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Exact Hamiltonian for an analytic correlated ground-state wave function for He-like ions
Physics Letters A 372 (2008) 4053–4056 www.elsevier.com/locate/pla
Exact Hamiltonian for an analytic correlated ground-state wave function for He-like ions C. Amovilli a,∗ , N.H. March b,c , I.A. Howard d , Á. Nagy e a Dipartimento di Chimica e Chimica Industriale, Università di Pisa, Via Risorgimento 35, 56126 Pisa, Italy b Department of Physics, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp, Belgium c Oxford University, Oxford, England, UK d Department of Chemistry (ALGC), Free University of Brussels (VUB), Pleinlaan 2, B-1050 Brussels, Belgium e Department of Theoretical Physics, University of Debrecen, H-4010 Debrecen, POB 5, Hungary
Received 12 November 2007; accepted 20 November 2007 Available online 19 March 2008 Communicated by V.M. Agranovich
Abstract A correlated three-parameter variational ground-state Ψ (r1 , r2 , r12 ) proposed by Chandrasekhar for helium-like ions gives a high percentage of the electron correlation energy resulting from the interaction energy e2 /r12 and also yields an analytic ground-state electron density ρ(r). Here, we extract via Schrödinger equation an exact Hamiltonian for which the Chandrasekhar wave function is the ground-state. Properties of the potential energy function in this Hamiltonian are quantified. Finally, kinetic energy densities are plotted and related to the Laplacian of ρ(r). © 2008 Elsevier B.V. All rights reserved. Keywords: Chandrasekhar wave function; Exact Hamiltonian
The purpose of this Letter is two-fold. First of all, we shall derive an exact Hamiltonian Hˆ for which the correlated threeparameter variational wave function Ψ (r1 , r2 , r12 ) constructed by Chandrasekhar [1] is the ground-state. This therefore connects the analytic ground-state electron density corresponding to Ψ (r1 , r2 , r12 ), obtained in earlier work [2] exactly with Hˆ above. Secondly, we analyze fully the properties of the potential energy function associated with this Hamiltonian. The Chandrasekhar wave function is one of the simplest approximations which can be used to approach the energy of atomic He in the ground state beyond the Hartree–Fock limit. In the literature one can find many other functional forms which can approach more closely the best energy and which differ by the number of parameters and of physical constraints imposed. Here, we mention, for example, the more complicate and accurate functions expanded in the so-called generalized Hylleraas
* Corresponding author.
E-mail address: [email protected] (C. Amovilli). 0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.11.075
basis set [3] or other compact forms which introduce electron– electron Kato and Rassolov–Chipman cusp conditions [4]. The Chandrasekhar wave function Ψ has the explicit unnormalized form Ψ = (1 + cr12 )(f1 + f2 )
(1)
where f1 = e−ar1 −br2
and
f2 = e−br1 −ar2
(2)
and r12 denotes the separation |r1 − r2 | between the two electrons, a, b and c the three parameters already referred to. To derive the desired exact Hamiltonian Hˆ , we use Eqs. (1) and (2) to obtain the local kinetic energy −(∇12 + ∇22 )Ψ/2Ψ as −
(∇12 + ∇22 )Ψ 2Ψ a 2 b2 b a f1 =− − + + 2 2 r1 r2 f1 + f2 b a 2c f2 + + − r1 r2 f1 + f2 r12 (1 + cr12 )
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C. Amovilli et al. / Physics Letters A 372 (2008) 4053–4056
+
c[(r1 − r2 μ)(af1 + bf2 ) + (r2 − r1 μ)(bf1 + af2 )] . r12 (1 + cr12 )(f1 + f2 ) (3)
In Eq. (3), μ denotes the cosine of the angle between r1 and r2 : μ=
r1 · r2 . r1 r2
(4)
The potential energy of the desired exact Hamiltonian is thus V (r1 , r2 ) = const +
(∇12 + ∇22 )Ψ 2Ψ
(5)
where the unknown constant can be fixed to −a 2 /2 − b2 /2 in order to have a zero value for such potential when all particles (including nucleus) are at infinite separation. With this choice, the Schrödinger equation for the ground-state of this Hamiltonian results 2 b2 a Ψ. Hˆ Ψ = − − (6) 2 2 While Eq. (3) is exact, we next attempt to gain insight into the nature of the potential energy function entering this equation. To do so, we appeal to the new non-variational procedure to determine the parameters a, b and c proposed by Howard and March [5]. From their Fig. 1, for two electron atomic ions with atomic number Z we can write the already useful approximations a(Z) ≈ Z − a1
