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A note on angular eigenfunctions for the ground states of the helium isoelectronic sequence
Volume 101A, number 2
PHYSICS LETTERS
12 March 1984
A NOTE ON ANGULAR EIGENFUNCTIONS FOR THE GROUND STATES OF THE HELIUM ISOELECTRONIC SEQUENCE Richard D. HARCOURT
Department of Physical Chemistry, University of Melbourne, Parkville, Victoria 3052, Australia Received 4 January 1984
For each angle 4~~-~2 - ~t, the orbital wave-function 1s (1)1s (2) = exp[-~ (r 1 + r 2)] is an eigenfunction for the atomic hamiltonian when r 1 = r 2 and ~ = Z - [4sin(lq~)] -1. As the atomic number Z increases from 2 to 100, the range of ¢ for which the two electrons must both be at the same distance from the nucleus increases from 73.6 ° -180 ° for He to 15.8°-180 ° for Fm 98+.
The purpose o f this communication is to demonstrate that the Schr6dinger equation for the ground state o f the helium atom iso-electronic sequence has a simple analytical solution for each angle ¢ = ¢2 - ¢1 when the two electrons are located at the same distance from the nucleus ,1, and to deduce the range o f angles for which this solution alone is appropriate. For a hydrogen-like atomic system with atomic number Z, the 1s wave-function = exp(-~r),
(1)
is an eigenfunction o f the hamiltonian H (in au) o f eq. (2),
I:t = - ~ V 2 - Zr -1
(2)
(3)
is an eigenvalue ( - ~ Z2), i.e. independent o f the distance r, for ~ = Z. The atomic hamiltonian for the helium isoelectronic sequence is
H=_lv with
(5)
in which ¢ is the instantaneous angle between vectors r 1 and r 2. The local energy for the 2-electron wavefunction ~k = l s ( 1 ) l s ( 2 ) = e x p [ - ~ ( r 1 + r 2 ) ]
(6)
is given by Etoc = _~2 + ~(ri-1 + r ~ l ) _ Z r ~ l
_ Z r ~ l + r ~ l , (7)
which reduces to eq. (8) when r 1 = r 2 = r, Eloc = _~2 + 2~r-1 _ 2Zr-1 + [2r sin(~-¢)] - 1 .
[ 7 2 = d2/dr 2 + (2/r)d/dr] , when ~ = Z. Specifically the local energy for eq. (1), namely Eloc =_/-~qj/~ = _½ ~2 + ~r-1 _ Zr-X
r12 = (r 2 + r 2 - 2 r l r 2 cos¢) 1/2 ,
l : 2 -Zr-{ 1 - Z r ~ 1 +r-{1 =I:I0 + r ; ~ , - ~V (4)
4:1 This result has been indicated briefly in the appendix of ref. [1]. 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(8)
For each value o f the angle ¢, Elo c is an eigenvalue, _~2, when =Z-
[4sin(½q~)]-1 = Z -
s.
(9)
Therefore eq. (6) is the exact (ground-state) solution o f the Schr6dinger equation for angle ¢ when ~ is given by eq. (9) and the two electrons are located at the same distance from the nucleus. In eq. (9), s =[4sin(½¢)] -1 is an angular-dependent screening constant. The 1- and 2-electron energy eigenvalues for the wave-functions o f eqs. (1) and (6) may also be obtained by requiring that for each angle ~b, the local energies of eqs. (3) and (8) are stationary, i.e. that
aEloc/ar = aEloc/a~ = 0 .
