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A generalized eigenvalue equation for the hydrogen atom
Volume 3, number 6
CHEMICAL PHYSICS LETTERS
A GENERALIZED
EIGENVALUE A. T.AMOS,
Department
of Mathematics,
EQUATION C. LAUGHLIN University
FOR
June 1969
THE
HYDROGEN
ATOM
and G.R.MOODY
of Nottzngham,
Nottmgham,
NC7 2RD,
UK
Received 1 May 1969
The Schrodmger equation for the hydrogen atom is .vrltten as a generalized elgenvalue equation for Lvhichthe spectrum is discrete. and the electric polarlzablllty of the atom 1s obtalned by applying perturbation theory. An extension of the method to many-electron systems 1s suggested.
1. THE GENERALIZED TION
EIGENVALUE
Consider the generalized
eigenvalue
FICl= ~AJ/
where the @ are eigenfunctions der equation
EQUA-
equation (1)
where F and A are Hermitian operators, + the eigenfunctlon and p the eigenvalue. This differs from the usual form of eigenvalue problem m quantum theory because of the operator A on the right hand side. The effect of this is that the eigenfunctlons of (1) do not form an orthonormal set of functions byt, rather, are orthonormal with respect to the weighting operator A, or, equrvalently w&h respect to F, i.e.
Now let us apply perturbation theory to eq. (1) by assuming that F can be expressed as F” f XF’ with X a perturbation parameter. Then expanding e and p in terms of h as
Ii”+;
of the zero-or-
= $AI);
(5)
with eigenvalues $ and $J: is taken as the zeroorder function. Interestingly enough, the form of equations (4) 1s precisely the text book result which is found by solving (1) urlth A F 1. Thus the presence of A is entirely taken care of by the unusual orthogonality property defined m (2) and does not appear exphcltly in solutions of the perturbation series. Notice that if the spectrum of (5) is not discrete then the summation sign in (4) means a sum over the discrete states and a~ integra*ion over continuum states. Even though they represent exphcit solutions of the equations of perturbation theory equations (4) are rarely appiied to atomic and molecular problems because the spectrum of the usual eigenvalue equation of the form Ho#i =
lE&
always contams a continuum, and the continuum integrations are difficult If not impossible in practice. and applying the usual perturbation obtain
methods we 2. THE ELECTRIC POLARIZABILITY HYDROGEN ATOM It has recently been perturbed Schrodinger atom can be written -h elgenvalue equation for discrete [1,2]. That IS
OF THE
recognized that the unequation for the hydrogen the form of a generalized which the spectrum is
CHEMICAL PHYSICS LETTERS
Volume 3. number 6
_=,) 9(r) =EJ/(r)
(49
(‘3)
can be written as $J/ = ~{-*v2+&2]*
(7)
where the eigenvalue E of (6) is replaced new variable IY defined by
by a
E=.-h2
TO Laguerre
3
n = 1,2,.
:-te2
tO(C3).
(15)
Thus the zero-order energy is -$ and the electric polarizability ! (in atomic units), the wellknown results. given m (10) were
(11)
3. POSSIBLE APPIJCATIONS TRON SYSTEMS
Thus we have changed the problem from that of solving (6) directly to that of solvmg (7) and then Imposing condition (9) to fmd the energy from (8), the advantage of this being that If we have a perturbation theory problem we have a discrete spectrum to deal with rather than a mixture of discrete and continuous. The removal of the continuum functions from the problem suggests tbat it should now be possible to use the perturbation eauations (4) hrectly. To examine tlus possib&y and to illustrate the metbod we consider the perturbation of a hydrogen atom in a static electric field of strength c. The equation correspondmg to (1) then reads $+ &r case J/ = n{-fv2+&21* (12) I i sothatX= &andF’=rcos6. Taking@Etobe the ground state function (i.e. with quantum numbers n = 1, Z = m = 0) the expressions for g’ and $ Hlstorlcally we might note that the mochfled elgenvalue equation for hydrogen (eq. (7)) was known to Hyilerzias (see, for example, ref. [3]). However, It. seems to have been forgotten unto the recent work cited in refs. [1] and [Z]. 412
EC-
(16) $. The
. .
