The Journal of Chemical Physics Volume 68 Issue 8 1978 [Doi 10.1063%2F1.436226] Bader, Richard F. W.; Srebrenik, Shalom; Nguyen-Dang, T. Tung -- Subspace Quantum Dynamics and the Quantum

Subspace quantum dynamics and the quantum action principleRichard F. W. Bader, Shalom Srebrenik, and T. Tung NguyenDang

Citation: J. Chem. Phys. 68, 3680 (1978); doi: 10.1063/1.436226 View online: http://dx.doi.org/10.1063/1.436226 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v68/i8 Published by the American Institute of Physics.

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Subspace quantum dynamics and the quantum action principle

Richard F. W. Bader, Shalom Srebrenik,a) and T. Tung Nguyen-Dang Department of Chemistry. McMaster University. Hamilton. Ontario. Canada L8S 4Ml (Received 31 October 1977)

Schwinger's quantum action principle is used to obtain a quantum mechanical description of a subspace and its properties. The subspaces considered are those regions (11) of real space as defined by a property of the system's charge distribution p(r), namely, that they be bounded by a surface S(r) through which the flux in vp(r) is zero at every point on S(r). Through the variation of the action integral, expressed in terms of the appropriately defined Lagrangian integral operator for a subspace, one obtains the quantum equations of motion and an expression for the change in the subspace action integral operator, a'\\? (11). This expression obeys a principle of stationary action, and thus, as for a system with boundaries at infinity, il defines the generators of infinitesimal unitary transformations. The change in a subspace property () (11) as induced by such an infinitesimal unitary transformation is investigated and related to the corresponding change as described by the calcul us of variations. The latter consists of two contributions, one from the variation over the domain of the subspace, the other from a variation of its surface. It is found that the change in [) (11) caused by the domain variation is expressible in terms of the commutator of!9 (11) and &(t), the generator of the infinitesimal unitary transformation. Through the definition of a subspace projector in the coordinate representation this commutator is shown to contain the subspace projection of the all space transformation and a term which corrects for the nonhermiticity of the projected generator. The contribution to the change in 0(11) from the surface variation is not expressible in terms of a commutator. However, this change is still quantitatively determined by the action of the generator as a result of the subspace boundary being defined in terms of the observable charge distribution. The Heisenberg equation of motion for a subspace property is a particular result obtained from the general analysis of the change induced in a subspace

A when the total system is subjected to a

canonical transformation. The effect of the temporal generator-Jell t on a subspace property is expressible as a projection of the usual all space result and a term describing the flux in the vector current of the property density across the boundary of the subspace (the domain variation). plus a contribution arising from the change in the boundary with time (the surface variation). Schwinger's quantum action principle is re-expressed as a sum of the changes in the action integral operator for each subspace in the system, changes which assume their simplest physical form for the particular class of subspaces studied here. Since the application of the zero-flux boundary condition to a molecular system partitions it into a collection of chemically identifiable atomiclike fragments, the total change in the transformation function as given by the action principle, may be expressed in terms of a sum over the change in action for each atom in a molec;ule.

I. INTRODUCTION

The purpose of this paper is to obtain a general quan-tum mechanical description of the properties of a sub-space. The subspaces of particular interest are those regions of real space bounded by a surface S(r) through which the flux of the gradient in the charge density p(r) is zero,

basis employing the correspondence principle and the associated assumptions based on classical Hamiltonian dynamics are replaced by a single dynamical prinCiple, the principle of stationary action. This approach yields the equations of motion and the commutation relation-ships.

Vp(r)n(r)=O VrES(r). (1) As shown earlier, 1 and in the preceding paper, 2 the quantum mechanical relationships governing the varia-tional properties of such subspaces exhibit maximum correspondence with the expreSSions of all space quan-tum mechanics, that is, with the expreSSions which de-scribe the variational properties of a system with bound-aries at infinity. The description of the subspace and its properties are obtained through the application of Schwinger's action principle. 3

A. Schwinger's action principle

This principle provides a formal axiomatic develop-ment of quantum mechanics in which the conventional

apresent address: Department of Physical Chemistry, Hebrew University, Jerusalem, Israel.

For a statement of the axiom upon which SchWinger's approach is based we quote Roman4 : "For every quan-tized system there exists an action-integral operator W constructed from the operators qj(t) and ql(t) in ex-actly the same manner as the corresponding classical action integral

'\\' = f2 (ql' qj, t) dt , t1

(2)

such that in performing an arbitrary general operator variation, the ensuing change in the action operator

(3) is the difference between the values at t2 and t1 of the generator of a corresponding unitary transformation, causing the change in the quantized system." This is a statement of the operator principle of stationary action. Formally, the change in the action integral as given by Eq. (3) is the same in the classical and quantized cases and thus Schwinger's action principle establishes the

3680 J. Chern. Phys. 68(8). 15 Apr. 1978 00219606178/68083680'$01.00 1978 American Institute of Physics


Bader, Srebrenik, and Nguyen-Dang: Subspace quantum dynamics. II 3681

correspondence between the generator of a claSSical canonical transformation and the generator of a unitary transformation of the equivalent quantized system.