(7)
and b(Z) ≈ Z + b1
(8)
while c increases from its initial value of 0.24 for the He atom to 0.46 for the two electron ion with Z = 10. Returning to V (r1 , r2 ), and for simplicity neglecting a1 and b1 in Eqs. (7) and (8) for a preliminary orientation, the first two terms on the right-hand side (RHS) of Eq. (3) become the hydrogenic limiting energy, in magnitude Z 2 . Furthermore, the ratios involving f1 and f2 become equal to 1/2 and so the next two terms, with a = b = Z then inserted, represent the external potential energy −Z/r1 − Z/r2 , in atomic units. The following term depends only on the interparticle repulsion e2 /r12 (e = 1 a.u. in Eq. (3)) but then the final contribution involving μ in Eq. (4) can be said to be a cross-term involving both r1 and r2 and also the interparticle interaction via r12 . Now, it is interesting to study the fluctuations of the potential defined in Eq. (5) from the real one, namely −Z/r1 − Z/r2 + 1/r12 . To do this, we have averaged the real He-like ion Hamiltonian over the square of the Chandrasekhar wave function by writing E = Ψ | − =−
∇12 2
−
∇22 2
−
Z 1 Z − + |Ψ r1 r2 r12
Z 1 Z a 2 b2 − V (r1 , r2 )|Ψ . − + Ψ | − − + 2 2 r1 r2 r12
(9)
Table 1 Quantum Monte Carlo energy, statistical error and contributions from Eq. (11) calculated using the Chandrasekhar wave function for the He-like series of atomic ions with atomic number Z ranging from 2 to 10. All data are in atomic units. The parameters defining the wave function are taken from the work of Howard and March [5] Z
E
error
E1
E2
E3
E4
2 3 4 5 6 7 8 9 10
−2.90031 −7.27614 −13.6493 −22.0224 −32.3924 −44.7663 −59.1396 −75.5170 −93.8857
0.00030 0.00044 0.00066 0.00086 0.0011 0.0014 0.0015 0.0018 0.0021
−3.45930 −8.39873 −15.3664 −24.3367 −35.3247 −48.3133 −63.3021 −80.2928 −99.2983
−0.41867 −0.50356 −0.55300 −0.59379 −0.61272 −0.63333 −0.64934 −0.66344 −0.66089
0.58070 0.68711 0.73793 0.80471 0.83066 0.85550 0.87645 0.89675 0.89193
0.39696 0.93902 1.53215 2.10338 2.71441 3.32481 3.93544 4.54250 5.18153
In particular, we have distinguished four contributions to E, namely E = E1 + E2 + E3 + E4
(10)
where a 2 b2 E1 = − − , 2 2 a b b a Z Z f1 f2 + + + − − , E2 = r1 r2 f1 + f2 r1 r2 f1 + f2 r1 r2 2c 1 − , E3 = − r12 (1 + cr12 ) r12 c[(r1 − r2 μ)(af1 + bf2 ) + (r2 − r1 μ)(bf1 + af2 )] . E4 = r12 (1 + cr12 )(f1 + f2 ) (11) From the above definition, it is important to remark that the terms E2 , E3 and E4 are directly related to the fluctuations of the model potential (5) with respect to the real one. In Table 1, for the He-like atomic ions sequence up to the atomic number Z = 10, we have collected numerical values which quantify the exact contributions of the four types of term entering Eq. (11). The three parameters of the Chandrasekhar wave function are taken from the recent study of Howard and March [5]. What is quite clear from the entries in Table 1 is that the cross term E4 referred to above makes a sizeable contribution. Data of Table 1 have been calculated by Monte Carlo integration using the Metropolis algorithm [6]. As a useful example, we show also in Fig. 1 the distribution of the local energy for the He atom when the state function is represented by the Chandrasekhar wave function used in this work. Such local energy is given by Eloc = −
(∇12 + ∇22 )Ψ Z 1 Z − − + 2Ψ r1 r2 r12
(12)
and assumes the distribution plotted in Fig. 1 when all possible electron configurations are generated according to the probability density function Ψ 2 , Ψ being the Chandrasekhar wave function. The particular shape of this curve, which is characterized by the width and by the position of the maximum, suffices
C. Amovilli et al. / Physics Letters A 372 (2008) 4053–4056
Fig. 1. Local energy distribution function for all possible He two electron configurations generated according to the density probability function Ψ 2 with Ψ the Chandrasekhar wave function.