(14)
.
used to fmd the polarizabihty of hydrogen in an example given by Pauling and Wilson in their book [4] as an illustration of the perturbation method of Epstein [5]. However, this method is based on the usual eigenvalue equation and is therefore completely different in concept to that used here.
polynomials
and spherical harmonics, respectively correspondmg eigenvalues are
=A
@5 = cY4 + ; c^2 -I-O(C3)
Notice that the functions
+
‘n’zsM =1n((n+Z)!)3I
bz,l,m
g e2+0($). (13) 4w.5 Now to find Q we must substitute into (9) which gives
(9)
where z is the nuclear charge. The complete set of discrete solutions of (7), orthogonahsed according to eq. (2), is
where 1; and Y refer
n=i+-
Expanding (Y as a power series in e and substituting in (14) shows that (Y’ = 1, o’ = 0, (Y” = f so that the energy given by E = -$a2 ~111 be
and (7) must be solved subject to the auxiliary condition
4o!(?z-Z-l)!
nn are easily computed since, as it turns out, there are only two non-zero terms in the sum giving Cl”. We then obtain that, through second order in &, the perturbed eigenvalue, p, is
(8)
P(4 = $
June 1969
TO MANY-ELEC-
The example given m the previous section is, in our opinion, rather neat and maybe deceptively so since clearly in most cases one will not be fortunate enough to have so few non-zero terms in the perturbation sums. Nevertheless it does seem clear that the method can be used for most If not all one-electron problems and in such situations it will always be possible to avoid the drfficulttes involved with continuum _tictions. However, the one-electron perturbation prob;drns have already been solved so that If the method is to have any real utility it will have to be applicable to many-electron systems. Here we shall indmate how it might be applied to a two-electron problem, the extension to larger systems being obvious. Let us consider the helium problem in which we take the zero-order equation to be p (- sv;
-*v;
+$(u2)*0(1,2)
= ($
I
+ $)
(c/“(l, 2) (16)
and treat the l/y12 term as a perturbation. Unfortunately it is not the case, as might perhaps
Volume 3. number 6
CHEMICAL PHYSICS LETTERS
be expected, that the solutions of the zero-order equation are simple products of the functions (10). These product functions do indeed form a complete set of discrete functions but since rhey do not in general satisfy eq. (16) they therefore do not obey the orthogonality relations (2). Consequently the simple form of the perturbation equations (4) is lost. Instead the complete set of solutions of (16) is (a) the &screte functions (~l,3cl,~l,rP1)~~,Z~rn2(~2,~2,~2,92) %Jl,ml where the parameters “1 and CY~satisfy (Y7.20
and (b) the mixed discrete
lX?zi
and contmuum set
~~l,zl,ml(‘Y1,?rltel,~1)5(k,
‘2,Qz,‘P2)
where t 1s the continuum hydrogen-like of energy +I ‘k2 and Q!1 and K satisfy 0;
+FE2 =c!
function
tions m some perturbation problems could lead to tractable expressions for the expansions given by equations (4) since they involve at most one mtegraYon over continuum fuactrcns rather than two wkch is the case when the usual method is used; Even so, since the main pbjecdve of the method presented m this paper is to avoid any such integration at all, it is obvious that if we are to do this an alternative zero-order equation must be used. While there &mayvery well be many ways to choose the zero-order equation so tkt it has a discrete set of solutions, one which we have found very useful IS
w’hich has m fact a &screte set of solutions since the equation 1s separable in terms of six-dimensional spherical coordinates [6]. FWI details of these zero-order solutions with an application to the helium problem will be published m due course.
2
so that the possible range of values for k 1s restricted, and (c) the reversed set corresponding to (b), 1-e 5 0%rl,e 1, @l)~~&m2(~2,
&me 1969
r2,Q2, 92) -
Notice that the set of functions represented by (a), (b) and (c) is certinly rather strange. There 1s a curious coupling between the two oneelectron functions 111(a) smce “1 depends on “2 and “2 on nl and in add&ion there are no functions in the set corresponding to t_le product of two continuum flmctlons. This docc suggest that the use of solutions to (16) as zero-order func-
REFEFUZNCES Mary Walmsley and C A.Coulson, Proc. Cambridge Phil. Sot. 62 (1966) 729, Mary Walmsley. Proc. Cambridge Phd. Sot. 63 (1967) 451. [21 . _ G.G.Hall, Chem. Phys. Letters 1 (1967) 495; G.G. Hall. J. Hyslop -&d D. Rees, to be pubhshed. [3] E.A.HylleIaas, 2. Physik 48 (1928) 469. [4] L. Paulmg and E.B. W&on. Introduction to quanturn mechanics (McGraw-Hill, New York. 1935) [l]
[5] ghi7&stem Phys Rev 28 U9%) 695 [S] P.M. Morse ‘and H.Feshbach: blethods-of theoretic& physics (McGraw-H111. New York. 1953) Ch. 12.
