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Studies in perturbation theory
IOLRNAI,
OF NOI,ECL’LAR
14, 131-144 (1(364)
SPECTROSCOPY
Studies
in Perturbation
Theory
Part VIII. Separation of Dirac Equation and Study of the Spin-Orbit Coupling and Fermi Contact Terms* PER-OLOV Liiwrm
I.
INTRODUCTION
In the treatment of atomic, molecular, and solid-state theory in the nonrelativistic approximation, it is often desirable to study also the relativistic corrections including the variation of the mass with velocity, spin-orbit coupling, Fermi contact interactions, and similar effects. In order to treat this problem, one has to start fronl the relativistic equation for the system under consideration in a Inore or less colnplete form and to go over to the llonrelativistic approximatioll. This procedure was first carried out for the one-electron case by Pauli (I ) by separating t.hc Dirac equation into two equations involving the large aud small components of the wave functions separately. Since t,hen, the connection bctlveeu the Dirac equation and t,he nonrelativistic approximation has been studied by a large Illullber of authors and, for a survey, we would like to refer to the, handbook art,icle by Bethe and Salpetcr (2). =1 very elegant method for deriving the nonrelativistic form by means of contact transformations has been developed by Foldy and Wouthuysen (3 j. I!ecent stlldies in the partitionin g technique reported in a series of papers ( d), which in the following will be referred to as PT I-T-II, indicate that this met.hod is pcbrhaps more powrful than previously expected. In this paper, we will reconsider the separation of the Dirac equation for a single-particle system floul this point of view. II.
F‘UNI~A:\1EiYTAL 1SQI~AT10NS
Let us start out from the Dirac aud charge e in an electromagnetic
equation for a single particle of rest-nlass VI field having the field strengths E and H and
* Spmsored in part by the King C+ust:rf 1’1 Adolf’s iO-Years Fund fw Swedish Cultuw. Knut and Alice Wallenberg’s Foundation and in part by the Aeronautical Research Lab~lrtrtar>-, OAR, th?nugh the European OKicr. derospace ICesearch, I-nited States Air Force. 131
132
LiiWDIN
the associated potentials A and 4: (-TO +
akTk
+
md)#
=
0.
(1)
In this equation (Ykand /3 are four-dimensional matrices of the form
where the submatrices are two-dimensional. For the kinematic momentum gp , one has the relation rP = -ifia/dx’ - (e/c)& , which gives rk
for the first three components. 4. = -4, and further ~0
=
=
pk
(e/c)Ak ,
-
L
For the fourth component,
-(l/c)(h
-
=
1,2,3
(3)
one has x0 = et,
(4)
e+),
is Schrodinger’s energy operator in the nonrelativistic where c,,, = - (h/2+3/3t quantum mechanics. It is often convenient to eliminate the rest-mass energy by the substitution 9 = exp { -imZt/fi)$, which leads to the relation ao# = exp { -imc’t/fi] the Dirac equation (1) leads to the form (-TO
+
ffkrk
+
mcp
+
(5) (TO -
mc)$
=
mc)$. Substitution into
(6)
0,
which will be our starting equation. For the electromagnetic field we will further assume that the Maxwell equations are valid: div H = 0,
div E = 4ap, curl E = - (l/c)
curlH
(aH/at),
= (4r/c)j
+
(l/c)(aE/at).
(7)
Here p and j represent the charge density and the current density of the electric field, respectively. The field strengths are derived from the potentials by means of the relations E = -grad
4 -
(l/c)(dA/at),
H = curl A,
(8)
and we note that these equations are invariant under a gauge transformation +’ = 4 + (llc)(axlat),
A’ = A -
grad x,
(9)
where the scalar x is the arbitrary gauge function. The potentials enter the Dirac equation (1) and for the wave function, the gauge transformation takes the form
STUDIES IN PERTURBATION
I$ =
THEORY.
exp (ie&/c)
VIII
.1c/*
1x:<
(gal
Front this one derives easily the relations P~‘#’ = exp (iefix/c) (r,&) for k = 0, I, 2, 3. Introducing the Lorentz condition div A + &$/CC% = 0, one finds that’ the potential:: satisfy the wave equations CIA = -(4a/c)j,
c&G = _4”P,
(10)
where 0 = V” - d2/c2dt2is the four-dinlensional wave operator. The standard solutions to Eq. (10) are, of course, given by the so-called retarded potentials. We note finally the existence of the conlrnutation relation x X x = i(efi/c)H
(,I])
for the three-dimensional part of the kinen~atic rnonlentunl which follows directly front the definitions and Maxwell’s equations. In the following it is often convenient to use the “Bohr magneton” pLR= efij2mc
(lla)
for a particle with charge e and rest-nlass m. There is also the well-known identity (d.a)(d.b) hfter start.
= a.b + id.(a
this brief survey of the fundanlental III. SEPARATION
X b).
equations,
(12) we are now ready to
OF 1)IRAC EQUATION
The Dirac wave function $ contains four colnponents, and WC will now writ,c this quantity in the fornl
[I
4=
K cp ’
(13)
where K and ‘p are containing two components each. Using the explicit nlatrices (2), we can now write the Dirac equation (6) in the forin
1 [I
---a0 + 2111c; d.lc a3.n
[ or more explicitly
;
K
-TO
=
0,
(0
(l-1)
: (-TO
+
2mC)K
+
d’zp
=
0,
=
0.
(15) d.ZK
-
Po(p
Solving K froln the first equation, we obtain t; = -(2,mc Suhstit.ution
-
7ro)-‘d.qo.
(16)
of this expression into the second of Eq. (15) gives -(d.x)(2mc
-
7ro)-‘(d.~)cp
-
nw = 0,
(17)
134
T,i)WDIN
and the partitioning is concluded. The quantities v and Kare called the large and small components, respectively, of the Dirac wave function 4; see Eq. ( 16). The relations (16) and (17) replace together completely the Dirac equation (6), and Eq. (17) gives us the differential equation which is characteristic for the large components cp alone. This equation is hence convenient for a study of the connection between the Dirac equation and the so-called nonrelativistic approximation. In the following, however, we will be careful to proceed in an exact manner and try to avoid any unnecessary approximations. For the discussion it is convenient to introduce the operator k,, = (1 -
7ro/2mc)-’ = [l + (cop -
e9)/2mcZ]-‘,
(18)
where we have defined the Schrodinger energy operator cop by means of the relation (4). In classical physics, the quantity (cop - e+) represents the kinetic energy mu’/2 of a particle and, in the nonrelativistic limit v << c, it is evident that the quantity 1~must have a value very close to 1. For most purposes, it seems convenient to expand the operator (18) into a power series, but this method has the disadvantage that it breaks down in regions where the electrostatic potential 4 becomes infinite. In such a region, the power series is actually divergent, and one has to be very careful to obtain meaningful results. In the following, we will hence try to avoid this approach. Using the definition ( 18), we can now rewrite the basic equations ( 16) and (17) in the form K = - (1/2mc)k(d.Z)50; ]%P -
ed -
(19)
(1/2m)(d.z)k(d.x)]9
= 0,
(20)
and these two equations will now be analyzed in some detail. By means of the identity (12), with the operator lc in the middle between a and b, we can rewrite Eq. (20) in the form [EOP- e+ -
(1/2m)Acx
-
(i/2m)d.