to show that a large amount of the correlation energy of He is taken into account. Moreover, in Fig. 2 we plot the contribution E4 of Table 1 against the atomic number Z. This term shows an almost linear dependence, the non-interacting −Z 2 value being subsumed in the contribution E1 . Here we stress that this curve has a slope very close to the large Z Schwartz limit, namely 5/8 [7]. As a final consequence of the Hamiltonian Hˆ uncovered here, along with the ground-state density ρ(r) [2,5] which it generates, we have used the wave function Ψ in Eq. (1) to calculate numerically kinetic energy densities tL (r) and tg (r) for the He atom itself, where the subscripts L and g refer respectively to the Laplacian and the gradient definitions from the wave function Ψ (see, for example, [8]). Fig. 3 shows the result for the He atom itself. While, of course, ∞
∞ tL (r)r 2 dr = 4π
4π 0
tg (r)r 2 dr = T
(13)
0
where T is the total kinetic energy corresponding to the wave function Ψ in Eq. (1), we have also plotted in Fig. 3 the radial difference 4πr 2 [tL (r) − tg (r)]. This, of course, must integrate to zero, while T itself is −2.918(1) a.u. If we were dealing with the exact wave function and its corresponding ground-state density, then
4055
Fig. 2. Depicts the fourth term of the RHS of Eq. (11) when averaged over the Chandrasekhar wave function Ψ . The abscissa is the atomic number Z. The linearity in Z is to be anticipated as correcting the bare ‘hydrogenic-like’ energy −Z 2 as become large.
1 tL (r) − tg (r) = − ∇ 2 ρ(r). 4
(14)
Thus, we have added to Fig. 3 a plot of the RHS of Eq. (14), but now with ρ(r) derived from Ψ in Eq. (1); ρ(r) being taken from [2]. We find, in fact, that to present numerical accuracy Eq. (14) is satisfied when both LHS and RHS are calculated from the wave function given in Eq. (1). In summary, we have demonstrated here that an exact Hamiltonian Hˆ exists for which the Chandrasekhar wave function is the ground-state. While there is an intimate connection with the exact non-relativistic Hamiltonian for the He-like series of atomic ions, there is a cross-term being an external potential contribution together with a two-particle interaction term mimicking 1/r12 . The accuracy of Eq. (14) has finally been explored, as this, at least in principle provides a route by which a differential equation for the ground-state density ρ(r) of the He-like atomic ions will finally emerge. Acknowledgements C.A. and Á.N. acknowledge the Bilateral Scientific Cooperation between Italy and Hungary sponsored by Consiglio Nazionale delle Ricerche and Hungarian Academy of Sciences.
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C. Amovilli et al. / Physics Letters A 372 (2008) 4053–4056
References [1] S. Chandrasekhar, Astrophys. J. 100 (1944) 176. [2] I.A. Howard, F. Bartha, N.H. March, Á. Nagy, Phys. Lett. A 350 (2006) 236. [3] W. Kołos, C.C.J. Roothan, R.A. Sack, Rev. Mod. Phys. 32 (1960) 178. [4] C. Amovilli, Á. Nagy, N.H. March, Int. J. Quantum Chem. 95 (2003) 21. [5] I.A. Howard, N.H. March, Phys. Lett. A 366 (2007) 451. [6] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller, J. Chem. Phys. 21 (1953) 1087. [7] C. Schwartz, Ann. Phys. 2 (1959) 156. [8] C. Amovilli, N.H. March, Int. J. Quantum Chem. 102 (2005) 132.
Fig. 3. Depicts form of kinetic energy densities tL (r) (1) and tg (r) (2) using Chandrasekhar wave function for the He atom. Difference tL (r) − tg (r) (3) is also shown confirming the equality (14) with the density ρ(r) derived from the same wave function.