The generator ~ of an infinitesimal unitary transfor-mation is defined in terms of the operator it,

(4) The operator iL causes a unitary transformation up to first-order in the infinitesimal t: if

3682 Bader, Srebrenik, and Nguyen-Dang: Subspace quantum dynamics. II

condition that ol/J vanish at the time end pOints, then oW yields in addition to the equations of motion the sec-ond integral on the rhs of Eq. (12). This was the pro-cedure used in the previous paper to obtain the subspace variational principle. The final term in Eq. (12) ob-tained from the general variation of W is the contribution to the change in the action resulting from the change in time at the time end points.

If the Lagrangian integral operator is taken to be

. = f drHi1i(l//~ - ~tl/J) - (lf2/2m)Vl/Jt . VI/J - V(r)l/Jtl/J} - ~ f dr f dr'l/Jt(r)l/Jt(r')U(r, r')I/J(r')I/J(r) ,

(13) then the variation of '\\, yields the following expression (and its adjoint) for the equations of motion - m(r, t) - (1i 2 /2m)V 21/J(r, t) + V(r )I/J(r, t)

+ f I/Jt(r', t)U(r, r' )I/J(r', t)dr' I/J(r, t) = 0 . (14) This is the field equation which is assumed to govern the matter field. It is Schrodinger's equation for a single particle moving in an external potential V(r) to-gether with a term U(r, r') representing the interaction of the field with itself. The latter term is required if the field is ultimately to describe the properties of a system of particles with two-body interactions.

Using the Lagrangian . given in Eq. (13), assuming6 the field equation, Eq. (14), the change in action is giv-en by

.:l'\\'={ f(7rOI/J- 01TI/J)dr+.(t)6t} I:: ' (15) Where, in correspondence with particle mechanics the momentum 1T(r, t) conjugate to the field variable I/J(r, t) is defined by

7r(r,t)=8L/a~=ti1fl/Jt(r,t). (16) By defining the Hamiltonian of the matter field :JeW in analogy with the classical expression for H(t)

(17)

to give7

JC(t) = f (7r~ - il/J jar - .(t) , one obtains

;;C(t) = f {(1i 2/2m)Vl/Jt. VI/J + Vl/Jtl/J

+tl/Jt(r)f I/Jt(r')U(r, r')I/J(r')I/J(r)dr'}dr. (18)

USing these definitions the change in the quantum action integral may be re-expressed as [using as well, Eq. (10)],

(19)

By appealing to Schwinger's action prinCiple, we may, from Eq. (19), identify the operator &(t),

(20)

as the generator of all possible infinitesimal unitary transformations, both spatial and temporal. For a pure-ly temporal change, .:l1/J = 0 and the generator of the in-finitesimal unitary transformation caused by a displace-ment of the time is -fc(f)ot. From Eq. (6), the change in any operator a caused by such a infinitesimal unitary transformation is

oa = - (i/n)[fc(t), a]Oi = - dot, (21) which yields Heisenberg's equation of motion from the action principle

a. = (i/n) [;;C(t), d] . (22) For a purely spatial change, 01/J is arbitrary, ot= 0

and the generator in Eq. (20) is given by

s(t) = S (7rol/J- o7rl/J)dr. (23) This generator of spatial changes is composed of two independent operators, 3

8001= f 7rol/Jdr,

which transforms I/J into I/J + ol/J/2 and

SI>r =- f o7rl/Jdr, which transforms 7r into 7r + 07r/2. The commutation re-lationships are derived by conSidering the separate ac-tions of 9011 and \1tir on I/J and 7r, respectively. One has

and

1 ') [ , ) ~ ] 2tlfOl/J(r ,t = I/J(r , t , So.

O=[I/Jt(r', f), SOoI] for the action of SOil and

tinol/Jt(r', t) = [I/Jt(r', f), So.] and

0= [S6r' I/J(r', f)]

(24)

(25)

for the action of Stir. Schwinger3 has shown that ol/J and 07r commute with the field operators I/J and 7r except when they deSignate a field with half-integral spin, in which case they anticommute. Thus, from Eqs. (24) and (25), one obtains, respectively,

i drfl/J(r', f), 7r(r, t)l.ol/J(r, t) =tinol/J(r', t) , o

L dr07r(r, t)[I/J(r, f), 7r(r', t)]. =tm07r(r', f) , where the - and + subscripts denote commutator and anticommutators, respectively. As 01/J and 07r are arbi-trary, one derives the commutation relations,

[I/J(r', f), 7r(r, t)J. = hno(r' - r) . (26)

J. Chem. Phys., Vol. 68, No.8, 15 April 1978



Thus, from a sing' e variational principle, one obtains the equations of motion, the canonical equations and the commutation relations. We now address ourselves to the problem of applying the same principle to a subspace.