(x X rC7c)]q= 0.
(21)
In the regions where eOp- e+ << 2mc2 the third term in the bracket corresponds apparently to the kinetic energy operator, whereas the last term according to ( 11) corresponds to the operator. (i/2m)d.(z
X Z) = - (efi/2mc)d.H
= --ped.H,
(22)
which gives the coupling between the spin of the particle and the outer magnetic field; the quantity pB is the Bohr magneton ( lla) It seems reasonable to expect that, for /c = 1, Eq. (21) would represent the nonrelativistic form of the wave equation for a single particle, and that the operator Icwould contain all the essential relativistic corrections. It is illustrative to consider the operator Ic for a stationary state of energy E’, in which case the operator simply becomes a multiplicative function, In a similar
STIJDIES
FIG.
1.
IN PERTURBATIOS
THEORY. VIII
135
The functions k(r) and k’(r) according to Blinder (5)
treatment, involving only stationary states, Blinder (5) has specially considered the case that the electrostatic potential 4 comes from an atomic point nucleus having the charge -Ze, so that e+ = -Z?/r, where Z is the atomic number and r is the distance to the nucleus. Introducing the notation rO = ZeZ/2mc2 = z. 1.4089 x lo-l3 cm, Rlinder
obtains,
instead
of the operator
(18),
(33)
the function
kc?,) = (1 + E/2mc’ + ‘*&)-l,
(24)
which is plotted in Fig. 1. Blinder points out that ~0is essentially of the order of magnitude of the diameter of a light atonlic nucleus, and that one can expect effects of the Is-dependence essentially in the neighborhood of the nucleus. Blinder uses the function X:(r) and its derivative in a study of the spin-orbit coupling and the Fermi contact term, and we will later rctum to these questions. In ou1 case, however, the quantity 1; is a more conlplicat,ed operator which contains the Schrtidinger energy operator cop = %J/c%. In order to investigate relation (21) in greater detail, it is necessary for us to bc able to move the operator /s at our convenience in the expressions, and this inlplies that we have to de&> its conm~utation relations. IV. COMMUTATION
RELATIONS
In order to derive the commutation relations for the operator k, we will start by studying the commutation relation for the operator I;-’ = 1 -l- ( E,~,- e+),&m’L = lao/2mc. If P is an arbitrary operator, we obtain the relation K’F
-
Fk-l = (1/2mc”)[~,,
-
e+, F],
(25)
136
LOWDIN
where we have used the bracket notation [A, B] = .#lB - BA. llultiplying tion (25) to the left and to the right by the operator k, we get kF = (1/2mc2)lc[e,,i -
Pk -
e+, F]k,
rela(262
which is the general commutation relation desired. For the special case F = X, one gets particularly 1% -
e+, P] = iefiE,
(27)
which relations may also be written in the form [x0 , x] = -i(ec/c)E. This conlmutation relation is completely analogous to ( 11)) and both are special cases of the general commutation relation T~?T~- r,r,, = i(efi/c)F,, , where FGy is the electromagnetic field tensor which contains the quantities E and H. Combining (26) and (27), we hence obtain the formula zlz -
kz = i(efi/2mc2)kEk
= i(~~/c)kEk,
(28)
by means of which we can move the operator k to the left and to the right in the expressions, as we desire. As an example we will consider the kinetic energy operator in (al), which is defined by the expression T = (1/2m)&cz. Using the commutation
(29)
(28), we get the two alternative forms
T = (l/2m)lcz2 (1/2m)z%
=
+ i(pB/2mc)kE.kz -
(301)
i(pB/2mc)zk.Ek.
(3%)
Since the operator (29) is self-adjoint, the same is true for the two expressions (30). Sometimes it is more convenient to use a third expression which is the arithmetic mean of the two forms (30)) namely : T = (1/4m)(k~‘+
‘x2/c) -
i(&4mc)(z.(IcEk)
-
(kEk).z].
(31)
The form of this expression is evidently self-adjoint and, by means of (28), one can easily transform it to the form 7’ = (1/4m)(k?e’
+ n2k) -
(H&/4mc)k(div
E)I; + (~B2/2mc2)kEkEk,
(32)
which is particularly convenient for the evaluation of expectation values of the operator T. As a second example we will consider the spin term in (21) represented by the operator &pin = (i/2m]d.(z Using the connnutation
X htj.
wj
relation (28), we get again two alternative forms for
STUDIES
I&- PERTURBATION
THEOIIT.
VIII
1317
the vectorial product ‘x X kn, namely: x X Xx = i( eli/c){kH + kE X kx/2mc}
= i(efi/c)
(HI; - zk X Ek/2mcJ.
(84)
Both forms arc, of course, self-adjoint, and it depends on the circumstances which form is most convenient in the applications. For thr: spin tern1 wc obtain hctlce the expression Xspin = -/.~gd. (kH •l- kE X l
-/&.(HX:
-
x/i X Ek/2mc).
(35)
From the theory of special relativit)y we know that, if a charged particle moves with the velocity v = z/m, it is subject to a magnetic field having the strength Hez = H + E X v/c = H + E X zlmc. It is remarkable that, in our relat’ivistic consideration, the magnetic field H in formula (22) is now simply replaced by the operator kH + kE X kx/2mc, where the factor f$ in the last term is the famous Thomas factor (6). In addition t.here is the operator I<,which accounts for the higher relativistic effects. 1’. 1TF:RATION
RELATIONS
;1 characteristic feature of the partition technique is that the energy operator G,~ or the corresponding eigenvalue occurs implicitly in the contracted Hamil tonian, so explicit forms or numerical values have to he found by iteration procedures. In the case of a stat’ionary state, the problem of the numerical evaluation of the energy eigenvalue has been considered in some details in a series of previous papers ( 4). In this section we will study the more general case when the energy is represented by an operator. In order to simplify the study of the forms of the iteratioti problem, it is convenient to introduce the scalar quantity
a = d.x/2mc. Instead of the fundamental
relations
( 36 j
(19) and (20),
we obtain
K = -kacp; (%p -
e$)(p = 2mc2akacp.
According to (US), one has (c,~ - e+)/2mc2 = k-l - 1. Multiplying (38) to the left by the operator k, one thus obtains the relation kcp = (1 -
kaka
jp,
(37) (38) relatioll
Wj
which forms the starting point for the iteration procedure. It should be observed that previously one has, in many cases, expanded the operator k in a power series in the quantity (cop - e+)/2mc2, but, since this often leads to singularities in the wave equation of the order l/r3 and higher, the expressions obtained have only had limited validity. We note that the goal for the calculations is to limit
138
LijWDIN
the implicit dependence on E,,,in the right-hand member of the expression (38). In order to iterate in formula (39), it is necessary to be able to move the operator k to the right or to the left in the expressions involved. By means of (28) and (36), we can now derive the commutation relation ak -
ka = i(~CLB/2m~2)X;d.Ek.
