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6-1991
Fock-Tani Hamiltonian for Reactions Involving Two-Electron
Atoms
Jack C. Straton
California State University
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Citation Details
Straton, Jack C. "Fock-Tani Hamiltonian for reactions involving two-electron atoms." Physical Review A 43 (1991): 6018-6026. DOI:
10.1103/PhysRevA.43.6018
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PHYSICAL REVIE%' A
Fock-Tani Hamiltonian
VOLUME 43, NUMBER 11
1
for reactions involving
JUNE 1991
atoms
tvvo-electron
Jack C. Straton
Department of Physics, Catifornia State University, Fresno, Fresno, California M7$0
(Received 11 July 1990; revised manuscript received 17 December 1990)
The I"ock-Tani Hamiltonian is found for scattering processes involving up to two ions and
two electrons. Possible bound-state species include one or tmo electrons bound to an ion fixed
at the origin, and a one-electron projectile atom. A diagrammatic technique is illustrated that
Coulomb- or plane-wave states are automatically
simplifies the algebra of the transformation.
generated by the same asymptotic Hamiltonian for all arrangement channels.
I. INTBODUCTION
II. THE DIACRAM
Fock- Tani
was
introduced
representation
by
Girardeau
as a means to extend field-theoretic techniques to include bound composites as well as their unThe resulting representation
bound constituents.
formal problems in
has resolved several outstanding
atomic scattering theory, as discussed previously.
In practical terms, the improved convergence of the
Fock-Tani perturbation series over the Born series (as
measured4 by Weinberg's criterion~~) has produced excellent results
in first-order calculations for charge
transfer involving one-electron atoms. In particular, it
may be seen7 that the first order F'ock--Tani differential cross section for H++H~H+H+ is virtually identical to that of the secoud-order boundary-corrected Born
approximation~2
(B2B) at 125 keV. The present paper
provides the Hamiltonian upon which first- and secondorder Fock- Tani calculations for two-electron atoms may
be based.
The major drawback of Fock-Tani representation lies in
the difIxculty of transforming the Fock Hamiltonian into
the Hilbert space that includes elementary bound-state
operators. One source of that difficulty is the tedium of
accounting for integrals and sums over dummy variables
that append the second-quantized operators. But it is
the contraction of these propagators that is the physical
content of the transformation, not the integrals in which
they reside. The present paper introduces a simplified
form of the diagrammatic notation of Gilbert, ~s which
greatly simplifies the multiplication and contraction of
field operators because all integrals, sums, and dummy
variables are suppressed. This notation is introduced in
Sec. II and utilized in Sec. III to find the Hamiltonian
for systems involving two electrons. The initial stage of
this transformation has been performed by Straton and
Girardeau and equations (K) of that paper will be refered to herein by (N-I). Finally, in Sec. IV the asymptotic
Hamiltonian is found to automatically generate states appropriate to whether or not there is a net Coulomb force
between target and projectile.
43
NOTATION
An algebraic second-quantized
representation always
includes integration over the position variables of free
particles and summation over the energies of the bound
states.
But the integrations and summations,
and
even the arguments themselves, are incidental to the
operator algebra of transforming from Fock to FockintroFor this reason Gilbert
Tani representations.
duced a diagrammatic notation in which a line represents a creation or annihilation operator, with implied
integration or summation over the dummy arguments.
have also used
and Brittin
Kvasnicka,
Fleckinger,
diagrammatic representations for bound-state problems.
For the present problem the very general notation of
Gilbert is not needed. By distinguishing ions from electrons and ordering diagrams with the former uppermost,
one may use a very simple set of diagram rules.
One first defines the correspondences between anuihilation operators and propagator lines,
&=p
fixed at the origin
nucleus
-- ———————-$— &=& projectile
X~
X Qo ~ ~ ~ ~ ~ ~ i
~
with
~
~ ~ ~ ~ ~~ ~~
fermionic
&=&
electron
&=&
(a+e } bosonic
(b'e e
(o ooo
"-"-"-- ~-
&=&
an implied
integral
bound-state
fermionic
(b
e)
propagator &&a(x}
propagator
propagator
&=&
&=&
a+
bosonic
propagator
&=&
fermionic
propagator
&=&
)
e(y}
P&
e„,
or sum over the ar gument,
wave functions
-X
(2)
(-&
6018
-u
(y)
1991
The American Physica1 Society
POCK- TANI HAMILTONIAN
FOR REACTIONS INVOLVING. . .