B. The subspace Lagrangian To determine the change in action over a subspace (0)

of the total system, one must construct a corresponding subspace Lagrangian (0), the variation of which yields the correct field equation, Eq. (14), and which reduces to the all space Lagrangian when the boundaries of (0)

are at infinity. This is accomplished by restricting the integration of the coordinate r in the all space Lagrang-ian, Eq. (13), to the region (0) and by the imposition of constraints on the variation. The constraints are an ex-pression of the fact that the definition of a subspace re-quires a partitioning of the potential energy of interac-tion between the subspace and the remainder of the sys-tem into contributions belonging to each component.

Consider the following subspace Lagrangian expressed in terms of the trial density cpt(r)cp(r),

(CP, 0, t) = f drHiJf(cptq> - 1> tcp) - (1f2/2m)Vcpt. Vcp}- f dr Vcptcp o 0

-{L drcpt(r) f dr'lj/(r')U(r, r')IjJ(r')cp(r) - Lo drljJt(r) f dr'ljJt(r')(- r' V'U(r, r'IjJ(r')IjJ(r)} . (27) In Eq. (27), IjJt(r) and ljJ(r) denote field variables which satisfy the field equation. They are not to be varied in the determination of ~W(O). Their appearance in Eq. (27) indicates the presence of constraints in the varia-tion of and on the value of the potential energy ariSing from the se If- interaction potential U.

In the variation of the all space t" Eq. (13), both ljJ(r) and ljJ(r') must be varied in the self- interaction term. The factor of t multiplying this term in will yield the correct contributions from the self-interac-tion term to both the field equation, Eq. (14) and to the field Hamiltonian, Eq. (18).8 Limiting the integration over r in this term to the region (0) in the case of a subspace, while yielding the required results for the field Hamiltonian, does not yield, upon variation, the result required for the field equation. This difficulty may be overcome by taking cognizance of the fact that in defining the energy of a subspace one must partition an energy of interaction of the subspace with the remainder of the system into separate contributions for both re-gions. This partitioning is accomplished through the use of the vi rial sharing operator1,2 - r' . V' which appears in the expression for t,(0) as given in Eq. (27). One may view the self-interaction of the field as one between the field to be varied, cpt(r)cp(r) and another IjJt(r')IjJ(r'), fixed by the field equation. The first of the self-inter-action term s in Eq. (27) states that cpt (r) and cp (r) are to be varied in the presence of a fixed repulsive poten-tial energy at r generated by

f dr' IjJt(r')U(r, r')IjJ(r') This variation yields the result required for obtaining the field equation from the variation of ~(O). Because of the identity

(-r V-r' V')U=U (28) which holds when U is derived from an inverse square force, one may interpret the operator - r' . V' U, appear-ing in the final term of Eq. (27) as one which determines the portion of the interaction U which belongs to the primed system. Thus this final term is a number which fixes the fraction of the self-interaction energy belong-

ing to the primed field. Its value is determined by de-manding that

L drljJt(r) (r V) J dr'/fi t(r')U(r, r' )1jJ(r')1/J(r) = i drljJt(r)f dr'1/Jt(r')(r'. V'U(r,r'))IjJ(r')1/J(r), (29a)

o

or equivalently

L drf(r)(r. V) fa. dr'/fit(r')U(r, r')IjJ(r')IjJ(r) = f dr/fit(r)i dr'ljJt(r')(r'. V'U(r, r'1/J(r')IjJ(r) (29b)

Cl 0'

where 0 and 0' denote the subspace and the remaining space of the total system, respectively. The condition given in Eq. (29) is a statement of the physical require-ment that the energy of interaction between identical particles, (or fields) one in 0 and the other in 0', be equally shared between them. Only if the restraint giv-en in Eq. (29) is satisfied do the separate portions of the same field in 0 and 0' exhibit identical average be-havior with respect to their interaction. 1

Because of the identity given in Eq. (28), the self-in-teraction terms in (0) reduce, at the point of variation, to the lhs of Eq. (30)

L dr IjJt(r)(_ r V) f dr' /fit(r')U(r, r')1/J(r')1/J(r) "'~t drljJt(r)f dr'1/Jt(r')U1/J(r')/fi(r). (30)

Finally, because of the constraint governing the sharing of the self-interaction energy between the subspaces 0 and 0' as given in Eq. (29), one obtains the identity giv-en in Eq. (30). When 0 refers to all space the rhs yields the correct expreSSion for the self-interaction energy as required for the field Hamiltonian. Thus (CP, 0) yields the field equation when varied, and .(CP,O) reduces to the all space Lagrangian integral when the boundaries of 0 occur at infinity.

In defining the energy of a subspace, one cannot make

J. Chern. Phys., Vol. 68, No.8, 15 April 1978



reference to an external potential. One must instead determine what fraction of the interaction of the source V with the system belongs to the system and what frac-tion belongs to the source. While such a partitioning of the contribution of V to (Ij;, n) is not required to obtain the field equations, (as it is in the case of the self-in-teraction energy), one may add the appropriate virial restraint to (Ij;, n) to obtain a field Hamiltonian which yields an expression for the energy of just the system rather than of the system and the source.