(40)
In order to proceed, it is convenient to distinguish between two cases: (1) E = 0. Let us first consider the case when there is no electric field present, so that E = 0. In this case the operators a and 17commute, so that aX: = lia, and formula (39) reduces to the form kp = (1 - a21i2)(o.By immediate use of this formula, one obtains the infinite series kp = ‘p -
a2k2;p= [l -
a2 + 2a4 -
5a6 + - . . .]p.
(41)
In each step in the iteration procedure, there is a remainder term, and we have here assumed that this remainder term goes to zero. In turns out that the operator k working on $ is equivalent with a power series k = k(a), which satisfies the relation (k - 1 + a2k2)(a = 0. Apparently there are two solutions k(a) = l&l + 4a” - 1]/2 a2 and, assuming that the positrons states are not available, we will here concentrate our interest on the root associated with the + sign, i.e., kp = {(dl
+ 4a’ -
1)/2a”}p.
(42)
It is easily checked that expression (42) leads to the power series (41), and by means of the explicit expression, one can now more easily discuss the remainder term. Combining (38) and (42), we obtain finally the relation ___+ Aa? - l)a, (43) (COP- e+)co = mc”(dl which would be the exact analog to the Schrodinger equation in the relativistic case. For a”, we have according to (12) the explicit expression a2
=
Cd.%)(d-n) = x2+ id.(x.n) 4V22C”
4m2c2
(44)
This gives (%P -
e$)cp = mc”{~l
+ x2/wiY -
(efi/m2c3)d.H
-
11%
(45)
which is the expression desired, in the case E = 0. We note that we have here not made any explicit assumption about the character of the vector potential A or the magnetic field H. l?ormula (45) has previously been derived by Huff (7) in the case of constant magnetic field H. A similar result has also been ob-
STtTDIES
IN l’ERTI~IIBATTO?;
THEORY.
13!1
\‘I11
tained by Johnson and Lipplnan (8), who have treated also the case of an inhornogeneous lnagnetic field. Expanding the right-hand member of (45) in a power series of a2 and including terms up to the order c-‘, we obtain (COP-
ed)(p =
[x’/Zm
-
(efi/2mc)d.H
-
~~/8m”c~ +
- .=.)+Q.
(46)
Here the first tern1 in the right-hand menlber represents the ordinary kinetic energy, the second the coupling between the spin and the magnetic field, and the third the relativistic correction associated with the fact that the Inass depends on the velocity, etc. It rnay be of sonle interest to conlpare fornlula (45) with the corresponding expression in the classical theory of relativity in the case of no field. One has the basic relations (4i)
froln which one derives the identity (E + me’)2 E = mc2(~1+x”lm2d”
c2a2 = m2c4and the fornlula -
1)
(48)
which is the classical analog to (45) in the case of no field. This shows that (45) gives the correct dependence of the relativistic mass on the velocity. (2). E # 0. Let us now consider the case when there is an electric field, E # 0. In order to iterate the relation (39) it is now necessary to nlove the operator 1~to the right in the right-hand Inember so that it works directly on the way function (o. This can be achieved by using the cornrnutation relation (40), but it is evident that there is no solution of the sanle sinlplicity as in the previous case. A systeniatic study of this problenl will be carried out in a forthcoming paper from the group. VI. THE
XONRELATIVISTIC
HAMILTONIAN
In the previous sections, we have shown that the original Dirac equation (I) nlay be replaced by two separate equations (37) and (38), which are still exact. Relation (38) is equivalent wit>h (21). Using the first of the forms (35) for the spin terni, we can express the Kave equation for the function cp in the fornl (top -
e$)co =
{ (1/2m)&x
-
~.l&:d.(H -t E X kx/2mc)}cp.
(49)
The first tern1 in the right-hand rnenlber represents the kinetic energy of the electron in the relativistic forIn, and it is easily checked that it gives the correct dependence of the lnass on the velocity. It contains further high-order ternls involving the electric field of the type occurring in (32). We will here concentrate our interest on the second terns involving the spin operat’er HsPin defined by (35). We note that this operator is self-adjoint and gauge-invariant as a whole, but we are now going t#osplit it into several parts:
l-10
LowurN
The first term -p&d.H gives the well-known coupling between the spin alid the magnetic field. The second term -(~B/2mc)1~d~ (E X k[p -
(e/c)A]l
(50)
is gauge-invariant as a whole, but it is usually divided into two terms without this property, namely, the “spin-orbit coupling” Xq3in-orbit
=
-
(/.&2mc)kd.
(E x kp)
(51)
and the so-called “contact interaction” XC&Z& = ( pBe/2mc)kd.
(E X kA).
(52)
The reason for the name will become clear in the following discussion. We observe that all these terms actually occur in Pauli’s nonrelativistic Hamiltonian (1) with the essential difference that we have here retained the operator k. In the following we will discuss the more detailed form of these operators in the case of a central-symmetric field. VII.
DISCUSSION SPIN-ORBIT
OF THE HYPERFINE SPLITTING HAMILTONIAN; COUPLING AND CONTACT INTERACTIONS
Let us consider an electron which moves in the field of a point nucleus atomic number 2 and a nuclear spin U. The electromagnetic field associated with this point nucleus and point dipole is characterized by the potentials 4 = ---e/r,
A = &yI X r/r3,
where y is the nuclear gyromagnetic given by the formula
(53)
factor. The magnetic field strength H is
H = curl A = Cy[-I/r3
+ 3(r/?)(I,r)],
(54)
and we note that it has a singularity of type l/r3 at the nucleus (r = 0). Introducing the electronic spin s = MU one obtains for the interaction between this spin and the magnetic field the expression Xspin-field
=
-p&d.H
= 2&+7c[s~I/r3
-
3(s.r)(I.r)/r5].
(55)
It is interesting to note that, if we put k = 1, this term would necessarily have singularity of type rM3at the origin, which is not permitted in a theory leading to stationary states. The nonrelativistic Hamiltonian can have singularities only up to the order r-‘, and the operator k plays a fundamental role in this connection. It has previously been pointed out that in the case of a central-symmetric electrostatic field, one can write the operator k in the form
mhere i’ ,, = Ze?;‘2mc2 = Z. 1.4089 X lo-‘” cm is indeed a very muall length. This gives t,hc forlliula
which is a very large but still finite quantity. This implies that the operator ( 5.5) has a singularity of the type F’) at the origin. Let us next’ consider the spin-orbit coupling operator, defined by (51) in the case of a central-synli~letric electrostatic field with the potential 4 = ~(,v’J. For t,hc electric field, one obtains E = -grad
d, = - 4’( /‘)r/‘r.,
(58)
and WCcau hence write the spin-orbit coupling iu the for111 ~:s~i”-“rl~it= (~H!21nc)I~2[~‘(1,)/l.]d.(r X p).
(59)
Iutroduciug the orbital angular-nlonlentulll 1 through the definition r X p = hl aud further t’he abbreviation 50~) = &~lc’$‘( /.),/wwr, we obtain Xs~)in_~rl,jt
=
((1,)S.l.
