and inserting them also into Eq. (7), straightforwardly one obtains Eq.(6).
(iii) Choose and f = F(r) and plug them into Eq. (8). Besides Eq. (9), since
ÑF(r) decays faster than 1/r2,
. (12)

Thus, the right-hand side of Eq. (8) is zero. Upon using Eqs. (10) and (11), Eq. (3) follows.
(iv) Choosing and f=F(r), where b is an arbitrary parameter, and inserting them into Eq. (8), by similar steps, one has Eq. (5).
Q.E.D.
It is interesting to notice that: (i) In each of the expressions (5) and (6), an arbitrary constant exists in the evaluation of any quantity at the r=0 point which is supposedly always constant and finite; (ii) The conditions that restrict the theorem, which state that a quantity and its derivative must decay faster than 1/r and 1/r2, respectively, are applicable to atomic and molecular systems, in which it is known that quantities, such as wave function, density and potentials, decay expotentailly.

7.3 The Electron Density and Its Derivative at a Nucleus
Setting F(r) = r(r), one obtains four expressions for evaluating the density at a nucleus:

, (13)

, (14)

, (15)

, (16)

where a and b are arbitrary constants. In the spherically-symmetrical case, by setting , one obtains four corresponding expressions for the density derivative at a nucleus:

, (17)

, (18)

, (19)

, (20)

where a and b are arbitrary constants, and r''(r) and r"'(r) stand for the second- and third-order density derivative with respect to r, respectively.
It is interesting to notice that Ghosh and Parr [7] used Eq. (2), and Zhou and Parr [11] used

, (21)

where k and k' are parameters to be determined, as constraints to force the density to satisfy both cusp and asympotic behavior conditions. They found much improvement both in density behavior and in energetics. The forms they introduced are much like the numerators of components in the right-hand side of Eq. (15). To see how one can obtain the constraint Eq. (2) from Eq. (15), one takes the derivative of both sides of Eq. (15) with respect to the parameter a. Since a is arbitrary, dr(0)/da = 0. Thus,

(22)

for any a. In this equation the first term on the left side is exactly the constraint of Eq. (2), while the second term is similar to Eq. (21). So, the present formulation for the density at nucleus is in accord with the previous applications of constraints. The difference lies mainly in the fact that the parameters k and k' were determined definitely by the previous authors, while in our expression, the constants a and b are actually arbitrary.
Results for the density and its derivative at a nucleus for the neutral atoms from He to Ar are listed in Table 1 and 2, respectively, in which, for the purpose of comparison, exact results and Kato's cusp conditon are also given. The calculations are performed using the Hartree-Fock densities of Clementi and Roetti [12]. Ñr and Ñ2r are computed analytically in terms of Slater-type orbitals. And the value of arbitrary constants a and b we choose here are just the nuclear charge Z of each atom. Excellent agreement between values from these expressions and exact values are observed. Accurate satisfaction of Kato's cusp condition between r(0) and r'(0) is seen in the last column in Table 2.
There have been recent works [13,14] dealing with the electron density near or at nuclear cusps. An interesting formula which has been used is the Hiller-Sucher-Feinberg identity [15], which takes the form

, (23)

where and , which is the summation of the electron-nuclear attractive and electron-electron repulsive potentials. This means that if one knows the wave function, hence the density, of the system and its potentials, the density at nuclei cusp can be determined. The advantage of this approach lies in that it explicitly employs the potential of the system. On the other hand, however, one still has to use wave functions to compute r(0) and, furthermore, calculations show that results are not good even when high-quality wave functions are employed. Some of the results are listed in Table 1 and 2 from reference [13], obtained by using 6-311++G basis sets. Our approach clearly is superior.