For an external potential V derived from an inverse square force, one may write2

(- r 1 V 1 +


(>It I 6 L VZp(r)drl>It) =0 (39b) for any variation of the field variables 1j/(r) and 1/J(r). That is, the only admisSible variations are those which satisfy Eq. (39). Therefore, in the variation of the field theoretic action integral, the variational constraint cor-responding to the zero flux condition on the gradient of p(r), may be restated as

6 L VZp(r)dr = 0 . (40) It is to be understood that Eq., (40) has phYSical meaning only when averaged over the state function >It(r, t) which determines the charge density and hence the surface.

The contribution of the surface variation to the change in action is given by the variation of the surface of ,c(0, t),

(41 )

where L is the Lagrangian density obtained by the sur-face variation: The surface of the subspace integral labelled by an 0 0, appearing in the constraint term of (0, t), Eq. (27), is not varied as it is determined by the correct field variables. Thus the Lagrangian den-sity appearing in Eq. (41), is expressible as

L=HiIit~-t[(-1/2/2m)V2- V- f dr'1/Jt(r') xU(r, r')1/J(r')(r)]}+c. c. - (hz/4m)Vz(t(r)=0 , (45) and hence Eq. (44) may be re-expressed as

~SW(O) = (1/z/4m) fZ i 6 (vzp(rdr dt . 11 0

(46)

Evaluation of the variation of the divergence term in Eq. (46), which contributes only a surface term, 2 and its addition to the result for ~DW(O), Eq. (34), yields

~W(O) ={ fo (1T~1j) - ~mp)dr - X(0)6t}I:: 1 lIZ A ~ }

-2" {s , (61/J, Vp n=0)+CP'(61/J, Vp. n=O) dt+c. c. 11 (47)

for the change in action for a subspace bounded by a surface satisfying the zero flux condition, Eq. (1). The surface integrals S' and It I Jet(O)1 >It) . Because of the zero flux boundary condition imposed on the subspace, the average value of its kinetic energy as determined by the averaging of the field Hamiltonian Jet(O) is the same as that obtained in the usual Schro-dinger representation. Thus, the definition of the ener-gy of a subspace of a total system, where the potential energy of the subspace is defined in terms of the virials of the forces exerted on it, may be obtained from the canonical formalism of field theory. 11

D. The subspace stationary action principle The expression for the change in action over a sub-

space, Eq. (47), is best interpreted using a four-dimen-



3686 Bader, Srebrenik, and Nguyen-Dang: Subspace Quantum dynamics_ II

y

FIG.!. Pictorial representation of the space-time develop-ment of a 2-D subspace (l(t) of a total system. The spacelike surfaces are denoted by (l(t), each being bounded by a surface S(r;t). The collection of the latter constitute a timelike sur-face, which, together with (l(t!) and (l(t2), defines the integra-tion volume in the subspace action integral w(fl). The prin-ciple of stationary action then states that the total change in wW) has only contributions from the space-time surface bounding that volume.

sional space-time reference system. ConSider, for simplicity, a system with two spatial dimensions. Take the time axis to be perpendicular to the spatial plane. The change of the system with time then generates a cylinder along the time axis (Fig. 1). The cylinder is capped by two "spacelike" surfaces, surfaces in which all points are defined by the single time tl or t2 The wall of the cylinder is a "time like" surface, the position of the wall being determined at each til11e t by the spatial boundary of the system.

For a system with boundaries at infinity, Schwinger's action principle, Eq. (3) or Eq. (15), states that the total change in action is equal to the difference in the values of the generator of a corresponding unitary tranS-formation in the two space like surfaces, that is, to the difference in the values of the spatial integral ~(t) eval-uated at tl and t2 The corresponding space like contri-butions appear in the expression for .6.'\~'(0), but the in-tegrations which determine the values of the generator are limited to the domain of the subspace in each of the space like surfaces. For a system with finite boundaries the total change in action includes a contribution from the timelike surface as well. That is, in the general case

.6.,\~, is determined by an integration over the total sur-face area of the cylinder swept out by the motion of the system through time. The integral over the timelike surface determines the change in action resulting from the infinitesimal transformations both in and on the spa-tial boundary of the system which occur between the times tl and t2.