(10) 83
Volume 101A, number 2
PHYSICS LETTERS • 2~1 4Z sin3(~-~ ' ) - 2 sm ~ ) -.,~
These give ~ = Z or ~ = Z - s as previously, and r = ~-1. (When r = ~-1, each 1 s electron is located at its most probable distance from the nucleus.) It is then easy to verify that the virial theorem E(¢) = - T(¢) = ~ V(¢) is obeyed at each angle, with E(~) = - ~ Z 2 or - ( Z - s) 2 as previously. To determine the range of angles for which eq. (6) is an angular eigenfunction for the hamiltonian of eq. (4), we proceed in the following manner. When r 1 :~r2, e x p [ - ( ~ l r 1 + ~2r2)] replaces exp [ - ~ ( r 1 + r2) ] as the 2-electron wave-function, and the local energy is given by
E = f - [Z - 1/4sin(l~)] 2P(~)d~b ¢, X
= aElo~/O~2
=
(/)_1 P(~)d~
,
(1 7)
;~EloJ~r I = OEloc/Or 2
: 0.
(12)
in which P(¢) is a weighting factor. Here we shall use two expressions for P(~b), namely: (a) P(¢) = r-2r212 o~ sin2(½~b); this is the simplest function of sin(½q~) which ensures that the integrand in the numerator o f eq. (1 7) remains finite as ~b'~ 0. (b) P(~b) = r2 2 oc r 2 sin2(~-~b) -- ~-2 sin 2 ({~b). The first-order perturbation energies (with ~ variation)
Application of eq. (12) gives ~1 = ri -1, ~2 = r2-1 and -~lr{ 2 +Zr{ 2 -(r2cosc~-rl)r-{ 3 =0,
(13)
- ~ 2 r ~ -2 + Zr~ 2 - (r 1 cos ~b - r2)r-~3 = 0 .
(14)
On substituting ~1 = r1-1, ~2 = r2-1, and r 2 = kr 1 into eqs. (13) and (14), and then eliminating r I we obtain
= ~2 _ 2Z~ + 0.625~,
= k 3 + (k 2 +k)(1 - cos $) + 1 .
(18)
and the exact energies [2] are also included in table 1 for comparison. As Z increases, the range of angle over which the two electrons are located at the same distance from the nucleus increases, i.e. ~bofeq. (6) [with given by eq. (9)] is an angular eigenfunction for an increasingly larger range of angles. For moderate Z,
Z ( k 2 - 2 k cos ¢ + 1) 1-5 (15)
When k = 1 at ¢ = ~b', eq. (15) reduces to
Table 1 Limiting angles (o) and energies (au) for Z = 2, 3, 10, 20 and 100.
He Li+ Nea+ Ca ts+ Fm 98+
~'
g a)
~ b)
Eexact
~ c)
73.556 60.000 36.125 27.858 15.796
-2.945 -7.333 -93.97 -387.67 -9937.0
-2.938 -7.321 -93.92 -387.63 -9936.9
-2.904 -7.280 -93.9• -387.66 -9937.7
-2.848 -7.223 -93.85 -387.60 -9937.6
a) Eq. (17) with P(~) = sin2(~), 84
(16)
7r
(11)
In order that the virial theorem is obeyed at each angle q~, we require that
OE~o~/~l
1 = 0.
For ~ > ~' and Z / > 2, it may be shown by substitution that eqs. (13) and (14) can only be solved when r 1 = r 2, i.e. eq. (15) has no real roots > 0. Therefore both electrons must simultaneously be at the same distance from the nucleus when ~ > ~'. Thus when ~' ~< ~ ~< ~r with ¢ ' calculated from eq. (•6), the orbital wave-function of eq. (6) is an eigenfunction of the 2electron hamiltonian for each angle ¢. In table 1, we report the ¢' for some representative values of Z, together with the average value o f energy (E) over the range of angle ¢' - 7r. As in ref. [ 1 ], the average energy is calculated from eq. (17)
Eloc = - ~11 2 + ~lri -1 - ~i 22 + ~2r~l
- Z r l 1 -Zr~ 1 +r{ 1.
12 March 1984
b) Eq. (17) with P(ck) = r]2.
c) Eq. (18).
