V.A. Fock
Zs. Phys.62, N11–12, 795, 1930 TOI5, N51, 29, 1931
1 A Form of Wave Functions
The number of electrons in a sodium atom is equal to 11; one of them is a valence electron. Ten inner electrons are distributed in pairs at five orbitals. Thus, we have six orbitals, and, hence, six different wave functions. In the equations of the previous article, it should be taken
N = 11, p= 5, q= 6.
Let us denote a wave function of a valence electron byψ6, and wave functions of inner electrons byψ1, . . . , ψ5. In the case of spherical sym- metry, these functions should have the form
ψ1= 1
rf1(r) 1
√4π, ψ2= 1
rf2(r) 1
√4π, ψ3= 1
rf3(r)
√3
√4πcosϑ, ψ4= 1
rf3(r)
√3
√4πsinϑcosϕ, ψ5= 1
rf3(r)
√3
√4πsinϑsinϕ, ψ6= 1
rf4(r) 1
√4πYl(ϑ, ϕ).
(1)
The spherical functionYl(ϑ, ϕ) is normalized by the following condi- tion:
1 4π
Z Z
|Yl(ϑ, ϕ)|2sinϑ dϑ dϕ= 1. (2)
The normalized condition forfi(r) is Z∞
0
[fi(r)]2dr= 1 (i= 1,2,3,4). (3) In order for the functions ψ1, . . . , ψ5 to be orthogonal, fi(r) should obey the following conditions:
Z∞
0
f1(r)f2(r)dr= 0; δl0ã Z∞
0
f1(r)f4(r)dr= 0;
δl0ã Z∞
0
f2(r)f4(r)dr= 0; δl1ã Z∞
0
f3(r)f4(r)dr= 0.
(4)
The functions f1, f2, f3, f4 correspond to different electronic shells.
For simplicity we denoted them by one subscript only; more complete notation would bef10, f20, f21, fnl, where two subscripts are nothing else but the quantum numbers of the electron shell under consideration.
2 Expressions for the Energy of an Atom
To obtain the equations, we will vary the expression for the energy of an atom derived in the previous article [formula (93)]. In this formula we should carry out integration over the variablesϑ andϕ. Let us denote a sum of simple volume integrals by W1 and a sum of double volume integrals byW2, so that
W =W1+W2. (5)
Evaluation ofW1is quite trivial; as the result, one gets W1=
Z∞
0
ãàdf1(r) dr
ả2 +
àdf2(r) dr
ả2 + 3
àdf3(r) dr
ả2 +1
2
àdf4(r) dr
ả2 +
+6
r2f32+l(l+1) 2r2 f42
á dr−
Z∞
0
¡2f12+ 2f22+ 6f32+f42¢11
r dr. (6) In this expression the first integral represents an average kinetic energy, and the second integral gives the average potential energy of electrons with respect to the nucleus (without the energy of interaction).
Evaluation of the interaction energyW2 is more complicated; it can be carried out as follows. Let us suppose in formulae (62) and (62∗) of the previous article
%2(x, x0) =%(r,r0); %1(x, x0) =%(r,r0) +σ(r,r0),
%(r,r) =%(r); σ(r,r) =σ(r),
)
(7) where
%(r,r0) and σ(r,r0) have the following values:
%(r,r0) = 1
4πrr0 (f1f10 +f2f20+ 3f3f30cosγ), σ(r,r0) = 1
4πrr0f4f40 Yl(ϑ, ϕ)Yl(ϑ0, ϕ0).
(8) Here we assumed for brevity that
fi=fi(r); fi0=fi(r0) and
cosγ= cosϑcosϑ0+ sinϑsinϑ0cos(ϕ−ϕ0). (9) We also suppose that
dω= sinϑ dϑ dϕ; dω0 = sinϑ0 dϑ0 dϕ0. (10) The interaction energyW2 will have the form
W2= Z Z £
2%(r)%(r0)− |%(r,r0)|2+2%(r)σ(r0)−
−%(r,r0)σ(r0,r)] r2dr dω r02 dr0 dω0 pr2+r02−2rr0cosγ.