(60)
Very ofteu the coefficient <( 1’) or the radial integral over this coefficient is tollsidercd as a semiempirical parallleter which can be detcmined from experimmts. The forln (60) forms the basis for the theory of spin-orbit coupling in conuection with atomic spectroscopy (9). It should be observed that this form of the spiuorbit’ coupling exists essentially alreadg iu Pauli’s nonrelativistic approximation iI>. It is illustrative to consider this spin-orbit coupling iu the case the atonlic uucleus is replaced by a point charge, so that $ = -eZ,~‘r. Suhstitutiou of this expression int,o (59) leads to the formula Gr,in-or1,it= 2&“k”s.
l/L
(61
j
LZgainit should be noted that the singularity at the origin is of the order F1( s. 1)) i.e., a permitted singularity. The spin-orbit coupling is usually treated in a senliempirical fashion, and it would certainly be highly worthwhile to study this interaction frownfirst principles in greater detail. Let us now consider t’he cont’act term define by (33 j, i.e., (E X /CA). Xcontact= (~~P~‘~vzc)/G~~ Using the elementary vector relation a X (b X c) = b(c.a) obtain inmediately the expression E X A = - (e_Z/lS”)r X (fir1 X r/F,‘) = -(efiZr/r”)(I(r.r)
-
r(r.I)},
(62) -
c(a.b),
WC
(63)
142
LiSWDIN
and the formula x contact= -2
eZy(~B2/c)k2(s~I/r4
-
(r.s)(r.I)/r’J.
(64)
Again, we notice that, since k2/t2 is regular at the origin, the singularity is of the order l/r”. It is possible to give a simplified form of this expression in the case of a stationary state. The operator k is now a function only of the radius r and, from ( IS), we obtain directly k’( r ) = k( e+‘/2 mc2)k = ( e2Z/2mc”) ( k2/r2). Substituting this expression into (64)) one obtains x contact= -2p,-yfik’(r)(s.I/r2
-
(r.s)(r.I)/r4}.
(65)
notice that one has the relation s: k’(r) dr = k( 00 ) k(0) = (1 + Gmc 2>-I M 1, so that k’(r) is essentially a d-function which is different from zero only in the neighborhood of the origin; compare (24) and Fig. 1. We observe further that, in calculating expectation values of the contact interaction (65), one obtains only contribution from s-part of the electron density. In the calculation of such expectation values, one is then only interested in the spherically symmetric part of the contact transformation, This means that the second term in the bracket of (65) may be replaced by the average value --$/3(.sV1)/r4. One has further the relation We
k’(r)
J
-;,dv
=
s0
k’(~)4?r dr = 4~,
(66)
which implies that one can introduce the three-dimensional b-function Jc’(r)/r’ + 4&(r).
(67)
Instead of (65), we obtain hence the expression X contact= ->~16~~s+.16(r)
7
(68)
which is the standard formula found in the literature (10). Under very special assumptions the contact term defined by (62) may hence be replaced by an interaction between the electronic and nuclear spins containing a d-function, which explains the name given to this interaction. We note that both the spinorbit coupling and the contact interaction originally occur in Pauli’s nonrelativistic Hamiltonian (I), but that they can be brought to a form which is correct in the neighborhood of the nucleus, only if one includes the operator k. In conclusion, a few words should be added about the treatment of the kinetic energy term Tp = (&t/2m)cp in the fine structure Hamiltonian. If the magnetic field is weak, the quadratic term in vector potential is usually neglected, and one concentrates the interest on the linear term. Considering the first term in (32), one obtains for the linear contribution in the vector potential -(e/4mc)(k(pA
+ Ap) +
(PA + Ap)k}
= -(e/27nc){lcAp
+ Apk),
(69)
where we have used the fact that div A = 0. Using the explicit form (53) the vector potential, one gets further
for
(70)
Hence one has the following contribution to the hyperfine splitting Hamiltonian from the linear terms in the kinetic energy -~&[kI.l/+~
+ (I.l/r3)k].
(71)
C’onsidcring the expressions ((is), (68), and (71), we thus obtain the following expression for the part of the hyperfine splitting Hamiltonian which depends on the nuclear spin I: X,,,(I)
= -2&yl;{I.(l
-
s)/:? + 3(s.r)(I.r)/?} - f~l6a&Cy(s.I)6(r).
(7”)
We note that the Bohr magneton is here defined by the expression ( lla), where e is the electronic charge with negative sign. This Hamiltonian is here derived under the assumption that the nucleus can be approximated by a point charge and a point dipole, and it is clear that the actual form and extension of thcb nucleus will influence the coupling between the nuclear spin and the magnetic ~~~on~cntsassociated with the electronic motion and spin. VIII.
DISCUSSION
111 this paper we have studied the separation of the Dirac relativistic wave equation for one electron by means of the partitioning technique. In a series of previous papers (4), we have learned that this technique is a powerful tool fol the evaluation of energy eigenvalues of the stationary states by means of a secondorder iteration procedure, which is closely connected with the quantun-nlcchanical variation principle. \Ve have here studied essentially the operatol aspects of the partitioning of the Dirac equation, and it seems hence WJrth while to s;tutly also the numerical aspects in greater detail. In this connection it is convenient to introduce a complete orthonormal basis and to introduce the tnatrix representation of the operators involved. A first partitioning leads to the separation of the Dirac equation into its large and small components, and a seco~~d partitioning takes out a suhspace which contains only a single element of the basis; see kf. .$. Yor certain systems, one bows the nonrelativistic eigenvalucs and eigenhn~ctions with a fair alnount of accuracy, and it seems plausible that this data would form a convenient starting point for the second-order iteration proccdurc connected with partitioning technique. In this way one should be able to obt’ain the relativistic corrections in a rather simple way. Studies of test prohlcn~s of tbis type arc in progress at the Uppsala group.
RECEIVED 1)ecember 7, 1963
144
LiiWDIN REFERENCES
1. W. PAULI, z. Ph@' 43, 601 (1927). 2. H. A. BETHE AND E. E. SALPETER, in “Handbuch der Physik” (S. Fliigge, ed.), Vol. 35, p. 88. Springer-Verlag, Berlin, 1957. 3. L. L. FOLDY AND S. A. WOUTHUYSEN, Phys. Rev. 78,29 (1950); L. L. FOLDY, Phys. Rev. 87, 688 (1952). 4. P. 0. LOWDIN, Studies in Perturbation Theory. I. An Elementary Iteration-Variation Procedure for Solving the Schrodinger Equation, J. Mol. Spectry. 10, 12 (19G3); II. Generalization of the Brillouin-Wigner Formalism. III. Solution of the Schrodinger Equation under a Variation of a Parameter J. Mol. Spectry. 13, 326; IV. Solution of Eigenvalue Problem by Projection Operator Formalism, J. Math. Phys. 3,969 (1962); V. Some Aspects on the Exact Self-Consistent-Field Theory, J. Math. Phys. 3, 1171 (1962); VI. Contraction of Secular Equations, preceding paper [J. Mol. Spectry. 13, 112-118 (1964)l; VII. Localized Perturbation, preceding paper 1.1. Mol. Spectry. 13, 119-130 (1964)]. 5. S. M. BLIK~EK, J. il101. Spectry. 6, 17 (1960). 6. L. H. THOMAS, Nature 107, 514 (1926). 7. L. D. HUFF, Phys. Rev. 38, 501 (1931). 8. M. H. JOHNSON AND B. A. LIPPMAN, Phys. Rev. 76,828 (1949); 77,702 (1950). 9. E. N. CONDON AND G. H. SHORTLEY, “Theory of Atomic Spectra.” Cambridge Univ. Press, London and New York, 1935, particularly pp. 1299131. 10. E. FERMI, Z. Physik 60, 320 (1930). E. FERMI AND E. SEC~I&, Z. Physik 82, 729 (1933). J. HAROREAVES, Proc. Roy. Sot. A124, 568 (1928); A127, 141, 407 (1930). G. BREIT, Phys. Rev. 36, 1447 (1930); 37, 51 (1931). G. BREIT AND F. W. DOERRMAN, Phys. Rev. 36, 1732 (1930). Vol. 24, Part I, p. 385. Springer, Berlin, 1933. H. BETHE, in “Handbuch der Physik,” H. G. B. CASIMIR, “On the Interaction between Atomic Nuclei and Electrons,” p. 36. Teyler’s Tweede Genootshaafs, Haarlem, 1936. R. A. FROSCH AND H. M. FOLEY, Phys. Rev. 88,1337 (1952). A. ABI~A~AM AND M. H. L. PRYCE, Proc. Roy. Sot. A206,135 (1951). R. ARNOWITT, Phys. Rev. 92, 1002 (1953).