Exact Hamiltonian for an analytic correlated ground-state wave function for He-like ions C. Amovilli a,∗ , N.H. March b,c , I.A. Howard d , Á. Nagy e a Dipartimento di Chimica e Chimica Industriale, Università di Pisa, Via Risorgimento 35, 56126 Pisa, Italy b Department of Physics, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp, Belgium c Oxford University, Oxford, England, UK d Department of Chemistry (ALGC), Free University of Brussels (VUB), Pleinlaan 2, B-1050 Brussels, Belgium e Department of Theoretical Physics, University of Debrecen, H-4010 Debrecen, POB 5, Hungary
Received 12 November 2007; accepted 20 November 2007 Available online 19 March 2008 Communicated by V.M. Agranovich
Abstract A correlated three-parameter variational ground-state Ψ (r1 , r2 , r12 ) proposed by Chandrasekhar for helium-like ions gives a high percentage of the electron correlation energy resulting from the interaction energy e2 /r12 and also yields an analytic ground-state electron density ρ(r). Here, we extract via Schrödinger equation an exact Hamiltonian for which the Chandrasekhar wave function is the ground-state. Properties of the potential energy function in this Hamiltonian are quantified. Finally, kinetic energy densities are plotted and related to the Laplacian of ρ(r). © 2008 Elsevier B.V. All rights reserved. Keywords: Chandrasekhar wave function; Exact Hamiltonian
The purpose of this Letter is two-fold. First of all, we shall derive an exact Hamiltonian Hˆ for which the correlated threeparameter variational wave function Ψ (r1 , r2 , r12 ) constructed by Chandrasekhar [1] is the ground-state. This therefore connects the analytic ground-state electron density corresponding to Ψ (r1 , r2 , r12 ), obtained in earlier work [2] exactly with Hˆ above. Secondly, we analyze fully the properties of the potential energy function associated with this Hamiltonian. The Chandrasekhar wave function is one of the simplest approximations which can be used to approach the energy of atomic He in the ground state beyond the Hartree–Fock limit. In the literature one can find many other functional forms which can approach more closely the best energy and which differ by the number of parameters and of physical constraints imposed. Here, we mention, for example, the more complicate and accurate functions expanded in the so-called generalized Hylleraas
* Corresponding author.
E-mail address: [email protected] (C. Amovilli). 0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.11.075
basis set [3] or other compact forms which introduce electron– electron Kato and Rassolov–Chipman cusp conditions [4]. The Chandrasekhar wave function Ψ has the explicit unnormalized form Ψ = (1 + cr12 )(f1 + f2 )
(1)
where f1 = e−ar1 −br2
and
f2 = e−br1 −ar2
(2)
and r12 denotes the separation |r1 − r2 | between the two electrons, a, b and c the three parameters already referred to. To derive the desired exact Hamiltonian Hˆ , we use Eqs. (1) and (2) to obtain the local kinetic energy −(∇12 + ∇22 )Ψ/2Ψ as −
(∇12 + ∇22 )Ψ 2Ψ a 2 b2 b a f1 =− − + + 2 2 r1 r2 f1 + f2 b a 2c f2 + + − r1 r2 f1 + f2 r12 (1 + cr12 )
4054
C. Amovilli et al. / Physics Letters A 372 (2008) 4053–4056
+
c[(r1 − r2 μ)(af1 + bf2 ) + (r2 − r1 μ)(bf1 + af2 )] . r12 (1 + cr12 )(f1 + f2 ) (3)
In Eq. (3), μ denotes the cosine of the angle between r1 and r2 : μ=
r1 · r2 . r1 r2
(4)
The potential energy of the desired exact Hamiltonian is thus V (r1 , r2 ) = const +
(∇12 + ∇22 )Ψ 2Ψ
(5)
where the unknown constant can be fixed to −a 2 /2 − b2 /2 in order to have a zero value for such potential when all particles (including nucleus) are at infinite separation. With this choice, the Schrödinger equation for the ground-state of this Hamiltonian results 2 b2 a Ψ. Hˆ Ψ = − − (6) 2 2 While Eq. (3) is exact, we next attempt to gain insight into the nature of the potential energy function entering this equation. To do so, we appeal to the new non-variational procedure to determine the parameters a, b and c proposed by Howard and March [5]. From their Fig. 1, for two electron atomic ions with atomic number Z we can write the already useful approximations a(Z) ≈ Z − a1