(11)
On account of
σ(r)σ(r0)− |σ(r,r0)|2= 0
the integrand does not contain terms quadratic with respect toσ, which greatly simplifies the calculations. Evaluating the integral, we will use the known expansion
p 1
r2+r02−2rr0cosγ = X∞
n=0
(2n+ 1)Kn(r, r0)Pn(cosγ), (12)
which assumed that
Kn(r, r0) = 1 2n+ 1
r0n
rn+1 for r0 ≤r, Kn(r, r0) = 1
2n+ 1 rn
r0n+1 for r0≥r,
(13)
as well as the integral property of the spherical functions 2n+1
4π Z
Yl(ϑ0, ϕ0)Pn(cosγ)dω0=δnl Yn(ϑ, ϕ). (14) Here we will write down the evaluation for only one term of expression (11) forW2, namely, for
J = Z Z
%(r,r0)σ(r0,r) r2dr dω r02 dr0 dω0 pr2+r02−2rr0cosγ . Let us write this integral in more detail:
J = 1 (4π)2
Z Z
f4f40 dr dr0ã Z Z
[f1f10 +f2f20 + 3f3f30cosγ]ã
ãYl(ϑ, ϕ)Yl(ϑ0, ϕ0) dω dω0
pr2+r02−2rr0cosγ . Using the known relation between the spherical functions
(2n+1)x Pn(x) = (n+1)Pn+1(x) +n Pn−1(x), one gets
f1f10+f2f20 + 3f3f30cosγ pr2+r02−2rr0cosγ =
X∞
n=0
ẵ
(2n+1)(f1f10 +f2f20)Kn+ +3f3f30[nKn−1+ (n+1)Kn+1]
ắ
ãPn(cosγ). Integration overdω0 anddωgives
J = Z Z
f4f40
ẵ
(f1f10+f2f20)Kl+ +3f3f30
ã l
2l+1Kl−1+ l+1 2l+1Kl+1
áắ dr dr0.
In order to carry out integration overr0, let us suppose Flik(r) =
Z∞
0
fi(r0)fk(r0)Kl(r, r0)dr0; (15) then we will finally get
J= Z∞
0
ẵ
f1f4Fl14+f2f4Fl24+ 3f3f4
ã l
2l+1Fl−134 + l+1 2l+1Fl+134
áắ dr .
The other integrals in expression (11) forW2can be evaluated in the same manner. As the result, one has
W2= Z∞
0
ẵ1
2(2f12+ 2f22+ 6f32+f42)(2F011+ 2F022+ 6F033+F044)−
−f12F011−f22F022−3f32(F033+ 2F233)−1
2f42F044−
−2f1f2F012−6f1f2F113−f1f4Fl14−6f2f3F123−
−f2f4Fl24−3f3f4
ã l
2l+1Fl−134 + l+1 2l+1Fl+134
áắ dr .
(16) In this formula the terms containing products of different functions fi(r) (the last two lines) represent the energy of quantum exchange.1
3 Variational Equations
To derive variational equations, we should vary the energyW =W1+W2
under additional conditions (3) and (4) ensued from orthogonalization and normalization of the wave functions. Constructing the variation of W2, we should take into account that the coefficientsFlik(r) also depend on the varied functions. We can distinguish between two kinds of W2
variation: total variation δW2, when both fi(r) andFlik(r) are varied, and partial variationδ∗W2, when onlyfi(r) is varied. It is not difficult to see that total and partial variations are related as follows:
δW2=δ∗W2. (17)
1According to preliminary calculations these terms are about 3% of the total value W2for the ground state of sodium. (V. Fock)
This remark simplifies constructing of the equations because a partial variation is evaluated easier than the total one.
The variational equations have the form:
−d2f1
dr2 + 2 à
−11
r +F011+ 2F022+ 6F033+F044
ả f1−
−2F021f2−6F131f3−Fl41f4=λ11f1+λ21f2+λ41δl0f4,
(18)
−d2f2
dr2 + 2 à
−11
r + 2F011+F022+ 6F033+F044
ả f2−
−2F012f1−6F132f3−Fl42f4=λ12f1+λ22f2+λ42δl0f4,
(19)
−3d2f3
dr2 + 6 à1
r2 −11
r + 2F011+ 2F022+ 5F033−2F233+F044
ả f3
−6F113f1−6F123f2−3 àl+1
2l+1Fl+134 + l 2l+1Fl−134
ả f4=
=λ33f3+λ43δl1f4,
(20)
−1 2
d2f4
dr2 +
ãl(l+1) 2r2 −11
r + 2F011+ 2F022+ 6F033
á f4−
−Fl14f1−Fl24f2−3
ã l+1
2l+1Fl+134 + l 2l+1Fl−134
á f3=
=λ14δl0f1+λ24δl0f2+λ34δl1f3+λ44f4.