Vol. 11,
OF NOI,ECL’LAR
14, 131-144 (1(364)
SPECTROSCOPY
Studies
in Perturbation
Theory
Part VIII. Separation of Dirac Equation and Study of the Spin-Orbit Coupling and Fermi Contact Terms* PER-OLOV Liiwrm
I.
INTRODUCTION
In the treatment of atomic, molecular, and solid-state theory in the nonrelativistic approximation, it is often desirable to study also the relativistic corrections including the variation of the mass with velocity, spin-orbit coupling, Fermi contact interactions, and similar effects. In order to treat this problem, one has to start fronl the relativistic equation for the system under consideration in a Inore or less colnplete form and to go over to the llonrelativistic approximatioll. This procedure was first carried out for the one-electron case by Pauli (I ) by separating t.hc Dirac equation into two equations involving the large aud small components of the wave functions separately. Since t,hen, the connection bctlveeu the Dirac equation and t,he nonrelativistic approximation has been studied by a large Illullber of authors and, for a survey, we would like to refer to the, handbook art,icle by Bethe and Salpetcr (2). =1 very elegant method for deriving the nonrelativistic form by means of contact transformations has been developed by Foldy and Wouthuysen (3 j. I!ecent stlldies in the partitionin g technique reported in a series of papers ( d), which in the following will be referred to as PT I-T-II, indicate that this met.hod is pcbrhaps more powrful than previously expected. In this paper, we will reconsider the separation of the Dirac equation for a single-particle system floul this point of view. II.
F‘UNI~A:\1EiYTAL 1SQI~AT10NS
Let us start out from the Dirac aud charge e in an electromagnetic
equation for a single particle of rest-nlass VI field having the field strengths E and H and
* Spmsored in part by the King C+ust:rf 1’1 Adolf’s iO-Years Fund fw Swedish Cultuw. Knut and Alice Wallenberg’s Foundation and in part by the Aeronautical Research Lab~lrtrtar>-, OAR, th?nugh the European OKicr. derospace ICesearch, I-nited States Air Force. 131
132
LiiWDIN
the associated potentials A and 4: (-TO +
akTk
+
md)#
=
0.
(1)
In this equation (Ykand /3 are four-dimensional matrices of the form
where the submatrices are two-dimensional. For the kinematic momentum gp , one has the relation rP = -ifia/dx’ - (e/c)& , which gives rk
for the first three components. 4. = -4, and further ~0
=
=
pk
(e/c)Ak ,
-
L
For the fourth component,
-(l/c)(h
-
=
1,2,3
(3)
one has x0 = et,
(4)
e+),
is Schrodinger’s energy operator in the nonrelativistic where c,,, = - (h/2+3/3t quantum mechanics. It is often convenient to eliminate the rest-mass energy by the substitution 9 = exp { -imZt/fi)$, which leads to the relation ao# = exp { -imc’t/fi] the Dirac equation (1) leads to the form (-TO
+
ffkrk
+
mcp
+
(5) (TO -
mc)$
=
mc)$. Substitution into
(6)
0,
which will be our starting equation. For the electromagnetic field we will further assume that the Maxwell equations are valid: div H = 0,
div E = 4ap, curl E = - (l/c)
curlH
(aH/at),
= (4r/c)j
+
(l/c)(aE/at).
(7)
Here p and j represent the charge density and the current density of the electric field, respectively. The field strengths are derived from the potentials by means of the relations E = -grad
4 -
(l/c)(dA/at),
H = curl A,
(8)
and we note that these equations are invariant under a gauge transformation +’ = 4 + (llc)(axlat),
A’ = A -
grad x,
(9)
where the scalar x is the arbitrary gauge function. The potentials enter the Dirac equation (1) and for the wave function, the gauge transformation takes the form
STUDIES IN PERTURBATION
I$ =
THEORY.
exp (ie&/c)
VIII
.1c/*
1x:<
(gal
Front this one derives easily the relations P~‘#’ = exp (iefix/c) (r,&) for k = 0, I, 2, 3. Introducing the Lorentz condition div A + &$/CC% = 0, one finds that’ the potential:: satisfy the wave equations CIA = -(4a/c)j,
c&G = _4”P,
(10)
where 0 = V” - d2/c2dt2is the four-dinlensional wave operator. The standard solutions to Eq. (10) are, of course, given by the so-called retarded potentials. We note finally the existence of the conlrnutation relation x X x = i(efi/c)H
(,I])
for the three-dimensional part of the kinen~atic rnonlentunl which follows directly front the definitions and Maxwell’s equations. In the following it is often convenient to use the “Bohr magneton” pLR= efij2mc
(lla)
for a particle with charge e and rest-nlass m. There is also the well-known identity (d.a)(d.b) hfter start.
= a.b + id.(a
this brief survey of the fundanlental III. SEPARATION
X b).
equations,
(12) we are now ready to
OF 1)IRAC EQUATION
The Dirac wave function $ contains four colnponents, and WC will now writ,c this quantity in the fornl
[I
4=
K cp ’
(13)
where K and ‘p are containing two components each. Using the explicit nlatrices (2), we can now write the Dirac equation (6) in the forin
1 [I
---a0 + 2111c; d.lc a3.n
[ or more explicitly
;
K
-TO
=
0,
(0
(l-1)
: (-TO
+
2mC)K
+
d’zp
=
0,
=
0.
(15) d.ZK
-
Po(p
Solving K froln the first equation, we obtain t; = -(2,mc Suhstit.ution
-
7ro)-‘d.qo.