In accordance with Schwinger's action principle, we identify the operator attached to the spacelike surfaces in the expression for .6.,\~, (0) as the generator ;J (t) of the

infinitesimal unitary transformation which causes the change in the quantized system. Thus, one obtains

(50)

which is identical to the all space result. Since the operator ff (t) generates an infinitesimal unitary trans-formation over the total system, the limits On the inte-grals appearing in Eq. (50) are not restricted to the re-gion (0) (which would imply incorrectly, that one per-forms a unitary transformation over the subspace alone). The integrations are so limited in the expres-sion for .6.,\\'(0), as the change in action is determined by the difference in the values of the generator for the system in question, namely one extending over (0), at the times tl and t2 Thus, the expression for the change in action is used, as it is for a system extending over all space, to define the generator of infinitesimal unitary transformations. The statement of the principle of sta-tionary action for a subspace bounded by a surface of zero flux in the gradient of p(r), Eq. (1), is as follows; the change in the subspace quantum action integral is the difference between the values of the generator of a uni-tary transformation as determined within those regions of the two space like surfaces defined by the boundary of the subspace at times tl and t2, plus an integral over the time like surface corresponding to the time integral of the change in the value of the generator as determined by the flux in its vector current across the spatial boundary of the subspace and by the change in the bound-ary with time.

E. Unitary transformations and subspace properties

Throughout this work, we have applied the calculus of variations in the evaluation of the changes in the sub-space energy or action2 caused by infinitesimal varia-tions in either the state fUnctions >J.!(r, t) and >J.!*(r, t), or in the field variables 1/i(r, t) and 1/it (r, t), which COnserve the zero flux condition. We noW focus our attention on changes in any subspace property induced by infinitesi-mal unitary transformations, changes which we shall ultimately require to equal those obtained through the use of the calculus of variations. The relationship be-tween the change in a subspace operator and the action of the generator of an infinitesimal unitary transforma-tion is determined by deriving the subspace analogues of the commutator expressions given in Eqs. (6) and (22).

This discussion will make use of the coordinate rep-resentation which is expressed in terms of the complete set of local state vectors

I q> = I r 1, r 2, ", rN> and their duals, with the operators 0 and ff: referring to observables and both expressible in terms of their ma-trix elements as

(ql&lq'>=o(q'7 V~O(q-q')=(q'10Iq>*, (51a) ( q I ff: I q' > = f (q, 7 v.) a (q - q') = ( q' I ff: I q> * (51b)




The operators o(q, (li/i)Vq ) and f(q, (1i/i)Vq ) are con-structed fz:om the position and momentum operators

q ={r t , r 2, ", rN}

(Ii/ i)V q = {~ V h ~ V 2, , ~ V N} l t l

acting on the delta function

o(q - q') = o(rt - r{)O(r2 - r~) ... o(rN - r~) so as to ensure the hermiticity of tJ and:t, respective-ly.

The changes induced in a state vector I t;) and an oper-ator 0, Eqs. (5) and (6), by an infinitesimal unitary transformation with generator :t as defined in Eq. (4) ensure that

In terms of the above representation, the identity stated in Eq. (52) can be rewritten in terms of averages over the wavefunctions t;(q) and t;*(q) as

f dq{ot;*(q)ot;(q) + t;*(q)o o (q)} = iE: f dq{t;*(q)[o, il t;(q)} , (53)

where

and

ot;(q) = id t;(q) , ot;*(q) = - iE:;*t;*(q) . When one coordinate among the set of N is selected for integration over a finite region 0, we shall write

I q) == I r t , r') . We are speCifically interested in finite regions which are bounded by a zero flux surface as defined in Eq. (1).

The subspace average of a property J

(6)0= L drtf' dr''lto'lt (54) can be written in terms of a projection operator n(o) as

(55) where ll(O) is defined as the sum of the local one-parti-cle operators rr(r) = Ir)(rl over 0:

1l(0) = i drtf' dr'lrhr')(rtor'l ={ drlr)(rl o 0

(56) and where

(57) By virtue of the identity in Eq. (52), one then has

(o( 'ltl )1l(O)e I 'It) =( 'ltl Il(O)O (0 I 'It)) =-iE:('ltI[:t,ii(o)o]I'lt), (58)

or, explicitly in the coordinate representation

10 drt r dr'{o'lt*O\J!' + 'It*8c5'lt} = - iE: So dr1 r dr'{f*'lt*~ - 'It*8i'lt} . (59)

From the calculus of variations as exemplified by the variation of V.'(O), the infinitesimal change in a sub-space operator 0(0) caused by a variation of the field variables IjJt(r) and ljJ(r) consists of the variation of its functional denSity 0(1jJt, 1jJ) over the domain of the sub-space and of the variation of the surface of the subspace. That is,

(60) which when averaged over the state in question yields the corresponding expression for the variation in a sub-space property

o( 6)0 = L dr1 f' dr'{o'lt*a'lt + 'It*oo'lt} + fdS(r1)f' dr'oS(r1){'lt*6'lt} . (61)

From a comparison of Eqs. (59) and (61) it is clear that the commutator relationship

(62) describes only the domain variation of 0(0) and of the subspace property (6) 0' This is understandable, as the transformation in Eq. (62) changes only the operator, leaving the state vector unaltered, and thus the boundary of the subspace is not varied.