(7)
and b(Z) ≈ Z + b1
(8)
while c increases from its initial value of 0.24 for the He atom to 0.46 for the two electron ion with Z = 10. Returning to V (r1 , r2 ), and for simplicity neglecting a1 and b1 in Eqs. (7) and (8) for a preliminary orientation, the first two terms on the right-hand side (RHS) of Eq. (3) become the hydrogenic limiting energy, in magnitude Z 2 . Furthermore, the ratios involving f1 and f2 become equal to 1/2 and so the next two terms, with a = b = Z then inserted, represent the external potential energy −Z/r1 − Z/r2 , in atomic units. The following term depends only on the interparticle repulsion e2 /r12 (e = 1 a.u. in Eq. (3)) but then the final contribution involving μ in Eq. (4) can be said to be a cross-term involving both r1 and r2 and also the interparticle interaction via r12 . Now, it is interesting to study the fluctuations of the potential defined in Eq. (5) from the real one, namely −Z/r1 − Z/r2 + 1/r12 . To do this, we have averaged the real He-like ion Hamiltonian over the square of the Chandrasekhar wave function by writing E = Ψ | − =−
∇12 2
−
∇22 2
−
Z 1 Z − + |Ψ r1 r2 r12
Z 1 Z a 2 b2 − V (r1 , r2 )|Ψ . − + Ψ | − − + 2 2 r1 r2 r12
(9)
Table 1 Quantum Monte Carlo energy, statistical error and contributions from Eq. (11) calculated using the Chandrasekhar wave function for the He-like series of atomic ions with atomic number Z ranging from 2 to 10. All data are in atomic units. The parameters defining the wave function are taken from the work of Howard and March [5] Z
E
error
E1
E2
E3
E4
2 3 4 5 6 7 8 9 10
−2.90031 −7.27614 −13.6493 −22.0224 −32.3924 −44.7663 −59.1396 −75.5170 −93.8857
0.00030 0.00044 0.00066 0.00086 0.0011 0.0014 0.0015 0.0018 0.0021
−3.45930 −8.39873 −15.3664 −24.3367 −35.3247 −48.3133 −63.3021 −80.2928 −99.2983
−0.41867 −0.50356 −0.55300 −0.59379 −0.61272 −0.63333 −0.64934 −0.66344 −0.66089
0.58070 0.68711 0.73793 0.80471 0.83066 0.85550 0.87645 0.89675 0.89193
0.39696 0.93902 1.53215 2.10338 2.71441 3.32481 3.93544 4.54250 5.18153
In particular, we have distinguished four contributions to E, namely E = E1 + E2 + E3 + E4
(10)
where a 2 b2 E1 = − − , 2 2 a b b a Z Z f1 f2 + + + − − , E2 = r1 r2 f1 + f2 r1 r2 f1 + f2 r1 r2 2c 1 − , E3 = − r12 (1 + cr12 ) r12 c[(r1 − r2 μ)(af1 + bf2 ) + (r2 − r1 μ)(bf1 + af2 )] . E4 = r12 (1 + cr12 )(f1 + f2 ) (11) From the above definition, it is important to remark that the terms E2 , E3 and E4 are directly related to the fluctuations of the model potential (5) with respect to the real one. In Table 1, for the He-like atomic ions sequence up to the atomic number Z = 10, we have collected numerical values which quantify the exact contributions of the four types of term entering Eq. (11). The three parameters of the Chandrasekhar wave function are taken from the recent study of Howard and March [5]. What is quite clear from the entries in Table 1 is that the cross term E4 referred to above makes a sizeable contribution. Data of Table 1 have been calculated by Monte Carlo integration using the Metropolis algorithm [6]. As a useful example, we show also in Fig. 1 the distribution of the local energy for the He atom when the state function is represented by the Chandrasekhar wave function used in this work. Such local energy is given by Eloc = −
(∇12 + ∇22 )Ψ Z 1 Z − − + 2Ψ r1 r2 r12
(12)
and assumes the distribution plotted in Fig. 1 when all possible electron configurations are generated according to the probability density function Ψ 2 , Ψ being the Chandrasekhar wave function. The particular shape of this curve, which is characterized by the width and by the position of the maximum, suffices
C. Amovilli et al. / Physics Letters A 372 (2008) 4053–4056
Fig. 1. Local energy distribution function for all possible He two electron configurations generated according to the density probability function Ψ 2 with Ψ the Chandrasekhar wave function.
