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Further solutions to Schrödinger's equation for the helium atom
REPORTS ON MATHEMATICAL PHYSICS
Vol. 67 (2011)
No. 1
FURTHER SOLUTIONS ¨ TO SCHRODINGER’S EQUATION FOR THE HELIUM ATOM C HRISTOPHER S. W ITHERS Applied Mathematics Group, Industrial Research Limited, Lower Hutt, New Zealand
and S ARALEES NADARAJAH School of Mathematics, University of Manchester, Manchester M13 9PL, UK (e-mail: [email protected]) (Received May 4, 2010 — Revised September 1, 2010) It is well known that Hylleraas’s series for the wave function of He does not satisfy Schr¨odinger’s equation. Kinoshita (1957) has given a class of solutions which do satisfy this equation as well as Kato (1957)’s four constraints. This note extends Kinoshita (1957)’s class of solutions. Keywords: helium atom, power series solutions, Schr¨odinger’s equation.
1.
Introduction
This note concerns solutions to Schr¨odinger’s equation for the helium atom. For about eighty years, starting with Hylleraas, such problems have been discussed. However, there are only few rigorous results available and many questions remain open, for example: 1. the lack of accessible computer software to calculate accurately and efficiently the solutions for the helium atom; 2. the lack of a unified framework to provide rigorous solutions to Schr¨odinger’s equation for all atoms (not just helium); 3. the lack of multidimensional extensions; 4. the lack of extensions involving both time and space; and so on. Kinoshita (1957) showed that the Schr¨odinger equation for He has a number of power series solutions in the variables s = r1 + r2 , p = r12 / (r1 + r2 ) , q = (r2 − r1 ) /r12 , [33]
34
C. S. WITHERS and S. NADARAJAH
where r1 and r2 are the distances from the electrons to the nucleus and r12 is the distance between electrons. Here, we give a more general class of solutions. 2.
Main results Kinoshita (1957) showed that ∞ ∞ clmn s l pm q n = exp(−s/2) clmn s l−m um−n t n ψ0 = exp(−s/2) l,m,n=0
(1)
l,m,n=0
satisfies Schr¨odinger’s equation for He with s, p, q as above and u = r12 , t = r2 −r1 if the coefficients {clmn } satisfy his recurrence relation (A.1): n(n − 1)clmn + (m − n + 2)(m + n − 1)cl,m,n−2 +(l − m − n + 4)(l + m − n + 3)cl,m−2,n−2 − (m − n + 2)(2l − m − n + 7)cl,m−2,n−4 −(l − m + 4)(l − m + 3)cl,m−4,n−4 − cl−1,m−1,n−2 − (l − n − 5)cl−1,m−2,n−2 +(m − n + 2)cl−1,m−2,n−4 + cl−1,m−3,n−4 + (l − m + 3)cl−1,m−4,n−4 + (E + 1/4) cl−2,m−2,n−2 − cl−2,m−4,n−4 = 0. (2) Starting with arbitrary α = {clm0 , clm1 }, the other coefficients are uniquely determined by this relation. However, the same is true of the more general type of solution ψ1 = s L pM q N ψ0
(3)
with c000 = 0. In this case, the {clmn } satisfy (2) with l, m, n in the multiplier functions n(n−1), . . . , (l −m+3) replaced by l = l +L, m = m+M, and n = n+N . We shall call this recurrence relation (A.1). Note that the factor s/2 in (1) can be replaced by any polynomial in s, p, q (since all such modified series are formally equivalent) – some of which should give better convergence. Kinoshita (1957) states that only {clm0 } are needed, but this is only so if odd powers of q are disallowed as in his (3.5). This restricts him to solutions symmetric in r1 and r2 . The wave equation has no Frobenius type solution (3) with (s, p, q) replaced by (r1 , r2 , r12 ), see Withers (1984). Let us write L = (L, M, N ), s = (s, p, q) and l = (l, m, n). Putting l = 0, we obtain N = 0 or 1. However, L and M are arbitrary. If N = 0 and α = {clm0 , clm1 } is specified, or if N = 1 and clm1 ≡ 0 and α = {clm0 } is specified, then (3) has a unique solution with coefficients cl (L, α), say, so that the wave equation has a solution of the form ∞ (4) ψ = ψ (s; L, α) = exp(−s/2) s l pm q n cl (L, α) . l=0
Kato (1957) noted that for atomic systems ψ must satisfy four constraints. Kinoshita (1957) labels them (A.7)–(A.10). In our more general case, we shall label them (A.7)–(A.10).