(21)
The last of these equations is the wave equation for a valence electron, and the first three are the ones for inner electrons. Constantsλik=λki
are nothing else but the Lagrangian factors. Off-diagonal terms (for example, the terms with f1, f2, f3 in the last equation) describe the influence of quantum exchange. Equations (18), (19), (20) are invariant with respect to substitutionf1, f2, f3, f4, keeping the quadratic form 2f12+ 2f22+ 6f32+f42 invariant. We can assume this substitution to be chosen so that
λik=λiδik.
4 Properties of the Coefficients of the Equations
The coefficients Flik(r) of our system of equations defined by formula (15) satisfy the differential equation
r2d2Flik
dr2 + 2rdFlik
dr −l(l+1)Flik=−fi(r)fk(r). (22) The functionFlik(r) can be determined as a solution of this equation, which remains finite atr = 0 and tends to zero at infinity. The value Kl(r, r0) is the Green’s function of the self-conjugate differential operator in the left-hand side of (22). Numerical integration of the differential equation (22) by the Adams–St¨ormer method gives a convenient way for calculating the functionsFlik(r) whenfi(r) are known.
Let us write down expression (15) forFlik(r) in more detail:
Flik(r) = 1 2l+1
1 rl+1
Zr
0
fi(r0)fk(r0)r0ldr0+rl Z∞
r
fi(r0)fk(r0) dr0 r0l+1
. (23) As we treat only those functions fi(r), which belong to a discrete spectrum and rapidly decrease at infinity, we can transform this expres- sion in the following manner:
Flik(r) = Clik
rl+1 −Rikl (r), (24) where it is supposed that
Clik= 1 2l+1
Z∞
0
fi(r)fk(r)rldr (25) and
Rikl (r) = 1 2l+1
Z∞
r
à r0l rl+1 − rl
r0l+1
ả
fi(r0)fk(r0)dr0. (26) This formula allows one to obtain an approximate expression for Flik(r) at large values of r. Let us suppose that for sufficiently large values ofr (in any case, larger than the largest root offi(r)) the func- tionfi(r) approximately equals2
fi(r) =Mi rαi e−βr
ã 1 +O
à1 r
ảá
, (27)
2On account of the fact that the functions fi(r) are bound by the system of equations, they have the equal coefficientsβin indices. (V. Fock)
where the symbolO à1
r
ả
denotes a value of the order of 1
r. In this case formula (26) gives the following approximate expression forRikl (r):
Rikl (r) =MiMk
4β2 rαi+αk−2 e−2βr
ã 1 +O
à1 r
ảá
. (28)
We see that this expression is notably small as compared with the first term of formula (24) and does not depend on the subscriptlin the approximation considered.
Thus, formula (24) can be treated as the asymptotic expression for Flik(r), such that the first term gives an approximate value of the func- tion, andRikl (r) does the remainder. Wheni=kandl= 0, the constant valueC0kkis equal to unity due to the normalization offk(r), so that the functions F0kk(r) are asymptotically equal to 1r, as should be expected, because they are the potential of a unit charge, the density of which decreases rather quickly with the separation from the origin.
5 Calculation of Terms
Numerical solution of the system of equations (18)–(21) can be carried out by means of consequent approximations. After getting the functions fi(r), as well as Flik(r), the energy of an atom W = W1+W2 can be found by means of formulae (6) and (16). Herewith, in order to control the calculations it is possible to use the following relation:
W1+1
2W2=λ11+λ22+λ33+λ44. (29) Besides, the calculations can be controlled by means of the virial theorem, according to which the double kinetic energy must be equal to the absolute value of the potential energy (including the energy of the electron interaction). As was shown by the author,3 this relation takes place not only for the exact solution of the Schr¨odinger equation, but also for an approximate solution obtained by the method described in the present article.
It is necessary to take into account that the total energy W of an atom does not coincide with the value of the term; the term is equal to the difference between the energies of an atom in the present state and
3V.A. Fock,Comment on the Virial Relation, JRPKhO62, N4, 379, 1930. (See [30-1] in this book. (Editors))
in the ionization state. In order to get the value of the energy in the ionization state, it is necessary to solve a new system of equations that is derived from the present one, if all functions having the symbol 4 are assumed to equal zero. However, the first-hand calculation of the term as the energy difference is disadvantageous in the sense that the term is obtained as a small difference between two large quantities. In view of this, it is more expedient to do as follows. Let us denote the solutions of the equation system for an ionized atom byfi0(r) (i= 1,2,3) and for an atom in the present state byfi(r), and construct the differences (for the first three functions)
δfi=fi(r)−fi0(r) (i= 1,2,3). (30) For these differences, it is possible to develop a system of equations that allows one to calculate them directly (i.e., without knowingfi(r)).