Volume 101A, number 2
PHYSICS LETTERS
12 March 1984
the energies calculated from eq. (17) using either form of P(~) are closer to the exact energies than are the first-order perturbation energies [eq. (18)]. When 0 ~< ¢ < ~b', r 1 need not equal r2, and therefore over this range of angle, the ff of eq. (6) is not an angular eigenfunction for the 2-electron hamiltonian. A technique for evaluating E over the full range of angle (0-rr) is described in ref. [1], and will not be discussed here. The special case o f Z = 1 ( H - ) is also described in ref. [1]. Our purpose here has been to show that the orbital wave-function of eq. (6) is an exact wave-function for each angle ~> ~' when the orbital exponent is given by eq. (9). This wave-function is therefore an eigenfunction of two hamiltonian operators, namely HO of eq. (4) when ~ = Z, and H0 + r]-~ when~=Z-sandr l = r 2.
References
Note added: On page 2654 o f ref. [1 ], Pb (¢) = r22 ~ 1 + 2r 1 cos ~bwas used to evaluate approximately the E b ( 0 - 5 0 . 5 °) for helium. An approach which is consistent with that used to evaluate the E"a and
[1] R.D. Harcourt, J. Phys. B16 (1983) 2647; corrigendum 16 (1983) 3685. [2] C.W. Scherr and R.D. Knight, Rev. Mod. Phys. 35 (1963) 436.
F c o f ref. [1] requires Pb (~b) = r~2r22, which approximates to (1 + 2r 1 cos ~b)-1 when r 2 >> r 1 . With Pb (40 = (1 + 2rlcos 40 -1 and r 1 = 0.48409 au [1], E ( 0 180 °) = - 2 . 9 0 7 au is then calculated when either one or two fb (¢) functions are used for the evaluation o f the/~c(50.5-73.6 °) in ref. [1]. As ~bincreases from 0 ° to 50.5 °, Eh(q~ ) decreases, and r~2r h rather then r22 increases with angle. There fore r~2r~2 is the better form o f P(~b). When r 1 = r 2 = r (as in table 1 o f the present paper), both forms o f P(¢) increase as ¢ increases. Retention o f higher powers of r 1 in the evaluation of E"b (0-50.5°), with Pb (c~) = r ~ 2r22, is being currently investigated.
85
PHYSICS LETTERS
12 March 1984
A NOTE ON ANGULAR EIGENFUNCTIONS FOR THE GROUND STATES OF THE HELIUM ISOELECTRONIC SEQUENCE Richard D. HARCOURT
Department of Physical Chemistry, University of Melbourne, Parkville, Victoria 3052, Australia Received 4 January 1984
For each angle 4~~-~2 - ~t, the orbital wave-function 1s (1)1s (2) = exp[-~ (r 1 + r 2)] is an eigenfunction for the atomic hamiltonian when r 1 = r 2 and ~ = Z - [4sin(lq~)] -1. As the atomic number Z increases from 2 to 100, the range of ¢ for which the two electrons must both be at the same distance from the nucleus increases from 73.6 ° -180 ° for He to 15.8°-180 ° for Fm 98+.
The purpose o f this communication is to demonstrate that the Schr6dinger equation for the ground state o f the helium atom iso-electronic sequence has a simple analytical solution for each angle ¢ = ¢2 - ¢1 when the two electrons are located at the same distance from the nucleus ,1, and to deduce the range o f angles for which this solution alone is appropriate. For a hydrogen-like atomic system with atomic number Z, the 1s wave-function = exp(-~r),
(1)
is an eigenfunction o f the hamiltonian H (in au) o f eq. (2),
I:t = - ~ V 2 - Zr -1
(2)
(3)
is an eigenvalue ( - ~ Z2), i.e. independent o f the distance r, for ~ = Z. The atomic hamiltonian for the helium isoelectronic sequence is
H=_lv with
(5)
in which ¢ is the instantaneous angle between vectors r 1 and r 2. The local energy for the 2-electron wavefunction ~k = l s ( 1 ) l s ( 2 ) = e x p [ - ~ ( r 1 + r 2 ) ]
(6)
is given by Etoc = _~2 + ~(ri-1 + r ~ l ) _ Z r ~ l
_ Z r ~ l + r ~ l , (7)
which reduces to eq. (8) when r 1 = r 2 = r, Eloc = _~2 + 2~r-1 _ 2Zr-1 + [2r sin(~-¢)] - 1 .