¨ FURTHER SOLUTIONS TO SCHRODINGER’S EQUATION FOR THE HELIUM ATOM
35
Firstly, continuity of ψ holds if and only if clmn = 0 for m ≤ 0 and n = 0,
(A.7)
clmn = 0 for l ≤ 0 and l = 0.
(A.8)
and
Since c000 = 0, M > 0 or N = 0, and L > 0 or L = 0. So, we have three cases: (I) L = 0; (II) N = 0, L > 0; (III) N = 1, L > 0, M > 0. Case I is Kinoshita (1957)’s case. In Case II, (A.8) is automatically satisfied. In Case III both (A.7) and (A.8) are satisfied. One can show that the only other restrictions on L come from l = (0, 0, 2) and (1, 1, 2), which imply that for Case II, M ≥ 0. This compatibility of (A.7), (A.8) with (A.1) is not obvious. In fact, in Kinoshita (1957)’s case it only works because in (2) for l = (0, 2, 2), l − m − n − 4 = 0; for l = (0, 2, 4), m − n + 2 = 0; and, for l = (0, 4, 4), l − m + 4 = 0. Conditions (A.7), (A.8) may now be viewed simply as restrictions on the choice of α as follows: (I) L = 0: cl01 = 0; c0m0 = 0 for m = 0; c0m1 = 0, (II) N = 0, L > 0, M ≥ 0: either M = 0 and cl01 = 0 or M > 0, (III) N = 1, L > 0, M > 0: α is unrestricted. Secondly, there are the two constraints from Kato (1957)’s second theorem: for all l ∞ l − m − n clmn = 0 (A.9) m,n=0
and ∞
(n + 1)−1 cl1n = 0.5cl−1,0,0 .
(A.10)
n=0
We now show that in each case the unrestricted elements of α, say β, may be chosen to satisfy these. Note that (A.10) may be viewed as determining {cl10 } as a function of L and the other components of β. Note that (A.9) may be viewed as determining in Case I: {cl00 l = 0}, and in Cases II and III: ⎧ ⎧ ⎨c ⎨ = 0 010 if l = 0 and M + 1 + N − L ⎩ c020 ⎩= 0
36
C. S. WITHERS and S. NADARAJAH
and
⎧ ⎨c
l00
⎩ cl10
if l = 0 and M + N − L
⎧ ⎨ = l ⎩= l
,
again as functions of L and the remaining β. In each case, let γ denote the unrestricted components of β. For example, for Case I, γ = {c000 } ∪ {clm0 , for l ≥ 1, m ≥ 2} ∪ {clm1 , l = 0}. Then in each case for given L and γ we obtain a unique solution ψ(s; L, γ ) of (A.1), (A.7)–(A.10). Varying γ provides a continuous energy spectrum. It would be very interesting to know if in Cases II and III a minimization as done by Kinoshita (1957) for Case I would give significantly lower energy for the ground state of He. We now indicate how to widen our class of solutions given above. We noted that for N = 0 or 1 a solution of the wave equation of the form (4) is available for any L, M, α. It follows that any derivatives of (4) with respect to (L, M) will also satisfy the wave equation. This gives for each integer A, B ≥ 0 a solution of the form ∞ AB (5) s l pm q n (ln s)i (ln p)j clmnij ψAB = exp(−s/2) (L, α) . l=0
For related work, see Leray (1983) and Morgan (1986). One may now argue as before that Kato (1957)’s four constraints may be satisfied by a suitable restriction of α. It would be interesting to know if the set of solutions so obtained contains Fock’s solutions of the wave equation (Fock, 1954, 1958, 2004). It would also be interesting to know if (5) can be extended to allow nonintegral powers of ln s and ln p. This in turn would suggest allowing powers of ln ln s and ln ln p. However, the recurrence relation for this more general problem looks