Iffi(r), as well asf4(r), are known with sufficient accuracy, it is possible to get the value of the energy difference, i.e., the value of a term, also with the same accuracy.
6 Intensities
Finally we need to obtain the formulae for intensities. For the general case, these formulae have been derived in our first article (formulae (99) and (100)).4 In them we will make the simplifications that follow from the assumption of a spherical symmetry.
Let us calculate a matrix element for the coordinatez=rcosϑ. The matrix entries forx=rsinϑcosϕand fory=rsinϑsinϕcan be written by analogy.
For convenience, we will present formula (100) of the previous article replacingfik byzik in it and assumingp= 5,q= 6:
hE|z|E0i =
¯¯
¯¯
¯¯
¯¯
a11 . . . a15
. . . . . . . . a51 . . . a55
¯¯
¯¯
¯¯
¯¯
ã X6
k=1
¯¯
¯¯
¯¯
¯¯
¯¯
a11 . . . a16
. . . . zk1 . . . zk6
. . . . a61 . . . a66
¯¯
¯¯
¯¯
¯¯
¯¯ +
4See [30-2] in this book. (Editors)
+
¯¯
¯¯
¯¯
¯¯
a11 . . . a16
. . . . . . . . a61 . . . a66
¯¯
¯¯
¯¯
¯¯
ã X5
k=1
¯¯
¯¯
¯¯
¯¯
¯¯
a11 . . . a15 . . . . zk1 . . . zk5
. . . . a51 . . . a55
¯¯
¯¯
¯¯
¯¯
¯¯
. (31)
The values aik and zik = fik are written in formula (99) of the previous article. They are equal to
aik= Z
ψi(r, E)ψk(r, E0)dτ; zik= Z
ψi(r, E)z ψk(r, E0)dτ . (32) Let us make a table of valuesaikandzikfor the case of spherical sym- metry under consideration. Because of the orthogonality of the spherical functions, many of these values will be equal to zero, and we will get
((aik)) =
a11 a12 0 0 0 a16
a21 a22 0 0 0 a26
0 0 a33 0 0 a36
0 0 0 a44 0 a46
0 0 0 0 a55 a56
a61 a62 a63 a64 a65 a66
,
((zik)) =
0 0 z13 0 0 z16
0 0 z23 0 0 z26 z31 z32 0 0 0 z36
0 0 0 0 0 z46
0 0 0 0 0 z56
z61 z62 z63 z64 z65 z66
. (33)
It is not difficult to see that all determinants in the second sum of formula (31) are equal to zero. In order to calculate the first sum, as well as the factor in front of it, we will introduce the values
βik= Z∞
0
fi(r, E)fk(r, E0)dr; γik= Z∞
0
fi(r, E)r fk(r, E0)dr (34)
and denote bybik andcik the matrix elements
((aik)) =
β11 β12 0 δ0l0β14
β21 β22 0 δ0l0β24
0 0 β33 δ1l0β34
δl0β41 δl0β42 δl1β43 β44
, (35)
((cik)) =
0 0 γ13 γ14
0 0 γ23 γ24
γ31 γ32 γ33 γ34
γ41 γ42 γ43 γ44
. (36)
Then the matrix element hE|z|E0icorresponding to the transition from the levelE to the levelE0 will be equal to
hE|z|E0i=Cã 1 4π
Z Z
Yl Yl0cosϑ sinϑ dϑ dϕ , (37) where
C= (b33)5ã
¯¯
¯¯b11 b12
b21 b22
¯¯
¯¯ã X4
k=1
¯¯
¯¯
¯¯
¯¯
¯¯
b11 . . . b14
. . . . ck1 . . . ck4
. . . . b41 . . . b44
¯¯
¯¯
¯¯
¯¯
¯¯
. (38)
The matrix elements for the coordinatesxandyare expressed abso- lutely analogously. Thus, we will have
hE|x|E0i=Cã 1 4π
Z
Yl Yl0 sinϑ cosϕ dω, hE|y|E0i=Cã 1
4π Z
YlYl0 sinϑ sinϕ dω, hE|z|E0i=Cã 1
4π Z
Yl Yl0 cosϑ dω.
(39)
These expressions have the same form, as in the usual theory where only a valence electron is treated. The selection rule remains valid with- out any changes. Here the distinction is only in the factorC, which has a slightly different meaning than in the usual theory, when it is equal to
C=c44= Z∞
0
f4(r, E)r f4(r, E0)dr . (40)
In our theory the equality C =c44 is only approximate. The exact calculation of the factor C by means of formula (38) has no problems, as many elements in the determinants in this formula are equal to zero.
Translated by A.K. Belyaev
30-4