7.4 Exchange-Only and Exchange-Correlation Potentials at a Nuclear Cusp
Although various approximate forms of the exchange-only and exchange-correlation energy functionals have been proposed in recent years, not until recently did we know the exact exchange-only potential and the exchange-correlation potential by numerical solution of the Kohn-Sham eqation. Surprisingly, it was not known until a very recent paper by Umrigar et al. [9] that, although approximate functionals for Ex[r] and Exc[r] do yield improvements over the conventional LDA approach in energetics, most of their potentials, however, are in poor agreement with the exact potentials both near or at nuclei cusps and in long-range regions. For example, as mentioned in the reference [9], all the GGA approximate potentials, such as those of Langreth-Mehl [16], Perdew-Wang '86 [17], Perdew-Wang '91 [18], Becke '88 [19] and so on, diverge at nuclear cusps. Only the LDA potential remains finite. This awkward situation stimulates the search for the new approximate functionals which behave properly both in cusp and in asymptotic regions. It also shows the need to find accurate data for these potentials at nuclear cusps, from which one may judge how well an approximate potential behaves at cusps.
Letting Vx or Vxc replace F in Eqs. (3)-(6), one obtains expressions for Vx(0) and Vxc(0). For example, assuming spherical symmetry,

. (24)

By employing the numerical data of Vx(r) and Vxc(r) from the Nagy method [20] and the Zhao-Parr method [21], we have calculated Vx(0) and Vxc(0) for five neutral atoms, He, Be, Ne, Mg and Ar, as shown in Tables 3 and 4, respectively. In the paper of Umrigar and coworkers [9], although the discrete values of the Vx(0) and Vxc(0) of helium were not given, they did plot the curves of Vx(r) and Vxc(r), which show that the value of Vx(0) is around -1.7, and Vxc(0) around -1.8, in excellent agreement with our results.
To check that these results at a nucleus are accurate, we note that the contributions to values of Vx(0) and Vxc(0) come from quite a long range. The results are relatively insensitive to the behavior of ÑVx(r) or Ñ2Vxc(r) near cusps, where errors are likely in numerical calculations. In Figure 1, the integrand of Eq. (24) is plotted for the exchange-correlation potential of helium.

7.5 Discussion
It can be seen from Table 3 that the values of Vx(0) and Vxc(0) both are close to minus the nuclear charge Z of the atom. This fact can serve as guidance in the finding of better approximate functionals for Ex[L 114 \f "Symbol"] and Exc[r].
To model this interesting behavior, it may assumed that near the nuclear cusp the main component in density and 1st-order density matrix is the 1s orbital contribution [23]. One then has an approximate expression for Vx(0):

, (25)

where J1s is the electron-electron repulsion energy between 1s orbitals. If one supposes that the density takes the form

, (26)

and the 1st-order density matrix has the form

, (27)

then one obtains the exchange energy functional

. (28)

The functional derivative of Ex[r] leads to the exchange potential, which turns out to be

. (29)

At r=0, we have Vx(0)= -Zeff. Values of -J1s/2 and -Zeff for a few neutral atoms are also listed in Table 3. Zeff was obtained by a least-square-fit process of Eq. (29) with first five points of Vx(r) for each atom. It is found that they are good approximations for Vx(0).
Another way to account for this is from the Kohn-Sham equation, which reads

, (30)

where

. (31)

Thus Vxc(r) can be expressed as

. (32)

Near the nuclear cusp, the most-important term is the 1s orbital, therefore, one has approximately

. (33)

Plugging Eq. (33) into Eq. (32), we derive an approximation for Vxc(0),

. (34)

By using Kato's cusp condition and supposing , one finally reaches

. (35)

Values from above equation for a few atoms are given in the last column of Table 4. Fair agreement with calculated Vxc(0) is found.

7.6 Concluding Remarks
In this paper, four rigorous expressions to obtain the value of any quantity at nuclear cusps have been given in terms of its gradients or Laplacians over the whole range of integration. Applications to electron density and its derivative for neutral atoms ranging from He to Ar, and to the exchange-only and exchange-correlation potentials of a few neutral atoms have been carried out. Excellent agreement with reference values are observed. As a first-order approximation, we have found that the value of Vx(r) and Vxc(r) at a nuclear cusp is minus the nuclear charge Z of the corresponding atom.
As a final remark, we would like to point out that, in terms of the gradient or Laplacian of any quantity, one actually can get the quantity itself anywhere in space from

, (36)

or

, (37)

or from other forms similar to Eqs. (5) and (6). The proof of this statement follows that given above, if one choose and in both identities of Green's theorem. Setting , one obtains integro-differential equations for the density. Bader and Beddall [24] first gave Eq. (36) for the density.

Table 1. Density at the nuclei cusp from eqs. (13)-(16) for neutral atoms ranging from He to Ar. For comparison, exact values and values using Hiller-Sucher-Feinberg identity are also shown.