413
CHEMICAL PHYSICS LETTERS
A GENERALIZED
EIGENVALUE A. T.AMOS,
Department
of Mathematics,
EQUATION C. LAUGHLIN University
FOR
June 1969
THE
HYDROGEN
ATOM
and G.R.MOODY
of Nottzngham,
Nottmgham,
NC7 2RD,
UK
Received 1 May 1969
The Schrodmger equation for the hydrogen atom is .vrltten as a generalized elgenvalue equation for Lvhichthe spectrum is discrete. and the electric polarlzablllty of the atom 1s obtalned by applying perturbation theory. An extension of the method to many-electron systems 1s suggested.
1. THE GENERALIZED TION
EIGENVALUE
Consider the generalized
eigenvalue
FICl= ~AJ/
where the @ are eigenfunctions der equation
EQUA-
equation (1)
where F and A are Hermitian operators, + the eigenfunctlon and p the eigenvalue. This differs from the usual form of eigenvalue problem m quantum theory because of the operator A on the right hand side. The effect of this is that the eigenfunctlons of (1) do not form an orthonormal set of functions byt, rather, are orthonormal with respect to the weighting operator A, or, equrvalently w&h respect to F, i.e.
Now let us apply perturbation theory to eq. (1) by assuming that F can be expressed as F” f XF’ with X a perturbation parameter. Then expanding e and p in terms of h as
Ii”+;
of the zero-or-
= $AI);
(5)
with eigenvalues $ and $J: is taken as the zeroorder function. Interestingly enough, the form of equations (4) 1s precisely the text book result which is found by solving (1) urlth A F 1. Thus the presence of A is entirely taken care of by the unusual orthogonality property defined m (2) and does not appear exphcltly in solutions of the perturbation series. Notice that if the spectrum of (5) is not discrete then the summation sign in (4) means a sum over the discrete states and a~ integra*ion over continuum states. Even though they represent exphcit solutions of the equations of perturbation theory equations (4) are rarely appiied to atomic and molecular problems because the spectrum of the usual eigenvalue equation of the form Ho#i =
lE&
always contams a continuum, and the continuum integrations are difficult If not impossible in practice. and applying the usual perturbation obtain
methods we 2. THE ELECTRIC POLARIZABILITY HYDROGEN ATOM It has recently been perturbed Schrodinger atom can be written -h elgenvalue equation for discrete [1,2]. That IS
OF THE
recognized that the unequation for the hydrogen the form of a generalized which the spectrum is
CHEMICAL PHYSICS LETTERS
Volume 3. number 6
_=,) 9(r) =EJ/(r)
(49
(‘3)
can be written as $J/ = ~{-*v2+&2]*
(7)
where the eigenvalue E of (6) is replaced new variable IY defined by
by a
E=.-h2
TO Laguerre
3
n = 1,2,.
:-te2
tO(C3).
(15)
Thus the zero-order energy is -$ and the electric polarizability ! (in atomic units), the wellknown results. given m (10) were
(11)
3. POSSIBLE APPIJCATIONS TRON SYSTEMS
Thus we have changed the problem from that of solving (6) directly to that of solvmg (7) and then Imposing condition (9) to fmd the energy from (8), the advantage of this being that If we have a perturbation theory problem we have a discrete spectrum to deal with rather than a mixture of discrete and continuous. The removal of the continuum functions from the problem suggests tbat it should now be possible to use the perturbation eauations (4) hrectly. To examine tlus possib&y and to illustrate the metbod we consider the perturbation of a hydrogen atom in a static electric field of strength c. The equation correspondmg to (1) then reads $+ &r case J/ = n{-fv2+&21* (12) I i sothatX= &andF’=rcos6. Taking@Etobe the ground state function (i.e. with quantum numbers n = 1, Z = m = 0) the expressions for g’ and $ Hlstorlcally we might note that the mochfled elgenvalue equation for hydrogen (eq. (7)) was known to Hyilerzias (see, for example, ref. [3]). However, It. seems to have been forgotten unto the recent work cited in refs. [1] and [Z]. 412
EC-
(16) $. The
. .