(16)
of this expression into the second of Eq. (15) gives -(d.x)(2mc
-
7ro)-‘(d.~)cp
-
nw = 0,
(17)
134
T,i)WDIN
and the partitioning is concluded. The quantities v and Kare called the large and small components, respectively, of the Dirac wave function 4; see Eq. ( 16). The relations (16) and (17) replace together completely the Dirac equation (6), and Eq. (17) gives us the differential equation which is characteristic for the large components cp alone. This equation is hence convenient for a study of the connection between the Dirac equation and the so-called nonrelativistic approximation. In the following, however, we will be careful to proceed in an exact manner and try to avoid any unnecessary approximations. For the discussion it is convenient to introduce the operator k,, = (1 -
7ro/2mc)-’ = [l + (cop -
e9)/2mcZ]-‘,
(18)
where we have defined the Schrodinger energy operator cop by means of the relation (4). In classical physics, the quantity (cop - e+) represents the kinetic energy mu’/2 of a particle and, in the nonrelativistic limit v << c, it is evident that the quantity 1~must have a value very close to 1. For most purposes, it seems convenient to expand the operator (18) into a power series, but this method has the disadvantage that it breaks down in regions where the electrostatic potential 4 becomes infinite. In such a region, the power series is actually divergent, and one has to be very careful to obtain meaningful results. In the following, we will hence try to avoid this approach. Using the definition ( 18), we can now rewrite the basic equations ( 16) and (17) in the form K = - (1/2mc)k(d.Z)50; ]%P -
ed -
(19)
(1/2m)(d.z)k(d.x)]9
= 0,
(20)
and these two equations will now be analyzed in some detail. By means of the identity (12), with the operator lc in the middle between a and b, we can rewrite Eq. (20) in the form [EOP- e+ -
(1/2m)Acx
-
(i/2m)d.
(x X rC7c)]q= 0.
(21)
In the regions where eOp- e+ << 2mc2 the third term in the bracket corresponds apparently to the kinetic energy operator, whereas the last term according to ( 11) corresponds to the operator. (i/2m)d.(z
X Z) = - (efi/2mc)d.H
= --ped.H,
(22)
which gives the coupling between the spin of the particle and the outer magnetic field; the quantity pB is the Bohr magneton ( lla) It seems reasonable to expect that, for /c = 1, Eq. (21) would represent the nonrelativistic form of the wave equation for a single particle, and that the operator Icwould contain all the essential relativistic corrections. It is illustrative to consider the operator Ic for a stationary state of energy E’, in which case the operator simply becomes a multiplicative function, In a similar
STIJDIES
FIG.
1.
IN PERTURBATIOS
THEORY. VIII
135
The functions k(r) and k’(r) according to Blinder (5)
treatment, involving only stationary states, Blinder (5) has specially considered the case that the electrostatic potential 4 comes from an atomic point nucleus having the charge -Ze, so that e+ = -Z?/r, where Z is the atomic number and r is the distance to the nucleus. Introducing the notation rO = ZeZ/2mc2 = z. 1.4089 x lo-l3 cm, Rlinder
obtains,
instead
of the operator
(18),
(33)
the function
kc?,) = (1 + E/2mc’ + ‘*&)-l,
(24)
which is plotted in Fig. 1. Blinder points out that ~0is essentially of the order of magnitude of the diameter of a light atonlic nucleus, and that one can expect effects of the Is-dependence essentially in the neighborhood of the nucleus. Blinder uses the function X:(r) and its derivative in a study of the spin-orbit coupling and the Fermi contact term, and we will later rctum to these questions. In ou1 case, however, the quantity 1; is a more conlplicat,ed operator which contains the Schrtidinger energy operator cop = %J/c%. In order to investigate relation (21) in greater detail, it is necessary for us to bc able to move the operator /s at our convenience in the expressions, and this inlplies that we have to de&> its conm~utation relations. IV. COMMUTATION
RELATIONS
In order to derive the commutation relations for the operator k, we will start by studying the commutation relation for the operator I;-’ = 1 -l- ( E,~,- e+),&m’L = lao/2mc. If P is an arbitrary operator, we obtain the relation K’F
-
Fk-l = (1/2mc”)[~,,
-
e+, F],
(25)
136
LOWDIN
where we have used the bracket notation [A, B] = .#lB - BA. llultiplying tion (25) to the left and to the right by the operator k, we get kF = (1/2mc2)lc[e,,i -
Pk -
e+, F]k,
rela(262
which is the general commutation relation desired. For the special case F = X, one gets particularly 1% -
e+, P] = iefiE,
(27)
which relations may also be written in the form [x0 , x] = -i(ec/c)E. This conlmutation relation is completely analogous to ( 11)) and both are special cases of the general commutation relation T~?T~- r,r,, = i(efi/c)F,, , where FGy is the electromagnetic field tensor which contains the quantities E and H. Combining (26) and (27), we hence obtain the formula zlz -
kz = i(efi/2mc2)kEk
= i(~~/c)kEk,
(28)
by means of which we can move the operator k to the left and to the right in the expressions, as we desire. As an example we will consider the kinetic energy operator in (al), which is defined by the expression T = (1/2m)&cz. Using the commutation
(29)
(28), we get the two alternative forms
T = (l/2m)lcz2 (1/2m)z%
=
+ i(pB/2mc)kE.kz -
(301)
i(pB/2mc)zk.Ek.
(3%)
Since the operator (29) is self-adjoint, the same is true for the two expressions (30). Sometimes it is more convenient to use a third expression which is the arithmetic mean of the two forms (30)) namely : T = (1/4m)(k~‘+
‘x2/c) -
i(&4mc)(z.(IcEk)
-
(kEk).z].
(31)
The form of this expression is evidently self-adjoint and, by means of (28), one can easily transform it to the form 7’ = (1/4m)(k?e’
+ n2k) -
(H&/4mc)k(div
E)I; + (~B2/2mc2)kEkEk,
(32)
which is particularly convenient for the evaluation of expectation values of the operator T. As a second example we will consider the spin term in (21) represented by the operator &pin = (i/2m]d.(z Using the connnutation
X htj.
wj
relation (28), we get again two alternative forms for
STUDIES
I&- PERTURBATION
THEOIIT.
VIII
1317
the vectorial product ‘x X kn, namely: x X Xx = i( eli/c){kH + kE X kx/2mc}
= i(efi/c)
(HI; - zk X Ek/2mcJ.
(84)
Both forms arc, of course, self-adjoint, and it depends on the circumstances which form is most convenient in the applications. For thr: spin tern1 wc obtain hctlce the expression Xspin = -/.~gd. (kH •l- kE X l
-/&.(HX:
-
x/i X Ek/2mc).
(35)
From the theory of special relativit)y we know that, if a charged particle moves with the velocity v = z/m, it is subject to a magnetic field having the strength Hez = H + E X v/c = H + E X zlmc. It is remarkable that, in our relat’ivistic consideration, the magnetic field H in formula (22) is now simply replaced by the operator kH + kE X kx/2mc, where the factor f$ in the last term is the famous Thomas factor (6). In addition t.here is the operator I<,which accounts for the higher relativistic effects. 1’. 1TF:RATION
RELATIONS
;1 characteristic feature of the partition technique is that the energy operator G,~ or the corresponding eigenvalue occurs implicitly in the contracted Hamil tonian, so explicit forms or numerical values have to he found by iteration procedures. In the case of a stat’ionary state, the problem of the numerical evaluation of the energy eigenvalue has been considered in some details in a series of previous papers ( 4). In this section we will study the more general case when the energy is represented by an operator. In order to simplify the study of the forms of the iteratioti problem, it is convenient to introduce the scalar quantity
a = d.x/2mc. Instead of the fundamental
relations
( 36 j
(19) and (20),
we obtain
K = -kacp; (%p -
e$)(p = 2mc2akacp.