[ 7 2 = d2/dr 2 + (2/r)d/dr] , when ~ = Z. Specifically the local energy for eq. (1), namely Eloc =_/-~qj/~ = _½ ~2 + ~r-1 _ Zr-X
r12 = (r 2 + r 2 - 2 r l r 2 cos¢) 1/2 ,
l : 2 -Zr-{ 1 - Z r ~ 1 +r-{1 =I:I0 + r ; ~ , - ~V (4)
4:1 This result has been indicated briefly in the appendix of ref. [1]. 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(8)
For each value o f the angle ¢, Elo c is an eigenvalue, _~2, when =Z-
[4sin(½q~)]-1 = Z -
s.
(9)
Therefore eq. (6) is the exact (ground-state) solution o f the Schr6dinger equation for angle ¢ when ~ is given by eq. (9) and the two electrons are located at the same distance from the nucleus. In eq. (9), s =[4sin(½¢)] -1 is an angular-dependent screening constant. The 1- and 2-electron energy eigenvalues for the wave-functions o f eqs. (1) and (6) may also be obtained by requiring that for each angle ~b, the local energies of eqs. (3) and (8) are stationary, i.e. that
aEloc/ar = aEloc/a~ = 0 .
(10) 83
Volume 101A, number 2
PHYSICS LETTERS • 2~1 4Z sin3(~-~ ' ) - 2 sm ~ ) -.,~
These give ~ = Z or ~ = Z - s as previously, and r = ~-1. (When r = ~-1, each 1 s electron is located at its most probable distance from the nucleus.) It is then easy to verify that the virial theorem E(¢) = - T(¢) = ~ V(¢) is obeyed at each angle, with E(~) = - ~ Z 2 or - ( Z - s) 2 as previously. To determine the range of angles for which eq. (6) is an angular eigenfunction for the hamiltonian of eq. (4), we proceed in the following manner. When r 1 :~r2, e x p [ - ( ~ l r 1 + ~2r2)] replaces exp [ - ~ ( r 1 + r2) ] as the 2-electron wave-function, and the local energy is given by
E = f - [Z - 1/4sin(l~)] 2P(~)d~b ¢, X
= aElo~/O~2
=
(/)_1 P(~)d~
,
(1 7)
;~EloJ~r I = OEloc/Or 2
: 0.
(12)
in which P(¢) is a weighting factor. Here we shall use two expressions for P(~b), namely: (a) P(¢) = r-2r212 o~ sin2(½~b); this is the simplest function of sin(½q~) which ensures that the integrand in the numerator o f eq. (1 7) remains finite as ~b'~ 0. (b) P(~b) = r2 2 oc r 2 sin2(~-~b) -- ~-2 sin 2 ({~b). The first-order perturbation energies (with ~ variation)
Application of eq. (12) gives ~1 = ri -1, ~2 = r2-1 and -~lr{ 2 +Zr{ 2 -(r2cosc~-rl)r-{ 3 =0,
(13)
- ~ 2 r ~ -2 + Zr~ 2 - (r 1 cos ~b - r2)r-~3 = 0 .
(14)
On substituting ~1 = r1-1, ~2 = r2-1, and r 2 = kr 1 into eqs. (13) and (14), and then eliminating r I we obtain
= ~2 _ 2Z~ + 0.625~,
= k 3 + (k 2 +k)(1 - cos $) + 1 .
(18)
and the exact energies [2] are also included in table 1 for comparison. As Z increases, the range of angle over which the two electrons are located at the same distance from the nucleus increases, i.e. ~bofeq. (6) [with given by eq. (9)] is an angular eigenfunction for an increasingly larger range of angles. For moderate Z,
Z ( k 2 - 2 k cos ¢ + 1) 1-5 (15)
When k = 1 at ¢ = ~b', eq. (15) reduces to
Table 1 Limiting angles (o) and energies (au) for Z = 2, 3, 10, 20 and 100.
