The well known relationship between gauge invariance and current conservation is exhibited within the usual quantum mechanical formalism. It is then shown that the use of gauge-invariant atomic orbitals (GIAO) does not necessarily lead to the expected current conservation. The reason is found to lie in the constrained nature of the gauge invariance which is provided by the use of GIAO's. It is concluded that this invariance is, of itself, no argument in favor of their use

Epstein, S. T.

English

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GAUGE INVARIANCE, CURRENT CONSERVATION, AND GIAO'S
by
*
S. T. Epstein
Theoretical Chemistry Institute
University of Wisconsin
Madison, Wisconsin 5370,6.
ABSTRACT
The well known relationship between gauge invariance and current,con-
servation is exhibited within the usual quantum mechanical formalism. It
is then shown that the use of Gauge Invariant Atomic Orbitals does not
necessarily lead to the expected current conservation. The reason is found
to lie in the constrained nature of the gauge invariance which is provided
by the use of GIAO's. It is concluded that this invariance is, of itself,
no argument in favor of their use.
*
Work partly done while a John S. Guggenheim Fellow, on leave from the
University of Wisconsin at the Mathematical Institute, Oxford, England.
Work also partially done under Grant NGL 50-002-001 from the National
Aeronautics and Space Administration.
I. INTRODUCTION
ii
If one views the Schroedinger equation from the point of view of
i i
Lagrangian field theory, then there is a well known connection between
gauge invariance and current conservation. In the next section of this
paper we will establish a similar connection within the ordinary quantum
mechanical formalism. In the third section we then show that the use of
2
gauge invariant atomic orbitals (GIAO's) does not necessarily lead to
the expected current conservation, and the reason is discussed. This
then leads us in the fourth section to examine in detail the nature of
the gauge invariance which is provided by the use of GIAO's. We conclude
that, contrary to what is probably the generally accepted view, this in-
variance is, of itself, no argument in favor of their use. The body of
the paper is concerned with static fields and with energy eigenfunctions.
An appendix is then devoted to a discussion of time dependent situations.
II. GAUGE INVARIANCE AND CURRENT CONSERVATION
We consider the problem of N non-relativistic electrons moving in
external static electric and magnetic fields, and we confine attention
to energy eigenstates. We now make use of the following general criterion
of invariance: Since the formalism of quantum mechanics is invariant to
unitary transformations of wave functions, one will have invariance to
particular transformations if these can be shown to lead to unitary trans-
formations. It is well known that gauge transformations meet this criterion.
Namely if the vector potential undergoes a gauge transformation
— 9« ><«•£>
where X is some real function, then the wave function undergoes the
unitary transformation
*"
and correspondingly the Hamiltoniah also undergoes a unitary transfor
mation
VV (A*) - (2)
where (we use atomic units and c is the velocity of light)
In particular we will be concerned with families of gauge transformations
•* (4)
where the ^J^ are arbitrary real numbers, and where the /^ o( are
some given real functions.
If now we expand rV^A'^ *h P0*7618 of the £^{ we find through
first order in the
where [ , ] denotes the commutator. Then from first order perturbation
T S> ~tytheory, the first order change in the energy is given by £* c^ £y
o<
where
< 6 >
and where &
A* - e fr ** ^
tt ^
However since (2) is a unitary transformation of Pr , the energy is
gauge invariant, whence *-«£ must vanish. Therefore we must have
I Ad U U)J^A) ) -=-0 (7)
which is of course true since Vst C A •) ^ C& } ::i ^ "^ ^^ J) • Thus
3
the hypervirial theorem (7) can be viewed as being a consequence of
gauge invariance.
It is now easy to show, after some integrations by parts, that (7)
can be written as
~
where j is the one-electron current. Then since (8) is to hold for
all Ac* we have
•^  ^V a \ "^ -G
J U (9)
4i.e., current conservation.
So far all our discussion has concerned exact wave functions. We
now quote two relevant results for variational wave functions:
(i) If the set of trial functions is invariant to the set of gauge
5
transformations & then the optimal trial functions will
satisfy the hypervirial theorems (7) and hence (8).