(14)
.
used to fmd the polarizabihty of hydrogen in an example given by Pauling and Wilson in their book [4] as an illustration of the perturbation method of Epstein [5]. However, this method is based on the usual eigenvalue equation and is therefore completely different in concept to that used here.
polynomials
and spherical harmonics, respectively correspondmg eigenvalues are
=A
@5 = cY4 + ; c^2 -I-O(C3)
Notice that the functions
+
‘n’zsM =1n((n+Z)!)3I
bz,l,m
g e2+0($). (13) 4w.5 Now to find Q we must substitute into (9) which gives
(9)
where z is the nuclear charge. The complete set of discrete solutions of (7), orthogonahsed according to eq. (2), is
where 1; and Y refer
n=i+-
Expanding (Y as a power series in e and substituting in (14) shows that (Y’ = 1, o’ = 0, (Y” = f so that the energy given by E = -$a2 ~111 be
and (7) must be solved subject to the auxiliary condition
4o!(?z-Z-l)!
nn are easily computed since, as it turns out, there are only two non-zero terms in the sum giving Cl”. We then obtain that, through second order in &, the perturbed eigenvalue, p, is
(8)
P(4 = $
June 1969
TO MANY-ELEC-
The example given m the previous section is, in our opinion, rather neat and maybe deceptively so since clearly in most cases one will not be fortunate enough to have so few non-zero terms in the perturbation sums. Nevertheless it does seem clear that the method can be used for most If not all one-electron problems and in such situations it will always be possible to avoid the drfficulttes involved with continuum _tictions. However, the one-electron perturbation prob;drns have already been solved so that If the method is to have any real utility it will have to be applicable to many-electron systems. Here we shall indmate how it might be applied to a two-electron problem, the extension to larger systems being obvious. Let us consider the helium problem in which we take the zero-order equation to be p (- sv;
-*v;
+$(u2)*0(1,2)
= ($
I
+ $)
(c/“(l, 2) (16)
and treat the l/y12 term as a perturbation. Unfortunately it is not the case, as might perhaps
Volume 3. number 6
CHEMICAL PHYSICS LETTERS
be expected, that the solutions of the zero-order equation are simple products of the functions (10). These product functions do indeed form a complete set of discrete functions but since rhey do not in general satisfy eq. (16) they therefore do not obey the orthogonality relations (2). Consequently the simple form of the perturbation equations (4) is lost. Instead the complete set of solutions of (16) is (a) the &screte functions (~l,3cl,~l,rP1)~~,Z~rn2(~2,~2,~2,92) %Jl,ml where the parameters “1 and CY~satisfy (Y7.20
and (b) the mixed discrete
lX?zi
and contmuum set
~~l,zl,ml(‘Y1,?rltel,~1)5(k,
‘2,Qz,‘P2)
where t 1s the continuum hydrogen-like of energy +I ‘k2 and Q!1 and K satisfy 0;
+FE2 =c!
function
tions m some perturbation problems could lead to tractable expressions for the expansions given by equations (4) since they involve at most one mtegraYon over continuum fuactrcns rather than two wkch is the case when the usual method is used; Even so, since the main pbjecdve of the method presented m this paper is to avoid any such integration at all, it is obvious that if we are to do this an alternative zero-order equation must be used. While there &mayvery well be many ways to choose the zero-order equation so tkt it has a discrete set of solutions, one which we have found very useful IS
w’hich has m fact a &screte set of solutions since the equation 1s separable in terms of six-dimensional spherical coordinates [6]. FWI details of these zero-order solutions with an application to the helium problem will be published m due course.
2
so that the possible range of values for k 1s restricted, and (c) the reversed set corresponding to (b), 1-e 5 0%rl,e 1, @l)~~&m2(~2,
&me 1969
r2,Q2, 92) -
Notice that the set of functions represented by (a), (b) and (c) is certinly rather strange. There 1s a curious coupling between the two oneelectron functions 111(a) smce “1 depends on “2 and “2 on nl and in add&ion there are no functions in the set corresponding to t_le product of two continuum flmctlons. This docc suggest that the use of solutions to (16) as zero-order func-
REFEFUZNCES Mary Walmsley and C A.Coulson, Proc. Cambridge Phil. Sot. 62 (1966) 729, Mary Walmsley. Proc. Cambridge Phd. Sot. 63 (1967) 451. [21 . _ G.G.Hall, Chem. Phys. Letters 1 (1967) 495; G.G. Hall. J. Hyslop -&d D. Rees, to be pubhshed. [3] E.A.HylleIaas, 2. Physik 48 (1928) 469. [4] L. Paulmg and E.B. W&on. Introduction to quanturn mechanics (McGraw-Hill, New York. 1935) [l]
[5] ghi7&stem Phys Rev 28 U9%) 695 [S] P.M. Morse ‘and H.Feshbach: blethods-of theoretic& physics (McGraw-H111. New York. 1953) Ch. 12.
413