Expanding the commutator in Eq. (62) yields OD0(0) = iE:n(O) [if, el + iE:[iF, IT (0)] 0. (63)

The separate averaging of the commutators in Eq. (63) demonstrates in a precise manner the important proper-ty of the domain variation of a subspace operator, name-ly that it consists of the simple projection of the sub-space contribution from the total transformation over all space,

('It I rr(o)[iF, ell 'It) = L dr1 f dr''lt*[r, a]'lt (64) plus a contribution which arises from the fact that f does not possess the property of hermiticity when averaged over a subspace,

('lti[iF, n(0)]61'lt) = fodrlfdr'{f*'lt*O'lt-'lt*ia'lt}. (65)

Because of the partitioning of oD6(0) given in Eq. (63), it is clear that if the property 0 is conserved over all space under the action of the generator ~, then the whole of 0D(t)O arises from the noncommutativity of if with n(O). For example, the number operator in the coordinate representation is NIl, where

ll=ll(O=RS)=l, and since the total number of particles is conserved over all space under any unitary transformation, one obtains from Eq. (63)

ODNIl(O) = iE:[&, Il(O)]N= OD~(O) . Thus the domain variation in a subspace population is given by

OD( it(O = - N( 'It I iE:[:t, n(o)] I 'It) = N L dr1 J' dr'{o'lt*'lt + 'It*o'lt} .



3688 Bader, Srebrenik, and Nguyen-Oang: Subspace quantum dynamics. II

In some specific cases the result of averaging the com-mutator of and n(O), the rhs of Eq. (65), may be transformed into an integral over the surface of the subspace. For example, when

one has

\ [

A A A \ 1f2 i S' - ('k JC, II(U)] t) 'k) = 2m rdS(rl) dr'

X{('Vl'k*)O'k- 'k*'v't(u'k)}. (66) The integral on the rhs of Eq. (66) is the surface inte-gral S'(6'k,Vp. n=O) as previously defined, 1,2 a quantity which represents the flux in the vector current density of the property () through the surface of the subspace. Since -:fcot is the generator of an infinitesimal temporal

,. change, one finds that the contribution to (0(U) from the temporal variation over the domain of the subspace is given by

(i/If)('k/rr(u)[:k,8}j'k)-S'(6'k,Vp. n=O). (67) If the results of the above transformations are to be

identical with those obtained through the calculus of var-iations as given in Eqs. (60) and (61), we must infer the existence of a surface variation term in both the average value and the operator expressions. That is,

= i( ('k/ [0(0), ] / 'k) +( 'k/ osIT(U) 0 / 'k) , and [compare with Eq. (60)],

o8(U)=iE:[,0(u)1-osn(u)8.

(68)

(69) The symbol osIT(O)O is to denote the surface variation resulting from the change in the domain of integration as caused by the unitary transformation. The quantity osll(U) denotes the variation in the projection operator Il(U) resulting from the change in the state vector l'k) and its dual as caused by the unitary transformation. We now proceed to derive its analytical expression for the case when the surface of U is one of zero flux in V p.

As noted above, the average of .Nii(0) in the state I 'k) gives the population of the subspace U, and the commutator iE:[, .Nii(U)] averages to the domain varia-tion of the population. Thus one must have

('k\ 0 sIT (0) \ 'k> = os( &(U) = ~ dS(r1 ) r dr' oS(rl)'k*'k , (70)

or equivalently

osll(U) = tdS(rl) f dr' oS(rl ) / rl> r') (rl> r' \ = tdS(r)os(r)\r)(r\ . (71)

Indeed, osr1(U) can be expressed as

OsIT(U)=IT(U')-fl:(U)= (Ir)dr(r\-i /r)dr(r/, (72) JOI 0

where U' is the volume bounded by the surface S such that

'tirES, Vp'(r)n(r)=O, (73) with

p' (r) = ( 'k' / r) ( r / 'k') , (74) where

IntrodUCing

I r') = {ell r) and its dual

( r' / = ( r I or I r) = = 'lil r') . We may now obtain an explicit formulation of the rhs

of Eq. (72), that is, of the variation of the surface in-duced by a unitary infinitesimal transformation on I >It). Performing the change in variable

r = r' + or

in the integral over U' and using

Ir'+or)(r'+or/ =V(!r)(rl)or+lr')(r'\, we obtain to first-order, and with oS(r) = or n(r) the result

osIT(U) = f dS(r)oS(r)I r) (r \ , (77) which is identical to the result given in Eq. (71), the form required to yield agreement with the calculus of variations. Therefore, the change induced in a surface of zero flux by an infinitesimal unitary transformation is of the precise form required by the calculus of vari-ations.