X--
where the minus signs keep the present notation consistent with Straton and Girardeau, and the Schrodinger
'
+
(=)
operators
(=) V(xy) =V„(xy) +
W (xy)
H„(xy) $ (xy)
=-(E„&„(Xy))
—
~ ~ ~4 ~
&0 ~ OO
V, b(y) +
T, (y)
Finally,
-Zb
crossed fermion lines give a factor of
—1
so
that
1
2pe
T, (x)
+ V b(x)
]
ZaZb X
&Pa
V(yy') = V (yy')
=
2
x
Thus, for example, the Fock bound-state
erators are
mj
-&„=-fdx dy 4„(xy) e(y) 6(x)
Iy - y'I
771/
+ 77lg
Closure relations for the three bound states are expressed diagramatically by
r
(=) P
dy dy'k~ (yy')e(y') e(y)
E
=
k
fdyU, (y)e(y),
where u{y) is the hydrogenic orbital involving the relative
coordinate y, g(yy') is the corresponding two-electron
orbital, and P(zy) is a traveling one-electron orbital.
The operation of multiplication is defined as performing all possible combinations of connecting propagators
from the two diagrams being multiplied (a normal ordering expansion) and putting unconnected lines in normal
order. For example, the commutator [A&, At] is
pav
P +
v
(=)
rI
op-
W, (yy')
The inertial potentials W, sometimes referred to as
"mass polarization terms, " and the external potentials
V,„are a result of the transformation to an accelerated
frame of reference in which the ion b is constrained to
remain at the origin, and is thus a nondynamical particle.
The two-body reduced masses are
Pj=
annihilation
P
/
~yv
+
and the operation of a Schrodinger Hamiltonian
eigenstate is expressed by
upon its
~
—
~~
(=)
~~v+
C@v
Note that the C„diagram is neglected because it is zero
on the subspace in which there is only one ion of type a.
JACK C. STRATON
6020
III. THE TRANSFORMATION
A. The
A.
is (57-I), the eighth is (55-I) (with the ninth as its Hermitian conjugate), and the tenth is (62-I). Finally, the last
transformation
diagram
The transformation from the Fock space of unbound
propagators to the Fock- Tani space that includes elementary composite propagators is most easily done in three
steps using a unitary operator defined as a product of
three transformations, each onto the subspace containing
one new bound-state operator. The ordering of the transformations must be such that the operator constituents
of the Fock bound states of the earlier transformations
are not subsets of the constituents of the Fock bound
states to be transformed later. ~ Thus the transformation
UE for the atom consisting of one electron bound to the
origin must come last. Since the operator constituents
of A~ and of B&~ are not proper subsets of one another,
there is a post-prior ambiguity in the choice of ordering,
U~UaUE or UaU~Uz.
for two electrons,
The present Fock Hamiltonian,
one projectile, and one ion fixed at the origin, i's exactly the same Hamiltonian that Straton and Girardeau
transformed. s In Eq. (2) of their paper, which will be
referred to as Eq. (2-I) herein, one must merely rename
the two "ions" to be "electrons" and the "electron" to be
the "projectile a." The undifFerentiated "fourth particle
fixed at the origin" is chosen here to be of positive charge
to bind the electrons.
Then the transformation U& y H~V~ has already been
completed. One may merely quote Eqs. (41-I) through
(63-I) after making the replacements X
y, A&
—At, at —+ —nt, and C&
C„' . With this work already done it is sensible to choose to find the post form of
the Fock-Tani Hamiltonian for two-electron atoms using
A
A
= U~ UBU~
.
UA
H. c
2
yl
~
Hc
~
H. c
I
A
(10)
(41-I) to fourth order in
The transformed Hamiltonian
bound-state wave functions, is
n, ~
Hc
~
Hc
~
~
U
43
I
+Hc
2
+
(12)
reaction term that was not included in the
Hamiltonian of Straton and Girardeau (41-I), but will
contribute to binary reactions after the subsequent transformations.
is a trinary
~
HF@
~0 ~0~ &
PI 1OON
~0 ~ 0 ~0
OO
~ ON
B. The B transformation
Turning now to the second step in the product transforrnation (10), we find that on the two-electron subspace,
the generator F~
~ 0 0 &10 +0
The algebraic translation of all but the last diagram
and will not
have been given in Straton and Giradreau
be repeated here. The first three diagrams are the first
three in the Fock Hamiltonian (41-I) and correspond to
The fourth and
(42-I) under the above replacements.
sixth diagrams are the two terms of (53-I). The fifth diagram is (44-I) including the external potentials that were
left out of the reduced form (50-I). The seventh diagram
~oe
of the unitary operator
U~
—e (vr/2)P
~
is simpler than the generator
(14)
FOCK-TANI HAMILTONIAN
43
I/ r
~l
d
I
y
2
..