difficult to solve. Finally, there is the problem of ensuring that ψ 2 dr < ∞ (cf. (12.6) and footnote 9 of Kinoshita (1957)). In the case of the H atom this requirement leads to the discrete spectrum of eigenvalues. The difficulties of a similar analysis for He are illustrated in Appendix C of Kinoshita (1957). One may also obtain E by minimizing ∞ s u 2 ds udu s − t 2 ψ 2 dt Q (L, γ , E) = ψ 2 dr = 0 0 0 −1 −1 =2 cl cl l + l + 2L + 4 ! m + m + 2M + 2 m + m + 2M + 4 l,l ≥0
at cl = cl (L, γ ) with respect to L, γ and E, where x! = (x + 1). For some recent references giving some other approaches and generalizations of the present problem, see Davis and Maslen (1983) and Nakashima and Nakatsuji (2008).
¨ FURTHER SOLUTIONS TO SCHRODINGER’S EQUATION FOR THE HELIUM ATOM
37
A future work is to explore the sets of functions in (1) for the considered cases I, II and III, that is to see what sort of linear independence, completeness/overcompleteness, etc. they represent and to see how the convergence of the Rayleigh-Ritz method may be affected. See Klahn and Bingel (1977) and Klahn (1981). Acknowledgments The authors would like to thank the Editor and the two referees for careful reading and for their comments which greatly improved the paper. REFERENCES [1] C. L. Davis and E. N. Maslen: On exact analytical solutions for the few-particle Schr¨odinger-equation .4. the asymptotic form and normalisability of the wavefunction, Journal of Physics A 16 (1983), 4255–4264. [2] V. A. Fock: On the Schr¨odinger equation of the helium atom, Izv. Akad. Nauk. SSSR, Ser. Fiz. 18 (1954), 161–172. [3] V. A. Fock: On the Schr¨odinger equation of the helium atom, Det Kongelige Norske Videnskabers Selskabs Forhandlunger, B 31 (1958), 138–152. [4] V. A. Fock: On the Schr¨odinger equation of the helium atom, In: V. A. Fock - Selected Works Quantum Mechanics and Quantum Field Theory, editors L. D. Faddeev, L. A. Khalfin and I. V. Komarov, Chapman and Hall/CRC, Boca Raton, Florida, pp. 525–538 (2004). [5] T. Kato: On the eigenfunctions of many-particle systems in quantum mechanics, Communications in Pure and Applied Mathematics 10 (1957), 151–177. [6] T. Kinoshita: Ground state of the Helium atom, Physical Review, 105 (1957), 1490–1502. [7] B. Klahn: Review of the linear independence properties of infinite sets of functions used in quantum chemistry, Advances in Quantum Chemistry 13 (1981), 155-209. [8] B. Klahn and W. A. Bingel: The convergence of the Rayleigh–Ritz method in quantum chemistry, Theoretica Chimica Acta 44 (1977), 9–26. [9] J. Leray: Sur les solutions de l’´equation de Schr¨odinger atomique et le cas particulier de deux e´ lectrons, Lecture Notes in Physics 195 (1983), 235–247. [10] J. D. Morgan: Convergence properties of Fock’s expansion for S-state eigenfunctions of the helium atom, Theoretica Chimica Acta 69 (1986), 181–223. [11] H. Nakashima and H. Nakatsuji: How accurately does the free complement wave function of a Helium atom satisfy the Schr¨odinger equation? Physical Review Letters 101, Article Number: 240406 (2008). [12] C. S. Withers: Schr¨odinger-equation for the Helium atom, Physical Review A 30 (1984), 1506–1506.