to show that a large amount of the correlation energy of He is taken into account. Moreover, in Fig. 2 we plot the contribution E4 of Table 1 against the atomic number Z. This term shows an almost linear dependence, the non-interacting −Z 2 value being subsumed in the contribution E1 . Here we stress that this curve has a slope very close to the large Z Schwartz limit, namely 5/8 [7]. As a final consequence of the Hamiltonian Hˆ uncovered here, along with the ground-state density ρ(r) [2,5] which it generates, we have used the wave function Ψ in Eq. (1) to calculate numerically kinetic energy densities tL (r) and tg (r) for the He atom itself, where the subscripts L and g refer respectively to the Laplacian and the gradient definitions from the wave function Ψ (see, for example, [8]). Fig. 3 shows the result for the He atom itself. While, of course, ∞
∞ tL (r)r 2 dr = 4π
4π 0
tg (r)r 2 dr = T
(13)
0
where T is the total kinetic energy corresponding to the wave function Ψ in Eq. (1), we have also plotted in Fig. 3 the radial difference 4πr 2 [tL (r) − tg (r)]. This, of course, must integrate to zero, while T itself is −2.918(1) a.u. If we were dealing with the exact wave function and its corresponding ground-state density, then
4055
Fig. 2. Depicts the fourth term of the RHS of Eq. (11) when averaged over the Chandrasekhar wave function Ψ . The abscissa is the atomic number Z. The linearity in Z is to be anticipated as correcting the bare ‘hydrogenic-like’ energy −Z 2 as become large.
1 tL (r) − tg (r) = − ∇ 2 ρ(r). 4
(14)
Thus, we have added to Fig. 3 a plot of the RHS of Eq. (14), but now with ρ(r) derived from Ψ in Eq. (1); ρ(r) being taken from [2]. We find, in fact, that to present numerical accuracy Eq. (14) is satisfied when both LHS and RHS are calculated from the wave function given in Eq. (1). In summary, we have demonstrated here that an exact Hamiltonian Hˆ exists for which the Chandrasekhar wave function is the ground-state. While there is an intimate connection with the exact non-relativistic Hamiltonian for the He-like series of atomic ions, there is a cross-term being an external potential contribution together with a two-particle interaction term mimicking 1/r12 . The accuracy of Eq. (14) has finally been explored, as this, at least in principle provides a route by which a differential equation for the ground-state density ρ(r) of the He-like atomic ions will finally emerge. Acknowledgements C.A. and Á.N. acknowledge the Bilateral Scientific Cooperation between Italy and Hungary sponsored by Consiglio Nazionale delle Ricerche and Hungarian Academy of Sciences.
4056
C. Amovilli et al. / Physics Letters A 372 (2008) 4053–4056
References [1] S. Chandrasekhar, Astrophys. J. 100 (1944) 176. [2] I.A. Howard, F. Bartha, N.H. March, Á. Nagy, Phys. Lett. A 350 (2006) 236. [3] W. Kołos, C.C.J. Roothan, R.A. Sack, Rev. Mod. Phys. 32 (1960) 178. [4] C. Amovilli, Á. Nagy, N.H. March, Int. J. Quantum Chem. 95 (2003) 21. [5] I.A. Howard, N.H. March, Phys. Lett. A 366 (2007) 451. [6] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller, J. Chem. Phys. 21 (1953) 1087. [7] C. Schwartz, Ann. Phys. 2 (1959) 156. [8] C. Amovilli, N.H. March, Int. J. Quantum Chem. 102 (2005) 132.
Fig. 3. Depicts form of kinetic energy densities tL (r) (1) and tg (r) (2) using Chandrasekhar wave function for the He atom. Difference tL (r) − tg (r) (3) is also shown confirming the equality (14) with the density ρ(r) derived from the same wave function.