The utility afforded by the above formalism in the




treatment of subspace properties is illustrated by a derivation of the subspace expression for the stationary state variational principle,

IlS'(a'1/!, O)=H([:if, a'])o +c. c.} discussed in the preceding pape~ and derived original-ly by the calculus of variations. 1 The energy functional S' (1/!, 0) may be expressed in terms of the proj ection operator IT(O) as [compare with Eq. (8) of the preceding paper ],

S'(1/! 0)=~{(1/!1~(0):JC11/!) +c.c.} +tfo v2p(r)dr . , 2 (1/! III (U) I1/! > ( 1/!, 1/! > 0

Because of the variational constraint on the volume 0, Eq. (39a) and the zero flux surface condition, the varia-tion of (1/!, 1/! >0 S' is equal to the variation of the average value of :iC(0) = Hrr(o)fc+:JCIT(O)), [compare with Eq. (11) of the preceding paper]. Thus, with ei' = iE:a, consid-ered as the generator of an infinitesimal unitary trans-formation, one obtains, through the use of Eq. (68), s' (a1/!, O){iE: (1/!1 [rr(O), a]l1/! >

+(1/!lll s IT(O)I1/!>}+( 1/!1 IT(O)I1/!> IlS'(cl1/!, 0) =i( 1/!1[:JC(O),cl] I1/!) +H(1/!lll sIT(o)i;I1/!) +c. c.} .

Expanding the commutator on the rhs of the above equa-tio~ and recalling that 9' = E when 1/! is an eigenfunction of JC, one obtains

(1/!1 IT(o)I1/!) Il S'(cl1/!, 0) = HiE:( 1/!1 IT (O)[JC, a]I1/!) + c. c.} or equivalently,

Il S' (


tation of the state at t1> through a unitary transforma-tion, to a new representation at tao The principle as given by Schwinger is3

6( a', tll a", t2 ) = (i/1i) ( a', tll 6'~hl a", f2 ) , (84) which states that changes in the transformation function are all derived from variations of one operator, name-ly the action operator. It was also shown3 that the ac-tion operators and their variations obey an additive law of composition with respect to the defining times:

W13 =W12 +W23 and

6W13 = 6-\'\'12 + 6W23

Exploiting the above additive property, one may con-struct the action operatorW12 as the time integral of a Lagrangian (t).

A J. t 2 A 'W12 = df(t) .

tl

Equivalently, the Lagrangian (f) can be defined through Wt, t+dt = (t)dt . (85)

On the other hand, the possibility of a well-defined partitioning scheme {Ol (t); i = 1,2, ... , n} at any time t by the zero-flux condition, ensures the spatially additive property of the action:

and its variation

where

A J. t2 A \\'(01)12 = df(OI(t)) , tl

(01 (t being the proj ection of (t) on the subspace 0 1 (f).

(86)

Thus the action principle may be expressed as a sum over the change in the action operator for each sub-space in the total system,

or

~(a', tll a", t2 ) = L: A(OI; a'; t1; a", t2 ) , (88) I

where A(OI) is the contribution to the total change in the transformation function as determined by the change in action for a particular region 0 1 of real space.

Consider the transformation function linking two states which are separated by an infinitesimal time interval. The contribution to the total change in the transforma-tion function from a particular subspace 0 is

A(O; a't', a", t+dt) =( a', tl ~'\\'(O)t,t+dtl a", t+dt) , (89) which, by virtue of Eq. (85) may be expressed (for an infinitesimal time increment) as

A(O; a', t; a", f+ df) =( a', tl 6(0). dtl a", t+ dt) (90) Equating the two operators in Eqs. (89) and (90) and using the expression for ~'\t, (0) in Eq. (47) one obtains

6.(0) = tf(O, t)-H~'(61j!, Vp n=O) + 6>' (61j!, Vp n=O)+c. c.}. (91 )

In Eq. (91), &(0, f) denotes the time derivative of the generator evaluated over the subspace. For a purely spatial change with 6 t = 0, the generator reduces to S(t), Eq. (23) and its time derivative evaluated over the subspace is

(92)

To obtain the equivalent of the commutator expression for 6(0) as given in the preceding paper2

6(0)=-H([x, al>n+c.c.}, (93) one must substitute for the generator an operator corre-sponding to some particular property, multiplied by an infinitesimal,

(l/iIi)S(t)=a(t)=hS (Ij!tfilj!+fitlj!tlj!)dr (94) As in the classical case, 2 when the property a has no explicit time dependence its time derivative can be equated to its commutator with:!C. With these restric-tions the term ~(O, t) in Eq. (91) can be replaced by the time derivative of the subspace property a(o, f),

(95) and Eq. (91) becomes

6(0)=-[:iC, a(o)]-HS'(nlj!, Vp. n=O)+c.c.}. (96) Recalling [see Eq. (67)] that the commutator expression for the temporal variation over the domain of a subspace implicity includes the surface term - Hs' + C. c.}, Eq. (96) is the analogue of the subspace variational principle for (0),2 Eq. (93).

III. SUMMARY AND CONCLUSIONS It is possible to use the quantum action principle and

the concepts of field theory to both define and deter-mine the properties of a subspace of a total system. The construction of the subspace Lagrangian which yields the quantum field equations from the principle of stationary action and which is applicable as well in the limiting situation of all space, requires that one define the energy of the subspace. Such a definition in turn re-quires a partitioning of the energy of interaction of the subspace with the remainder of the system into a part belonging to the subspace and a part belonging to the remainder of the system. As previously discussed2 this is accomplished through the definition of a single-par-ticle potential energy, a concept which finds a natural expression in the language of field theory.