FOR REACTIONS INVOLVING.
- (t) =-
Y-
Y
V
/
%
6021
+
r
=~
r=
------(t)
+
2
that appears in the unitary operator of the A transfor-
««« ~ ~ ~ I« ~ ~ ~ ~ ~ «
---"-- (t)
mation,
U
(x/2)
PA.
(16)
This is because the bound-state kernel terms Cp~ ~~ arising from [B~, Bt] = b~~+C~~ ~ would give terms in the
generatori
annihilating three (bound or free) electrons,
and thus give no contribution on the two-electron subspace.
The transformed Hamiltonian
II~ = U~'II„U~
(17)
may be found by solving the equation of motion
0(
)
=-2~"---
cos t
—
2
sin t ,
(24)
where we have used the symmetry
to obtain the factor of 2. Integrating
e(y) gives
- (t)
=
2
r=
of the wave function
and using e(y, 0) =
« ~ ~~
«s«««««««««EO
O(t)=U, 'OU, =e ' 'Oe' '
( t)
-
and
----"
cos t
~--- —-.-
&Qs
Vg
(26)
Note that this transformation
is exact (tiue to ail Qr
ders in bound-state wave functions) on the two-electron
subspace. Then the transformation of H~ consists of replacing all electron field operators by (26), performing
all possible contractions of lines, and neglecting terms
that annihilate three or more electrons. The resulting
Hamiltonian will not be displayed here since it will be
contained in the final Hamiltonian.
C. The E transformation
which clearly have the solution
~ ~ «g««
electron anni-
~ « ~ ~ ~ ~ Eo ~ « ~ « ~ « ~ Cl ~
Us
=B
(20)
~«««««««$ «
~
(25)
Finally, setting t = vr/2, the transformed
hilation operator is
js any of tlie operators pz(t), Bz(&), and e(y, t) if s
Then, on the two-electron subspace
$q f
(I- cos t)
where
Q «««««««««
+
t+
t
(22)
sin t
(23)
s~n
The final stage of the product transformation
(10)
closely follows that of the previous section. Since the
Fock operators for one electron bound to the origin anticommute, (E„,Et} = b„~, there are no bound-state kernel terms, C„~ ~, that enter the generator,
so
~ ««««««««««««««$«
satisfying the initial condition Pp(0) = Pg and B(0) = B.
Then (22) substituted into the equation for e(y, t) gives
~ Q««««
~
«««««« ~ «
(27)
Thus the last transformation of the electron propagators
is identical to the B transformation (26), but with the E
bound states.
6022
JACK C. STRATON
The final post Fock-Tani Hamiltonian on the twoelectron, one-projectile subspace in a coordinate system
with a fourth particle fixed at the origin to bind the elec-
43
aIld
Viz
00 0 0 00 ~ 0 ~
~ 0 ~ 0000
~0 ~ 0000 ~ ~
E
I ~ 0 0000 ~
+
H. c.
+
H. c.
tI'OIls IS
II~U
U
= U~
IIBUE
—II0+V01+V02+V11+V12
~0000
where
+ H. c.
+
H0
r
OOOO OO OOO
r
~0
L
~/000
+
0
OOOOOO
00 OO
+ H. c. +
Soo eooooooooooewOoo~
F ——
L~
J
+ H. c. +-----
0
+
H. c.
monopole
-00000 ~ 000000000000+
00+0 000 ~ 00 ~ 0 ~ ~ ~ ~ ~ ~ 00 ~ 00 ~ ~ 0
Q
+ H. c.
~00004
~ \ ~ ~ 0~ ~ ~
~ 00 ~ ~ ~ ~ ~ 0 ~ \
~ ~0 ~ \ ~
00
The matrix elements are given in terms of the matrix elements of the previous sections. In U02 the renormalized (orthogonalized)
matrix element for electronelectron scattering is
~ ~ 0 ~ 0 ~ 00 ~
C
monopole
rno
—
Ole,
H. c. +
voi
2
~00 0 ~
2
=
where
~ 00 00
~
. C.
~ 0000 ~
Vow
00000 ~ 0
~ 0 00000
The renormalized matrix element for creating a, oneelectron bound state in the presence of a free electron
is given in terms of (35),
--+
E
~ 000 ~ ~ 00 ~
+
H. c.
+
H. c.
~ 00000000
~000000
+ H. c.
Note that the monopole part of the elastic scattering
term for one bound and one free electron, the contribution of the third term of (42-1) to the last term of
(31), has been moved to the twelfth term in asymptotic
Harniltonian (29). The nonphysical states involving e e
(or e„e ) are neglected.