	Density at Nucleusa
	Formula I	Formula II	Formula III	Formula IV	rHSF(0)b	Exactc
He	3.597	3.597	3.597	3.597	3.562	3.60
Li	13.834	13.834	13.834	13.834
Be	35.428	35.428	35.428	35.428	35.311
B	71.985	71.985	71.985	71.985
C	127.556	127.555	127.556	127.556		127.56
N	206.135	206.135	206.135	206.135
O	311.975	311.975	311.975	311.975
F	448.710	448.710	448.710	448.710
Ne	620.146	620.146	620.146	620.146	619.13	620.15
Na	833.833	833.833	833.833	833.833
Mg	1093.731	1093.731	1093.731	1093.731		1093.73
Al	1402.907	1402.906	1402.907	1402.907
Si	1765.708	1765.707	1765.708	1765.708		1765.71
P	2186.309	2186.308	2186.309	2186.309
S	2670.098	2670.098	2670.098	2670.098
Cl	3218.036	3218.036	3218.036	3218.036
Ar	3840.223	3840.222	3840.223	3840.223		3840.22

a Formulas I to IV represent Eqs. (13)-(16), respectively.
b From Reference [13]
c From Reference [22]

Table 2. Density derivative at the nuclei cusp from eqs. (17)-(20) for neutral atoms ranging from He to Ar. For Comparison, values using Hiller-Sucher-Feinberg identity and Kato's cusp condition are listed. The exact value of r'(0)/2Zr(0) is 1.

	Density Derivative at Nucleusa					Kato's cond.
	Formula I	Formula II	Formula III	Formula IV	r'HSF(0)b	r'(0)/2Zr(0)
He	-14.420	-14.420	-14.420	-14.420	-12.744	1.002
Li	-83.479	-83.479	-83.479	-83.479		1.006
Be	-284.792	-284.792	-284.792	-284.792	-267.48	1.005
B	-722.601	-722.601	-722.601	-722.601		1.004
C	-1535.738	-1535.738	-1535.739	-1535.738		1.003
N	-2894.996	-2894.996	-2894.997	-2894.996		1.003
O	-5009.895	-5009.894	-5009.896	-5009.896		1.004
F	-8101.860	-8101.858	-8101.862	-8101.860		1.003
Ne	-12425.078	-12425.076	-12425.082	-12425.078	-11629	1.002
Na	-18364.846	-18364.843	-18364.852	-18364.846		1.001
Mg	-26274.491	-26274.487	-26274.501	-26274.491		1.001
Al	-36511.207	-36511.200	-36511.221	-36511.207		1.001
Si	-49489.964	-49489.953	-49489.985	-49489.964		1.001
P	-65638.236	-65638.222	-65638.266	-65638.236		1.001
S	-85545.142	-85545.121	-85545.184	-85545.142		1.001
Cl	-109422.248	-109422.219	-109422.304	-109422.248		1.000
Ar	-138373.992	-138373.954	-138374.067	-138373.992		1.001

a Formulas I to IV reprsent Eqs. (17)-(20), respectively.
b From Reference [13].

Table 3. Values of the exchange-only potential at the nuclear cusp Vx(0) for neutral atoms of He, Be, Ne and Ar.(a.u.)a

Atom	I	II	III	IV	-Zeff	-J1s/2
He	-1.708	-1.697	-1.704	-1.703	-1.668	-1.688
Be	-3.353	-3.351	-3.357	-3.358	-3.015	-3.684
Ne	-7.424	-7.399	-7.406	-7.409	-7.502	-9.622
Ar	-13.670	-13.665	-13.668	-13.677	-13.346	-17.555

a Formulas I to IV denote for Eqs. (3) - (6), all with F = Vx.

Table 4. Values of the exchange-correlation potential at the nulcear cusp Vxc(0) for a few neutral closed-shell atoms. (a.u.)a

Atom	I	II	III	IV	Estimatedb
He	-1.815	-1.755	-1.815	-1.817	-2.29
Be	-3.123	-3.055	-3.122	-3.116	-4.50
Ne	-8.208	-8.154	-8.209	-8.215	-11.80
Mg	-9.703	-9.645	-9.703	-9.707	-
Ar	-17.587	-17.505	-17.581	-17.567	-22.10

a Formulas I to IV denote for Eqs. (3)-(6), all with F = Vxc.
b Eq. (35) in the text.

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