Vol. 67 (2011)
No. 1
FURTHER SOLUTIONS ¨ TO SCHRODINGER’S EQUATION FOR THE HELIUM ATOM C HRISTOPHER S. W ITHERS Applied Mathematics Group, Industrial Research Limited, Lower Hutt, New Zealand
and S ARALEES NADARAJAH School of Mathematics, University of Manchester, Manchester M13 9PL, UK (e-mail: [email protected]) (Received May 4, 2010 — Revised September 1, 2010) It is well known that Hylleraas’s series for the wave function of He does not satisfy Schr¨odinger’s equation. Kinoshita (1957) has given a class of solutions which do satisfy this equation as well as Kato (1957)’s four constraints. This note extends Kinoshita (1957)’s class of solutions. Keywords: helium atom, power series solutions, Schr¨odinger’s equation.
1.
Introduction
This note concerns solutions to Schr¨odinger’s equation for the helium atom. For about eighty years, starting with Hylleraas, such problems have been discussed. However, there are only few rigorous results available and many questions remain open, for example: 1. the lack of accessible computer software to calculate accurately and efficiently the solutions for the helium atom; 2. the lack of a unified framework to provide rigorous solutions to Schr¨odinger’s equation for all atoms (not just helium); 3. the lack of multidimensional extensions; 4. the lack of extensions involving both time and space; and so on. Kinoshita (1957) showed that the Schr¨odinger equation for He has a number of power series solutions in the variables s = r1 + r2 , p = r12 / (r1 + r2 ) , q = (r2 − r1 ) /r12 , [33]
34
C. S. WITHERS and S. NADARAJAH
where r1 and r2 are the distances from the electrons to the nucleus and r12 is the distance between electrons. Here, we give a more general class of solutions. 2.
Main results Kinoshita (1957) showed that ∞ ∞ clmn s l pm q n = exp(−s/2) clmn s l−m um−n t n ψ0 = exp(−s/2) l,m,n=0
(1)
l,m,n=0
satisfies Schr¨odinger’s equation for He with s, p, q as above and u = r12 , t = r2 −r1 if the coefficients {clmn } satisfy his recurrence relation (A.1): n(n − 1)clmn + (m − n + 2)(m + n − 1)cl,m,n−2 +(l − m − n + 4)(l + m − n + 3)cl,m−2,n−2 − (m − n + 2)(2l − m − n + 7)cl,m−2,n−4 −(l − m + 4)(l − m + 3)cl,m−4,n−4 − cl−1,m−1,n−2 − (l − n − 5)cl−1,m−2,n−2 +(m − n + 2)cl−1,m−2,n−4 + cl−1,m−3,n−4 + (l − m + 3)cl−1,m−4,n−4 + (E + 1/4) cl−2,m−2,n−2 − cl−2,m−4,n−4 = 0. (2) Starting with arbitrary α = {clm0 , clm1 }, the other coefficients are uniquely determined by this relation. However, the same is true of the more general type of solution ψ1 = s L pM q N ψ0
(3)
with c000 = 0. In this case, the {clmn } satisfy (2) with l, m, n in the multiplier functions n(n−1), . . . , (l −m+3) replaced by l = l +L, m = m+M, and n = n+N . We shall call this recurrence relation (A.1). Note that the factor s/2 in (1) can be replaced by any polynomial in s, p, q (since all such modified series are formally equivalent) – some of which should give better convergence. Kinoshita (1957) states that only {clm0 } are needed, but this is only so if odd powers of q are disallowed as in his (3.5). This restricts him to solutions symmetric in r1 and r2 . The wave equation has no Frobenius type solution (3) with (s, p, q) replaced by (r1 , r2 , r12 ), see Withers (1984). Let us write L = (L, M, N ), s = (s, p, q) and l = (l, m, n). Putting l = 0, we obtain N = 0 or 1. However, L and M are arbitrary. If N = 0 and α = {clm0 , clm1 } is specified, or if N = 1 and clm1 ≡ 0 and α = {clm0 } is specified, then (3) has a unique solution with coefficients cl (L, α), say, so that the wave equation has a solution of the form ∞ (4) ψ = ψ (s; L, α) = exp(−s/2) s l pm q n cl (L, α) . l=0
Kato (1957) noted that for atomic systems ψ must satisfy four constraints. Kinoshita (1957) labels them (A.7)–(A.10). In our more general case, we shall label them (A.7)–(A.10).