FOCK-TANI HAMILTONIAN
In Vii the renormalized
matrix element is
~ O O ~ ~ ~ ON
— H.
~ O O O OOO
6023
scattering
0
E
~ OOOOONO
projectile-electron
FOR REACTIONS INVOLVING. . .
c.
~ O ~ O ~O
—H.
c.
(37)
where in the first term of (37) the monopole part of the
first term of (45-I) has been moved to the last term in
the asymptotic Hamiltonian.
The renormalized matrix
element for formation of an (a, e) bound state in the field
of the nuclear fixed at the origin is
~ NOON
(38)
NOONO
This will be needed to find the Fock-Tani Hamiltonian for
three-electron atoms. Likewise, there is a renormalized
matrix element for formation of the (a, e) bound state in
the presence of a free electron and the ion fixed at t, he
origin,
The (in)elastic scattering matrix element for the (a, e)
bound state from the ion fixed at the origin, the fourth
term of (32), is unchanged from (53-I) except that the
monopole part of the second term in (53-I) has been
moved to the tenth term in the asymptotic Hamiltonian. Likewise, the seventh term of Vi~, corresponding to
(in)elastic scattering of a projectile from the electron
bound to the origin, has the monopole part of the first
term in (45-I) moved to the fourteenth term in the asymptotic Hamiltonian. The matrix element for binding an
electron to the origin in the presence of the projectile is
(42)
where
and the last two terms of (32) represent charge transfer.
Vj2 contains the matrix for scattering two electrons
and one projectile ion (of different species) from the ion
fixed at the origin,
- H. c.
—
H. c.
and an (a, e) scattering matrix element in the presence
of a free electron and the charge at the origin,
E
+
+ H. c.
O\
~ ~ IO
~ ~O~O ~ ~
~0 ~
'I
~
~ ~
(44)
(4O)
where
Vj 2 also contains a re'normalized term describing the double ionization of the two-electron bound state,
JACK C. STRATON
6024
(45)
+
e Iso oops
where
and a term representing single ionization of the doublybound state in the presence of the projectile,
and a renormalized
transfer-ionization
matrix element
(47)
Single electron transfer between the two-electron atom
and the projectile is given explicity in the last two terms
of Vjg, (33).
Note that the seventh term in Vj2 (33), the scattering
matrix of a projectile from a two-electron atom, is shown
with a term subtracted,
where
~
tlP
~ $ ~ ~ ~ ~ ~ 11 ~ Q
~
N1 ~
(48)
There is a matrix element for binding an electron to the
origin in the presence of a free electron and projectile,
~0
~~~
le
0410
'
~
Q
~
I~
~
(49)
a scattering matrix for that bound state,
(5o)
a matrix for charge transfer in the presence of a free electron,
This monopole term in the elastic Coulomb scattering of
the projectile from the two bound electrons is instead
shifted to the seventh term in the asymptotic Hamiltonian Ho (29). Likewise, the elastic part of the third
to last term in (33), represent, ing the scattering of one
species of one-electron atom from another species of oneelectron atom, has the monopole part of (58-I) shifted to
the eighth and ninth terms in the asymptotic HamiltoIl ian.
IV. THE ASYMPTOTIC HAMILTQNIAN
AN ELASTIC SCATTERING
The Fock-Tani Hamiltonian developed herein is appropriate for two-electron, one-projectile (ion) problems
in which one or both electrons may be bound to a ion
fixed at the origin and one electron may be bound to
the projectile. Physically, one may have either Coulombor plane-wave asymptotic states, depending on the net
charge of the bound states, and the asymptotic Hamiltonian has been chosen to automatically generate the appropriate type of wave. Consider first an initial state
consisting of two electrons bound to the origin and a
projectile,
FOCK-TANI HAMILTONIAN
43
FOR REACTIONS INVOLVING. . .
iAk)
40 ~ 0 ~ t 00 04 N \ ~ i
(54)
I
1
=
d
6025
R y(R, , k) at(R. )Pqt~0)
The action of the asymptotic
Hamiltonian
.
(55)
on this stat;e
is
which has algebraic translation
Ho 1~k)
(56)
monopole
fHO ~ 0+ 0 QOOOiOtO
lg
~ OOSOOO ~ 0
[In forming the product of the sixth term of Ho (29) and the initial state (54), the third term of (5Q), t}ie dotted lines
(the ion Axed at the origin) have been superimposed (identiAed) and the closure relation (5) has been inserted for
visual clarity. ] This has algebraic translation
W
H, (Ak)
=
1
d
R a~(R.) P~t
Eg-
d
2p~
x
We retain only the monopole term of an expansion of
I/)R. —r~ and place all remaining terms back into the
interaction potential (53). Then the initial state (54) is
an eigenstate of the asymptotic Hamiltonian Ho, with
eigenvalue
(58)
if y is an eigenstate of
1,
2pg
Z (Z|,
—2))
~g(R, , k)
=
k'
2@a
y(R, , k).