(ii) We will have gauge invariance in the sense that if we use this
same set of trial functions with both "'A/ and *•' Mf, / -. the.n we
will get the same energies,-and also -the optimal trial functions will be
related by formulae like (1).
ar •••rS'i'^.iu- erfi /. "?'-•'• ic> i^ir: '•:-;-.;<••->:•. <v;'-:?.'; ;/jts.ila:j & B/.- U) 3%n.:.rt M3V3>ao.n
In particular it is easy to show ' that unrestricted.coupled Hartree-
'.:;• ^sij;;: *.-~.r £j"jo'i . r :Vr'i . i i l ' .-.[air. •?'« j -^Lf _.j ••"-" sio.rrsrfv/
 c;fn:J..;^r:fcv^.i a.viT.i;.
Fock theory meets these conditions for all Ao< > and indeed in this
theory the energies'-arfe 'gauge' invar iah't,-;iarid -'the^ brie -'el'e'ctron current is
8
conserved., ^, ,How.ever in mps,t# variational calculations this will .not be
"• 9'-
the case. . f . . . .
III. GIAO'S AND CURRENT CONSERVATION •.,_-.-.•;•• .-.y^ . :
 ;_;,,.--,.
For clarity we will consider a very simple system: a.single electron
moving in the field of two fixed centers, and in a uniform magnetic
field B . Further, in accord with what seems to be t'hfe custom in quantum
chemistry, we will confine attention to a very special class:;of gauge
transformations. Namely for the vector potential we take /
'.:*r >-v (10)
and then confine attention to the gauge transformations
- • • • ' > < f '
where d is an arbitrary constant vector. From (11) we have
33 "..: •. ej" ~ :' 'T-~ • ••. - ;'Yc -n.c r.-?. -• .'-.-.r.' --1-- - '>:£-, .": :•'- 1.-- ,..-.- •• '.
• : . . - • • •:: fa)
-can
ef the
--,c° be -
A
. .
Now since the use of GIAO's is known to give gauge invariant energies
(we discuss this further in the next section) one might suppose, from
our earlier discussions, that one would necessarily have
.--and
We will now show that this is not the case.
First we consider (18) . One readily finds that
fc
Our assumption that «£" and <fe^ are s-states then makes the first
-line of (19) vanish, and allows the second line to be written as
2C
We now note that for B = 6 the expression in the brackets becomes
which in general doesn't vanish. Therefore , (19) will in general vanish
only if 0^  is real.
We now turn attention to Vy 1 '' ^  Nf / • Then assuming C\
4' j and -^3^ - one readily finds
that
l^ u,*^ ) °3-2»o& I (^  //&^ TO|»E^ ') ^.T%.1 4-"'(4* ^^ ""^ l-Cn/t E^  "VU/
and in order for this to vanish the bracket must vanish (unless 0? ^
which seems unlikely. In particular if we expand in powers of B the
leading term in the bracket is
' " life T C<b ' v* ^4> \ » f
 L /•" "I (20)
. . . . . . 3Z. L % ? 53= ) 4- C* G-f)%.! 1
. • • • • • - ' • - ' • • • • ' • • V g r • J ' 'l<~ / • * - « • • • • • •
.and there seems no reason for even this to vanish, and,indeed a calculation
with Gaussians
yields a non vanishing, result (unless & '^ ~Q ) . Therefore we conclude
that in this particular example we cannot, have both (17) and (18), and
hence we conclude that in general the use of GIAO's does not guarantee
such theorems. '
The reason for this failure is hot hard to find. As stated in (i)
of Section. II, the theorems will hold if .the set of trial functions is
invariant to the appropriate gauge transformation. However our" set,
.which (for fixed <$ 9 T^ and 4'-^ — ) consists of the single function
, evidently changes under the gauge transformation (12): Namely
from (15) we have
, w *
- c- -e
1
 12Thus the conditions of (i) are not; met and so the theorems don't hpld.
Another, equivalent , reason for failure, one which makes direct
contact with (6) (t;he argument above of cqurse dealt directly with (7)),
is the following: First order perturbation theory in the form of equa-
tion (6), is a special case of the generalized Hellmann^Feynman theorem
f 13for the parameter **d. . Now one knows that this theorem will also
hold in v^riational calculations provided that the set of trial func^ l^ons
is independent; of the &ck . Evidently this is not the case here.
Thus the first orde? energy change is not given by a formuia like (6)
but rather by one which contains extra terms, and therefore although thp
first order energy change does vanish in accord with gauge invar iance,
that vanishing does not lead to (7).
These considerations naturally raise the question: what t^ en is the
nature of the gauge invariance which is provided by the use of GIAO's?
It; is th^ .s question that we now turn.