¨ FURTHER SOLUTIONS TO SCHRODINGER’S EQUATION FOR THE HELIUM ATOM
35
Firstly, continuity of ψ holds if and only if clmn = 0 for m ≤ 0 and n = 0,
(A.7)
clmn = 0 for l ≤ 0 and l = 0.
(A.8)
and
Since c000 = 0, M > 0 or N = 0, and L > 0 or L = 0. So, we have three cases: (I) L = 0; (II) N = 0, L > 0; (III) N = 1, L > 0, M > 0. Case I is Kinoshita (1957)’s case. In Case II, (A.8) is automatically satisfied. In Case III both (A.7) and (A.8) are satisfied. One can show that the only other restrictions on L come from l = (0, 0, 2) and (1, 1, 2), which imply that for Case II, M ≥ 0. This compatibility of (A.7), (A.8) with (A.1) is not obvious. In fact, in Kinoshita (1957)’s case it only works because in (2) for l = (0, 2, 2), l − m − n − 4 = 0; for l = (0, 2, 4), m − n + 2 = 0; and, for l = (0, 4, 4), l − m + 4 = 0. Conditions (A.7), (A.8) may now be viewed simply as restrictions on the choice of α as follows: (I) L = 0: cl01 = 0; c0m0 = 0 for m = 0; c0m1 = 0, (II) N = 0, L > 0, M ≥ 0: either M = 0 and cl01 = 0 or M > 0, (III) N = 1, L > 0, M > 0: α is unrestricted. Secondly, there are the two constraints from Kato (1957)’s second theorem: for all l ∞ l − m − n clmn = 0 (A.9) m,n=0
and ∞
(n + 1)−1 cl1n = 0.5cl−1,0,0 .
(A.10)
n=0
We now show that in each case the unrestricted elements of α, say β, may be chosen to satisfy these. Note that (A.10) may be viewed as determining {cl10 } as a function of L and the other components of β. Note that (A.9) may be viewed as determining in Case I: {cl00 l = 0}, and in Cases II and III: ⎧ ⎧ ⎨c ⎨ = 0 010 if l = 0 and M + 1 + N − L ⎩ c020 ⎩= 0
36
C. S. WITHERS and S. NADARAJAH
and
⎧ ⎨c
l00
⎩ cl10
if l = 0 and M + N − L
⎧ ⎨ = l ⎩= l
,
again as functions of L and the remaining β. In each case, let γ denote the unrestricted components of β. For example, for Case I, γ = {c000 } ∪ {clm0 , for l ≥ 1, m ≥ 2} ∪ {clm1 , l = 0}. Then in each case for given L and γ we obtain a unique solution ψ(s; L, γ ) of (A.1), (A.7)–(A.10). Varying γ provides a continuous energy spectrum. It would be very interesting to know if in Cases II and III a minimization as done by Kinoshita (1957) for Case I would give significantly lower energy for the ground state of He. We now indicate how to widen our class of solutions given above. We noted that for N = 0 or 1 a solution of the wave equation of the form (4) is available for any L, M, α. It follows that any derivatives of (4) with respect to (L, M) will also satisfy the wave equation. This gives for each integer A, B ≥ 0 a solution of the form ∞ AB (5) s l pm q n (ln s)i (ln p)j clmnij ψAB = exp(−s/2) (L, α) . l=0
For related work, see Leray (1983) and Morgan (1986). One may now argue as before that Kato (1957)’s four constraints may be satisfied by a suitable restriction of α. It would be interesting to know if the set of solutions so obtained contains Fock’s solutions of the wave equation (Fock, 1954, 1958, 2004). It would also be interesting to know if (5) can be extended to allow nonintegral powers of ln s and ln p. This in turn would suggest allowing powers of ln ln s and ln ln p. However, the recurrence relation for this more general problem looks