He Li+ Nea+ Ca ts+ Fm 98+
~'
g a)
~ b)
Eexact
~ c)
73.556 60.000 36.125 27.858 15.796
-2.945 -7.333 -93.97 -387.67 -9937.0
-2.938 -7.321 -93.92 -387.63 -9936.9
-2.904 -7.280 -93.9• -387.66 -9937.7
-2.848 -7.223 -93.85 -387.60 -9937.6
a) Eq. (17) with P(~) = sin2(~), 84
(16)
7r
(11)
In order that the virial theorem is obeyed at each angle q~, we require that
OE~o~/~l
1 = 0.
For ~ > ~' and Z / > 2, it may be shown by substitution that eqs. (13) and (14) can only be solved when r 1 = r 2, i.e. eq. (15) has no real roots > 0. Therefore both electrons must simultaneously be at the same distance from the nucleus when ~ > ~'. Thus when ~' ~< ~ ~< ~r with ¢ ' calculated from eq. (•6), the orbital wave-function of eq. (6) is an eigenfunction of the 2electron hamiltonian for each angle ¢. In table 1, we report the ¢' for some representative values of Z, together with the average value o f energy (E) over the range of angle ¢' - 7r. As in ref. [ 1 ], the average energy is calculated from eq. (17)
Eloc = - ~11 2 + ~lri -1 - ~i 22 + ~2r~l
- Z r l 1 -Zr~ 1 +r{ 1.
12 March 1984
b) Eq. (17) with P(ck) = r]2.
c) Eq. (18).
Volume 101A, number 2
PHYSICS LETTERS
12 March 1984
the energies calculated from eq. (17) using either form of P(~) are closer to the exact energies than are the first-order perturbation energies [eq. (18)]. When 0 ~< ¢ < ~b', r 1 need not equal r2, and therefore over this range of angle, the ff of eq. (6) is not an angular eigenfunction for the 2-electron hamiltonian. A technique for evaluating E over the full range of angle (0-rr) is described in ref. [1], and will not be discussed here. The special case o f Z = 1 ( H - ) is also described in ref. [1]. Our purpose here has been to show that the orbital wave-function of eq. (6) is an exact wave-function for each angle ~> ~' when the orbital exponent is given by eq. (9). This wave-function is therefore an eigenfunction of two hamiltonian operators, namely HO of eq. (4) when ~ = Z, and H0 + r]-~ when~=Z-sandr l = r 2.
References
Note added: On page 2654 o f ref. [1 ], Pb (¢) = r22 ~ 1 + 2r 1 cos ~bwas used to evaluate approximately the E b ( 0 - 5 0 . 5 °) for helium. An approach which is consistent with that used to evaluate the E"a and
[1] R.D. Harcourt, J. Phys. B16 (1983) 2647; corrigendum 16 (1983) 3685. [2] C.W. Scherr and R.D. Knight, Rev. Mod. Phys. 35 (1963) 436.
F c o f ref. [1] requires Pb (~b) = r~2r22, which approximates to (1 + 2r 1 cos ~b)-1 when r 2 >> r 1 . With Pb (40 = (1 + 2rlcos 40 -1 and r 1 = 0.48409 au [1], E ( 0 180 °) = - 2 . 9 0 7 au is then calculated when either one or two fb (¢) functions are used for the evaluation o f the/~c(50.5-73.6 °) in ref. [1]. As ~bincreases from 0 ° to 50.5 °, Eh(q~ ) decreases, and r~2r h rather then r22 increases with angle. There fore r~2r~2 is the better form o f P(~b). When r 1 = r 2 = r (as in table 1 o f the present paper), both forms o f P(¢) increase as ¢ increases. Retention o f higher powers of r 1 in the evaluation of E"b (0-50.5°), with Pb (c~) = r ~ 2r22, is being currently investigated.
85