According to (US), one has (c,~ - e+)/2mc2 = k-l - 1. Multiplying (38) to the left by the operator k, one thus obtains the relation kcp = (1 -
kaka
jp,
(37) (38) relatioll
Wj
which forms the starting point for the iteration procedure. It should be observed that previously one has, in many cases, expanded the operator k in a power series in the quantity (cop - e+)/2mc2, but, since this often leads to singularities in the wave equation of the order l/r3 and higher, the expressions obtained have only had limited validity. We note that the goal for the calculations is to limit
138
LijWDIN
the implicit dependence on E,,,in the right-hand member of the expression (38). In order to iterate in formula (39), it is necessary to be able to move the operator k to the right or to the left in the expressions involved. By means of (28) and (36), we can now derive the commutation relation ak -
ka = i(~CLB/2m~2)X;d.Ek.
(40)
In order to proceed, it is convenient to distinguish between two cases: (1) E = 0. Let us first consider the case when there is no electric field present, so that E = 0. In this case the operators a and 17commute, so that aX: = lia, and formula (39) reduces to the form kp = (1 - a21i2)(o.By immediate use of this formula, one obtains the infinite series kp = ‘p -
a2k2;p= [l -
a2 + 2a4 -
5a6 + - . . .]p.
(41)
In each step in the iteration procedure, there is a remainder term, and we have here assumed that this remainder term goes to zero. In turns out that the operator k working on $ is equivalent with a power series k = k(a), which satisfies the relation (k - 1 + a2k2)(a = 0. Apparently there are two solutions k(a) = l&l + 4a” - 1]/2 a2 and, assuming that the positrons states are not available, we will here concentrate our interest on the root associated with the + sign, i.e., kp = {(dl
+ 4a’ -
1)/2a”}p.
(42)
It is easily checked that expression (42) leads to the power series (41), and by means of the explicit expression, one can now more easily discuss the remainder term. Combining (38) and (42), we obtain finally the relation ___+ Aa? - l)a, (43) (COP- e+)co = mc”(dl which would be the exact analog to the Schrodinger equation in the relativistic case. For a”, we have according to (12) the explicit expression a2
=
Cd.%)(d-n) = x2+ id.(x.n) 4V22C”
4m2c2
(44)
This gives (%P -
e$)cp = mc”{~l
+ x2/wiY -
(efi/m2c3)d.H
-
11%
(45)
which is the expression desired, in the case E = 0. We note that we have here not made any explicit assumption about the character of the vector potential A or the magnetic field H. l?ormula (45) has previously been derived by Huff (7) in the case of constant magnetic field H. A similar result has also been ob-
STtTDIES
IN l’ERTI~IIBATTO?;
THEORY.
13!1
\‘I11
tained by Johnson and Lipplnan (8), who have treated also the case of an inhornogeneous lnagnetic field. Expanding the right-hand member of (45) in a power series of a2 and including terms up to the order c-‘, we obtain (COP-
ed)(p =
[x’/Zm
-
(efi/2mc)d.H
-
~~/8m”c~ +
- .=.)+Q.
(46)
Here the first tern1 in the right-hand menlber represents the ordinary kinetic energy, the second the coupling between the spin and the magnetic field, and the third the relativistic correction associated with the fact that the Inass depends on the velocity, etc. It rnay be of sonle interest to conlpare fornlula (45) with the corresponding expression in the classical theory of relativity in the case of no field. One has the basic relations (4i)
froln which one derives the identity (E + me’)2 E = mc2(~1+x”lm2d”
c2a2 = m2c4and the fornlula -
1)
(48)
which is the classical analog to (45) in the case of no field. This shows that (45) gives the correct dependence of the relativistic mass on the velocity. (2). E # 0. Let us now consider the case when there is an electric field, E # 0. In order to iterate the relation (39) it is now necessary to nlove the operator 1~to the right in the right-hand Inember so that it works directly on the way function (o. This can be achieved by using the cornrnutation relation (40), but it is evident that there is no solution of the sanle sinlplicity as in the previous case. A systeniatic study of this problenl will be carried out in a forthcoming paper from the group. VI. THE
XONRELATIVISTIC
HAMILTONIAN
In the previous sections, we have shown that the original Dirac equation (I) nlay be replaced by two separate equations (37) and (38), which are still exact. Relation (38) is equivalent wit>h (21). Using the first of the forms (35) for the spin terni, we can express the Kave equation for the function cp in the fornl (top -
e$)co =
{ (1/2m)&x
-
~.l&:d.(H -t E X kx/2mc)}cp.
(49)
The first tern1 in the right-hand rnenlber represents the kinetic energy of the electron in the relativistic forIn, and it is easily checked that it gives the correct dependence of the lnass on the velocity. It contains further high-order ternls involving the electric field of the type occurring in (32). We will here concentrate our interest on the second terns involving the spin operat’er HsPin defined by (35). We note that this operator is self-adjoint and gauge-invariant as a whole, but we are now going t#osplit it into several parts:
l-10
LowurN
The first term -p&d.H gives the well-known coupling between the spin alid the magnetic field. The second term -(~B/2mc)1~d~ (E X k[p -
(e/c)A]l
(50)
is gauge-invariant as a whole, but it is usually divided into two terms without this property, namely, the “spin-orbit coupling” Xq3in-orbit
=
-
(/.&2mc)kd.
(E x kp)
(51)
and the so-called “contact interaction” XC&Z& = ( pBe/2mc)kd.
(E X kA).
(52)
The reason for the name will become clear in the following discussion. We observe that all these terms actually occur in Pauli’s nonrelativistic Hamiltonian (1) with the essential difference that we have here retained the operator k. In the following we will discuss the more detailed form of these operators in the case of a central-symmetric field. VII.
DISCUSSION SPIN-ORBIT
OF THE HYPERFINE SPLITTING HAMILTONIAN; COUPLING AND CONTACT INTERACTIONS
Let us consider an electron which moves in the field of a point nucleus atomic number 2 and a nuclear spin U. The electromagnetic field associated with this point nucleus and point dipole is characterized by the potentials 4 = ---e/r,
A = &yI X r/r3,
where y is the nuclear gyromagnetic given by the formula
(53)
factor. The magnetic field strength H is
H = curl A = Cy[-I/r3
+ 3(r/?)(I,r)],
(54)
and we note that it has a singularity of type l/r3 at the nucleus (r = 0). Introducing the electronic spin s = MU one obtains for the interaction between this spin and the magnetic field the expression Xspin-field
=
-p&d.H
= 2&+7c[s~I/r3
-
3(s.r)(I.r)/r5].