(59)
[Note that we must make the ad hoc replacement of
(4) by the four-body reduced mass M = (mt, +
2m, )/(m + mb + 2m, ) to obtain the exact eigenstates
for this problem. s] If the nucleus of the two-electron atom
has charge Zb —2 then y is the corresponding plane-wave
eigenstate of (59). With no net charge between the projectile and the target, atom, this is the correct asymptotic
state. If the target nucleus has charge Zb = 1, 3, 4, . . . then
g is the corresponding Coulomb-wave eigenstate of (59),
which is again the correct asymptotic state for the physical system. Thus the Fock-Tani Hamiltonian automatically accommodates the asymptotic state appropriate to
the boundary conditions without resorting to the introduction of a distorted wave formalism that is required
when the conventional Born series is used.
Now consider the corresponding final state for charge
transfer, with an electron bound to each ion. Because
an (0~a~ a„ final state does not couple to at(R)Pt, the
p
rd r' g'„(rr')
2
—4
'
fR. —i.
gg(rr') y(R, , k)~0).
(57)
/
above terms will play no part. The terms in the asympthat will determine the Coulomb- or
plane-wave nature of the final state are the eighth and
ninth terms in (29). These yield a long-range potential
( —1)(Z —1)/R, which properly gives a plane wave when
Za = 1. This is consistent with the presumption in (8)
that only one electron may bind to the projectile. A
larger Z, would require a more general derivation.
Next consider a physical system in which there is only
one electron to bind. Then an initial state
at(R) ct&~0)
will couple only to the first, fifth, sixth, and fourteenth terms in (29), yielding a long-range potential
Z (Zt, —1)/R that correctly determines the Coulombor plane-wave nature of the asymptotic state, depending on whether or not Zb —1. Likewise, for an initial
state
et(r)e&~0), the twelfth and thirteenth terms in
Ho yield a long-range potential (1 —Zb)/R that properly
determines the wave nature of the eigenstate.
Consider a final state for charge transfer (O~a„. The
third, ~ tenth, and eleventh terms of (29) correctly choose
the nature of the asymptotic state depending on the
charges in the long-range potential Zq(Z, —1)/Ry. The
thirteenth term will contribute if one adds an additional
free electron, giving continuum electron states in addition to a plane-wave (a, e) state if Z = 1 (or one must
develop a three-body continuum state if Z
1). Finally, the sixth, thirteenth, and last term will contribute
to the asymptotic Green's operator for ionization of oneelectron atoms.
Thus one sees the advantage of working in a representation in which the states representing various species
are exactly orthogonal. The same asymptotic Hamiltonian can accommodate all possible arrangement channels
totic Hamiltonian
j
JACK C. STRATON
6026
This leads to the additional benefit for
elastic scattering that the singular terms
I/R in the
interaction potentials are automatically and consistently
removed.
simultaneously.
43
asymptotic Hamiltonian may be used for all these arwave functions for
channels,
rangement
appropriate
Coulomb- or plane-wave boundary conditions are automatically generated.
V. CONCr, USIONS
ACKNOWLEDC MENTS
The post form of the Fock-Tani Hamiltonian for two
electrons, one projectile ion, and a ion fixed at the
noorigin has been calculated using a diagrammatic
t ation.
The resulting terms in the interaction potential have visually obvious physical interpretations
such as (in)elastic scattering, single and double ionizatransfer-excitation, ionizationtion, transfer-ionization,
Because the same
excitation, and charge transfer.
Discussions with Dennis Gilbert on diagram formalisms are gratefully acknowledged. This work was begun while the author was a guest scientist conducting
research through the National Research Council —NASA
Research Associateship Program, and was also supported
by the Division of Chemical Sciences, Office of Basic Energy Science, Office of Energy Research, U. S. Department
of Energy.
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Phys. Rev. A 41, 15g
(1990).
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j. Phys.
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(1988).
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de Toulouse, 1979.
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(unpublished).
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Note that the replacement HP„(xy) = E„P„(xy) (6) must
include the long-range potential in the present case since P
includes the plane or Coulomb wave describing the motion
of the (a, e) bound state.
N. F. Mott and H. S.W. Massey, The Theory of Atomic
Colh'sions, 3rd ed. (Cambridge University Press, Oxford,
1965), pp. 327 —329.