IV. G|AO'S AND GAUGE INVARIANCE
• ^ ' - ^ • - • ' - " ' . '
Clearly the fact that T changes with A , the fact which led
to the failure in the previous section, is precisely the feature of
A. • -
which leads to the gauge invariance of the energy. Thus the
transformation (21) is exactly the transformation (1) (for a one electron
problem) and hence, one has
i.e., the energy is gauge invariant. However having recognized that the
gauge invariance is guaranteed by the way in which the orbitals change
14
with gauge, we then note that by a similar device we can make any
calculation equally gauge invariant. Thus for example consider instead
of 4"* * the function
-4
where §., and f^ are any given vectors. Then since we have
$* -= t ** "V
a calculation based on ^f will also yield a gauge invariant energy.
In particular if we choose jPj •= §^, = 0 then evidently we could con-
clude that even the use of simply
-5- \ r> - >-is
with A = vJ&Tt-* and e ° C^1 V<* otherwise, will yield gauge
invariant results.
More generally it is clearly always possible to get such, what we
might call "enforced gauge invariance" or "constrained gauge invariance",
for the energy calculated using any set of trial functions whatsoever in
any one gauge simply by changing the trial function appropriately as one
changes gauges (or equivalently, by always returning to the original
gauge to do the calculation) . Further it would seem that if one proceeds
in this way, and as we have seen in using GIAO's one does precisely this,
10
that then the resulting gauge invariance should not be thought to lend
any special credance to the.calculation since requiring it has simply led
to a specified connection:between the sets of trial functions to be used
in different gauges. Rather one's a priori, expectations concerning the
accuracy of the calculation, i.e., one's a priori expectations concerning
the physical completeness of the set jm any one gaugQ, must clearly be
"based on other considerations; In contrast, if in the spirit of (i) and
(ii) of Section II, the set: of:trial functions is invariant to gauge
" - '* { j • ' • ,. • • . : ' ' ' ' -
transformation^ then equally clearly this is a relevant piece of informa-
tion since evidently it does supply some information concerning the
'completeness of the set. '
Now in fact, as is well known, there is a real physical basis for
• " ' . • ' • " -i&ty)'^ " . ' ; : '•
the use of GIAO's: The^function & ® ^l is, through
; ' • ' ' • ' - i * ' : " '
first order in B ., the exact eigenfunction of.a one-electron one-center
• "^  «4teplus uniform magnetic field problem which for B = 0 has ^L as an
eigenfunction. Moreover we would emphasize that nothing we have said
, ' ! " ' ' ; "''
about "constrained" or "enforced" gauge invariance should obscure th>£
point. Rather we are saying that this, and not the resulting gauge
invariance, is what suggests that the use of GIAO's may be an effective
tool in dealing with problems of molecules in uniform magnetic fields.
On the other hand, of course, the failures which we have discussed .are
also real and should be weighed in the judgement.
-*•
Returning to our special, example, since any choice of ® and
^ ^J<&, will yield a gauge invariant energy .Which however depends on J>.
^ '- •£ -*
and it
 ;is natural to ask for the best J and in the
11
sense of the variation principle. To find out we of course simply let
& • ' • ' • • : '
and j^. be variational parameters. It; is then; of J
to note that the resultant set of trial functions is now invariant to
-* &! ••'•*'• V
gauge transformations since ^ can be gotten from j1 by
5replacing f and ^, by J\4-<i and >4^  respeqtlve,ly
so that a gauge transformation simply turns one trial function into
another. Thus from (i) and (ii) of Section II it follows tl>at by
ducing this extra freedom we will ensure that (17) and (18) are now
I- • t'. • . i . ' • ' • • ' . ; • ' • • • • •' .•">•-; ri'«V ' • ' -df- ' • •'
satisfied (evidently therefore in our example the optimal f, and ^r
are not (0,0,0) and (0,0,R)), and also we will gt^ll have gaug^ invariant
energies and properly gauge variant wave functions, but now in an un-
forced way (one uses the same set of trial functions in all gauges) rather
than by constraint.