(55)
It is interesting to note that, if we put k = 1, this term would necessarily have singularity of type rM3at the origin, which is not permitted in a theory leading to stationary states. The nonrelativistic Hamiltonian can have singularities only up to the order r-‘, and the operator k plays a fundamental role in this connection. It has previously been pointed out that in the case of a central-symmetric electrostatic field, one can write the operator k in the form
mhere i’ ,, = Ze?;‘2mc2 = Z. 1.4089 X lo-‘” cm is indeed a very muall length. This gives t,hc forlliula
which is a very large but still finite quantity. This implies that the operator ( 5.5) has a singularity of the type F’) at the origin. Let us next’ consider the spin-orbit coupling operator, defined by (51) in the case of a central-synli~letric electrostatic field with the potential 4 = ~(,v’J. For t,hc electric field, one obtains E = -grad
d, = - 4’( /‘)r/‘r.,
(58)
and WCcau hence write the spin-orbit coupling iu the for111 ~:s~i”-“rl~it= (~H!21nc)I~2[~‘(1,)/l.]d.(r X p).
(59)
Iutroduciug the orbital angular-nlonlentulll 1 through the definition r X p = hl aud further t’he abbreviation 50~) = &~lc’$‘( /.),/wwr, we obtain Xs~)in_~rl,jt
=
((1,)S.l.
(60)
Very ofteu the coefficient <( 1’) or the radial integral over this coefficient is tollsidercd as a semiempirical parallleter which can be detcmined from experimmts. The forln (60) forms the basis for the theory of spin-orbit coupling in conuection with atomic spectroscopy (9). It should be observed that this form of the spiuorbit’ coupling exists essentially alreadg iu Pauli’s nonrelativistic approximation iI>. It is illustrative to consider this spin-orbit coupling iu the case the atonlic uucleus is replaced by a point charge, so that $ = -eZ,~‘r. Suhstitutiou of this expression int,o (59) leads to the formula Gr,in-or1,it= 2&“k”s.
l/L
(61
j
LZgainit should be noted that the singularity at the origin is of the order F1( s. 1)) i.e., a permitted singularity. The spin-orbit coupling is usually treated in a senliempirical fashion, and it would certainly be highly worthwhile to study this interaction frownfirst principles in greater detail. Let us now consider t’he cont’act term define by (33 j, i.e., (E X /CA). Xcontact= (~~P~‘~vzc)/G~~ Using the elementary vector relation a X (b X c) = b(c.a) obtain inmediately the expression E X A = - (e_Z/lS”)r X (fir1 X r/F,‘) = -(efiZr/r”)(I(r.r)
-
r(r.I)},
(62) -
c(a.b),
WC
(63)
142
LiSWDIN
and the formula x contact= -2
eZy(~B2/c)k2(s~I/r4
-
(r.s)(r.I)/r’J.
(64)
Again, we notice that, since k2/t2 is regular at the origin, the singularity is of the order l/r”. It is possible to give a simplified form of this expression in the case of a stationary state. The operator k is now a function only of the radius r and, from ( IS), we obtain directly k’( r ) = k( e+‘/2 mc2)k = ( e2Z/2mc”) ( k2/r2). Substituting this expression into (64)) one obtains x contact= -2p,-yfik’(r)(s.I/r2
-
(r.s)(r.I)/r4}.
(65)
notice that one has the relation s: k’(r) dr = k( 00 ) k(0) = (1 + Gmc 2>-I M 1, so that k’(r) is essentially a d-function which is different from zero only in the neighborhood of the origin; compare (24) and Fig. 1. We observe further that, in calculating expectation values of the contact interaction (65), one obtains only contribution from s-part of the electron density. In the calculation of such expectation values, one is then only interested in the spherically symmetric part of the contact transformation, This means that the second term in the bracket of (65) may be replaced by the average value --$/3(.sV1)/r4. One has further the relation We
k’(r)
J
-;,dv
=
s0
k’(~)4?r dr = 4~,
(66)
which implies that one can introduce the three-dimensional b-function Jc’(r)/r’ + 4&(r).
(67)
Instead of (65), we obtain hence the expression X contact= ->~16~~s+.16(r)
7
(68)
which is the standard formula found in the literature (10). Under very special assumptions the contact term defined by (62) may hence be replaced by an interaction between the electronic and nuclear spins containing a d-function, which explains the name given to this interaction. We note that both the spinorbit coupling and the contact interaction originally occur in Pauli’s nonrelativistic Hamiltonian (I), but that they can be brought to a form which is correct in the neighborhood of the nucleus, only if one includes the operator k. In conclusion, a few words should be added about the treatment of the kinetic energy term Tp = (&t/2m)cp in the fine structure Hamiltonian. If the magnetic field is weak, the quadratic term in vector potential is usually neglected, and one concentrates the interest on the linear term. Considering the first term in (32), one obtains for the linear contribution in the vector potential -(e/4mc)(k(pA
+ Ap) +
(PA + Ap)k}
= -(e/27nc){lcAp
+ Apk),
(69)
where we have used the fact that div A = 0. Using the explicit form (53) the vector potential, one gets further
for
(70)
Hence one has the following contribution to the hyperfine splitting Hamiltonian from the linear terms in the kinetic energy -~&[kI.l/+~
+ (I.l/r3)k].
(71)
C’onsidcring the expressions ((is), (68), and (71), we thus obtain the following expression for the part of the hyperfine splitting Hamiltonian which depends on the nuclear spin I: X,,,(I)
= -2&yl;{I.(l
-
s)/:? + 3(s.r)(I.r)/?} - f~l6a&Cy(s.I)6(r).
(7”)
We note that the Bohr magneton is here defined by the expression ( lla), where e is the electronic charge with negative sign. This Hamiltonian is here derived under the assumption that the nucleus can be approximated by a point charge and a point dipole, and it is clear that the actual form and extension of thcb nucleus will influence the coupling between the nuclear spin and the magnetic ~~~on~cntsassociated with the electronic motion and spin. VIII.
DISCUSSION
111 this paper we have studied the separation of the Dirac relativistic wave equation for one electron by means of the partitioning technique. In a series of previous papers (4), we have learned that this technique is a powerful tool fol the evaluation of energy eigenvalues of the stationary states by means of a secondorder iteration procedure, which is closely connected with the quantun-nlcchanical variation principle. \Ve have here studied essentially the operatol aspects of the partitioning of the Dirac equation, and it seems hence WJrth while to s;tutly also the numerical aspects in greater detail. In this connection it is convenient to introduce a complete orthonormal basis and to introduce the tnatrix representation of the operators involved. A first partitioning leads to the separation of the Dirac equation into its large and small components, and a seco~~d partitioning takes out a suhspace which contains only a single element of the basis; see kf. .$. Yor certain systems, one bows the nonrelativistic eigenvalucs and eigenhn~ctions with a fair alnount of accuracy, and it seems plausible that this data would form a convenient starting point for the second-order iteration proccdurc connected with partitioning technique. In this way one should be able to obt’ain the relativistic corrections in a rather simple way. Studies of test prohlcn~s of tbis type arc in progress at the Uppsala group.
RECEIVED 1)ecember 7, 1963
144
LiiWDIN REFERENCES
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