difficult to solve. Finally, there is the problem of ensuring that ψ 2 dr < ∞ (cf. (12.6) and footnote 9 of Kinoshita (1957)). In the case of the H atom this requirement leads to the discrete spectrum of eigenvalues. The difficulties of a similar analysis for He are illustrated in Appendix C of Kinoshita (1957). One may also obtain E by minimizing ∞ s u 2 ds udu s − t 2 ψ 2 dt Q (L, γ , E) = ψ 2 dr = 0 0 0 −1 −1 =2 cl cl l + l + 2L + 4 ! m + m + 2M + 2 m + m + 2M + 4 l,l ≥0
at cl = cl (L, γ ) with respect to L, γ and E, where x! = (x + 1). For some recent references giving some other approaches and generalizations of the present problem, see Davis and Maslen (1983) and Nakashima and Nakatsuji (2008).
¨ FURTHER SOLUTIONS TO SCHRODINGER’S EQUATION FOR THE HELIUM ATOM
37
A future work is to explore the sets of functions in (1) for the considered cases I, II and III, that is to see what sort of linear independence, completeness/overcompleteness, etc. they represent and to see how the convergence of the Rayleigh-Ritz method may be affected. See Klahn and Bingel (1977) and Klahn (1981). Acknowledgments The authors would like to thank the Editor and the two referees for careful reading and for their comments which greatly improved the paper. REFERENCES [1] C. L. Davis and E. N. Maslen: On exact analytical solutions for the few-particle Schr¨odinger-equation .4. the asymptotic form and normalisability of the wavefunction, Journal of Physics A 16 (1983), 4255–4264. [2] V. A. Fock: On the Schr¨odinger equation of the helium atom, Izv. Akad. Nauk. SSSR, Ser. Fiz. 18 (1954), 161–172. [3] V. A. Fock: On the Schr¨odinger equation of the helium atom, Det Kongelige Norske Videnskabers Selskabs Forhandlunger, B 31 (1958), 138–152. [4] V. A. Fock: On the Schr¨odinger equation of the helium atom, In: V. A. Fock - Selected Works Quantum Mechanics and Quantum Field Theory, editors L. D. Faddeev, L. A. Khalfin and I. V. Komarov, Chapman and Hall/CRC, Boca Raton, Florida, pp. 525–538 (2004). [5] T. Kato: On the eigenfunctions of many-particle systems in quantum mechanics, Communications in Pure and Applied Mathematics 10 (1957), 151–177. [6] T. Kinoshita: Ground state of the Helium atom, Physical Review, 105 (1957), 1490–1502. [7] B. Klahn: Review of the linear independence properties of infinite sets of functions used in quantum chemistry, Advances in Quantum Chemistry 13 (1981), 155-209. [8] B. Klahn and W. A. Bingel: The convergence of the Rayleigh–Ritz method in quantum chemistry, Theoretica Chimica Acta 44 (1977), 9–26. [9] J. Leray: Sur les solutions de l’´equation de Schr¨odinger atomique et le cas particulier de deux e´ lectrons, Lecture Notes in Physics 195 (1983), 235–247. [10] J. D. Morgan: Convergence properties of Fock’s expansion for S-state eigenfunctions of the helium atom, Theoretica Chimica Acta 69 (1986), 181–223. [11] H. Nakashima and H. Nakatsuji: How accurately does the free complement wave function of a Helium atom satisfy the Schr¨odinger equation? Physical Review Letters 101, Article Number: 240406 (2008). [12] C. S. Withers: Schr¨odinger-equation for the Helium atom, Physical Review A 30 (1984), 1506–1506.