V. ACKNOWLEDGEMENTS . .
I am indebted to audiences in England (Nottingham, Oxford, and
Sheffield) and in my own univeristy for having suffered through., ai»d fpp
having commented on, various versions of this material. V
APPENDIX . . ;
Although in the body of the paper we are concerned only with station-
ary states it would seem useful to add some comments on the time dependent
case. Here under a gauge transformation the vector and scalar potentials
change according to
12
-
Again invariance is expressed by the fact that the wave function undergoes
a unitary transformation
~ •*•""• f of,
and correspondingly
c.
where «\
' "
 A
 "5, >(
If now we choose
"5*>C*;<t>^ £.
then the analogue of the result (6) for the first order energy change is
i~"-," '-"•- .. -. . • ••- ,. -.-. - • '••••• . • 18 ' - '
provided by the time dependent Hellmann-Feynman theorem which yields
a relation between the change in T and the change in H , due to a
change in a parameter, whatever the nature of that change may be. As we
have emphasized, gauge invariance is expressed by the particular nature
of the changes (jV^ J and (frfyt in particular by the fact that (jtV-1^  is a unitary
transformation. For our particular case then the general theorem
13
where Q-* is a real parameter yields
f)
This result, a special case of the generalized Ehrenfest's theorem or
time dependent hypervirial theorem, is of course a well known consequence
of the Schroedinger equation. However we now see that it can also be
regarded as a consequence of gauge invariance, and evidently in the time
independent case, and with the €:$ = 0 , it reduces to (7). Further it
is easy to show that (A-4) can be rewritten as
JJ?-( V? ^.;J ) Xrf =o (A-5>
>-c
where P is the one electron density. Then since (A-5) is true for
all /sot, we have
i.e., current conservation.
All this is for exact wave functions. For variational wave functions
19
one. can show in complete analogy to the static case, that if the set
of trial functions is invariant to the set of gauge transformations then
the optimal trial functions will satisfy (A-4) . This condition is met
by unrestricted coupled Hartree-Fock theory for all AQ{ and hence
one has current conservation in that theory.
14
FOOTNOTES AND REFERENCES
1. See for example W. Pauli, Rev. Mod. Phys. 13, 203 (1941).
2. GIAO's were first introduced by F. London, J. Phys. Rad. £, 307 (1937).
The recent paper by R. Ditchfield, J. Chem. Phys. _56, 5688 (1972)
contains an extensive bibliography.
3. J. 0. Hirschfelder, J. Chem. Phys. 33. 1762 (1960).
4. One can of course consider more general unitary transformations. For
example, at the other extreme, with J^, an arbitrary function of
all coordinates, one is led to
and hence
L y\ *T ~^ if *r~" 3 * *^" ^O
? .is the N-electron current.
5. S. T. Epstein and J. 0. Hirschfelder, Phys. Rev. 123. 1495 (1961).
6. S. T. Epstein, J. Chem. Phys.; _42, 2897 (1965).
7. One uses the same set of trial functions - the set of all Slater
determinants - in any gauge, and this set is invariant to all gauge
transformations. (See Ref. 6).
8. Our statement of course refers to total current. The individual
orbital currents are not in general conserved.
9. The work reported in this section stems in part from a conversation
with Professor R. McWeeny in which he raised the question of current
conservation in variational calculations.
10. Strictly speaking these symmetry restrictions on & and ^^
fr
are irrelevant for the questions we are considering. Also irrelevant is
the question whether or not Q£
 t (p and (p are dettrmined variaiionally,
so we will treat them as given.
15
11. R. McWeeny, Chem. Phys. Lett. 9_, 341 (1971) has discussed some aspects
of the question of invariance to choice of coordinates (rotating or
translating the coordinate system of course also involves unitary
transformations). We will not enter into a discussion of such matters
here except to note that similar considerations applying as in the
case of gauge invariance
12. As derived in Ref. 5 these conditions are only sufficient conditions.
However in practice they, and others like them, seem usually to be
necessary unless symmetry and reality considerations intervene.
13. A. C. Hurley, Proc. Roy. So a. A226, 179 (1954).
14. They might therefore be better called gauge varian^ atomic orbitals.
15. Unless the set of atomic orbitals is essentially complete, butj in
such a case the use of the gauge factors become irrelevant anyway.
16. That is with the gauge origin at the center, the eigenfunctions of
the one electron problem are unchanged through first order in B
(more .precisely they are the zero field eigenf unctions whose orb'ital
-^
angular momenta are quantized along B ).
17. For the purpose of invariance of the set, it would already suffice
-.iM?) *+i»jk . j-i •»to use <2, & • L,^ 4-c* fsj with & as a variational parameter,
18. E. F. Hayes and R. G. Parr, J. Chem. Phys. ^ 3, 1831 (1961).
19. S. T. Epstein, J. Chem. Phys. 45. 385 (1966); S. T. Epstein and
R. .E. Johnson, J. Chem. Phys. 51, 188 (1969).


Gauge invariance, current conservation, and GIAO's

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