The restraint ariSing from the zero flux surface con-dition defines a subspace for which the principle of sta-tionary action assumes its simplest physical form. The change in action for a subspace must necessarily include a contribution from the timelike surface, which is phys-




ically interpretable as resulting from the infinitesimal flux in the current of the generator across the spatial boundary and from changes in the boundary of the sys-tem with time. Without the zero flux restraint there is an additional contribution to the change in action from the time like surface, namely,

(1I2/4m) f2\S i V'2(ljitlji)dr}dt. t1 l 0

This quantity is of arbitrary value and appears to have no physical meaning other than the one obtained by the requirement that it vanishes as it does in all space phys-ics. Finally, we note that the set of surfaces as defined in Eq. (1) by a property of the charge denSity, partitions a molecule into a collection of chemically identifiable atomiclike fragments. 2,14 These are the subspaces which are summed over in Eq. (87). Thus, the quantum action principle is expressible as a sum of the separate changes in action over each atom in a molecule.

ACKNOWLEDGMENTS We wish to thank Professor Y. Nogami and Professor

J. P. Carbotte for a number of valuable discussions concerning this work.

IS. Srebrenik and R. F. W. Bader, J. Chern. Phys. 63, 3945 (1975).

2S. Srebrenik, R. F. W. Bader, and T. T. Nguyen-Dang, J Chern. Phys. 68, 3667 (1978), preceding paper.

3J Schwinger, Phys. Rev. 82, 914 (1951). 4p. Roman, Advanced Quantum Theory (Addison-Wesley, Read-

ing, Mass., 1965), p. 37, p. 81. 5J Schwinger, Brandeis Summer Inst. Theoret. Phys. 2, 157

(1964). sIn deriving the principle of stationary action, 8chwinger3 was

able to demonstrate from general considerations of the prop-erties of infinitesimal unitary transformations that the change in action is given solely by an expression of the form of Eq. (3) or Eq. (15). A comparison of Eqs. (12) and (15) for .a.~, then shows that the equations of motion must be obtained as a

consequence of the principle of stationary action. TThere are two independent fields employed in the definition of

.c , Eq. (13). They are 1/i and 1/i t The momentum conjugate to 1/i is, by Eq. (16), rr=!ifi/2)1/it. Correspondingly, the momen-tum conjugate to 1/it is, again using Eq. (16) but this time dif-ferentiating with respect to ~t, rrt = - (ifi/2)1j:. Thus, the quan-tum analogue of pq for the two field case is rr~ -;1/i. Alterna-tively, one may consider simply adding the divergence term

-~d(l/JtljJ)/dt to a Lagrangian density defined for a single field, i. e., without the term -ifi~t1/i. This yields the Lagrangian de-fined in Eq. (13) without altering the equations of motion ob-tained from the variational principle.

8Roman4 defined a Lagrangian density for a system with inter-actions without the factor of ~ in the interaction term, claim-ing that it will yield the correct field equation by virtue of the Euler-Lagrange equations. The factor of ~ is then (rather arbitrarily) added to the interaction term in the definition of :!CW. However, in the interaction case, the Lagrangian den-sity is afunctional of 1/;. Thus, in using the Euler-Lagrange equations both 1/;(r) and 1/;(r') must be varied. Thus the factor of ~ must appear in the Lagrangian itself.

9p M. Morse and H. Feshbach, Methods of Mathematical PhYSics (McGraw-Hill, New York, 1953), Vol. I.

IOR.F. W. Bader and G. R. Runtz, Mol. Phys. 30, 117 (1975). lilt is assumed that the origin chosen to fix the value of the ex-

ternal constraint in Eq. (32) is the same as that required to satisfy the value of the constraint placed on the self-interac-tion energy, Eq. (29). In molecular systems described by the Born-Oppenheimer approximation this assumption is numeri-cally justified for Hartree-Fock wavefunctions if the subspace is bounded by a zero-flux surface. 1,12

128 8rebrenik and R. F. W. Bader, J. Chern. Phys. 61, 2536 (1974).

13The surface variation Os e(m arises from the dependence of the surface S(r) on the real charge distribution per) as de-s'cribed in Eqs. (36) and (38). A change in the state vector I >It), or in the operator per) induced by S, causes the shift

OS = 0 r' n, in the surface coordinates where, as before,

Ir+or) = (l+iS') Ir) =U I r) vr ES(r) We also note that the relationship between the transformed and the original surfaces, Eq. (76), remains formally un-changed when stated in terms of the operators per) and {I (r) =Up(r)U-I

I4R . F. W. Bader, Acc. Chern. Res. 8, 34 (1975).



Documents

The Journal of Chemical Physics Volume 68 Issue 8 1978 [Doi 10.1063%2F1.436226] Bader, Richard F. W.; Srebrenik, Shalom; Nguyen-Dang, T. Tung -- Subspace Quantum Dynamics and the Quantum