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An atomic population as the expectation value of a quantum observable
Volume 19 1, number
I,2
CHEMICAL
PHYSICS
LETTERS
27 March
1992
An atomic population as the expectation value of a quantum observable R.F.W.
Bader
and P.F. Zou
Department of Chemistry, McMaster University, Hamilton, Ontario, Canada L8S 4Ml Received
30 October
199 1; in final form 18 December
199 1
Dirac defines an observable to be a real dynamical variable with a complete set of eigenstates. It is shown that the density operatorb= Z, 6( i$ - r), is a quantum-mechanical observable whose expectation value is the particle density and that the integral form of this operator, the number operator fi, is also a quantum-mechanical observable whose expectation value is the average number of particles. The principle of stationary action defines the expectation value and the equation of motion for every observable. Using this principle it is demonstrated that an atomic population is the expectation value of the observable fi when p is the electron density operator. An atom and its population are defined in terms of experimentally measurable expectation values of the observables b and @.
1. Introduction One frequently encounters the statement that “an atomic population is not a quantum-mechanical observable”. This statement certainly applies to populations based upon a partitioning of the basis functions in the Hilbert space of the orbitals, as found in the Mulliken and other orbitally based definitions of atomic populations. It does not however, apply to populations determined by the integration of the electronic charge density over certain bounded regions of real space. To prove this assertion one must first show that the measurable electronic charge density is the expectation value of a quantum mechanical observable, the density operator. This is followed by the demonstration that an atomic population is in turn obtained as the expectation value over a quantum mechanical subsystem of the corresponding integral operator, the number operator, which is also a quantum observable.
a real dynamical variable whose eigenstates form a complete set. This definition is preceded by a discussion of how the result of the measurement of a real dynamical variable is one of its eigenvalues. Dirac states that while not all observables are measurable in practice, any quantity that can be measured is an observable. We show that the density operator is an observable whose expectation value for a given state is the measurable charge density, the expectation value being the average value obtained by the measurement of the observable a large number of times [ 11. The density of a classical particle can be expressed as )
p(r)&(r’-r)
(1)
where r’ is the position of the particle. Corresponding to this definition one has the quantum density operator [ 2 ] jkD(r)=d(i-r)
.
(2)
2. The density operator as a quantum-mechanical observable
The operator p is linear and is clearly Hermitian, as it is a real function of the position operator. To complete Dirac’s definition of an observable, it must be shown that (p), the eigenstates of p
Dirac [ 1 ] terms a linear Hermitian operator a real dynamical variable. He defines an observable to be
BIP) =PlP)
54
0009-2614/92/$
T
05.00 0 1992 Elsevier Science Publishers
(3) B.V. All rights reserved.
Volume 19 1, number I,2
form a complete set. This one can do by showing that the set { ]p)} satisfies the completeness relation for any state ) !P),
IV=
J
27 March 1992
CHEMICAL PHYSICS LETTERS
=
IP)~P(PIw,
(4)
J
dr’ (Ylr’)
(r’(d(r’-r)l
Y)
.
(10)
In terms of the wavefunction One sees from the definition of j? that the coordinate state I r, ) is one of its eigenstates and the corresponding eigenvalue is 6( rl - r ), j31rl )=6(+-r)
Ir, ) =6(r,
-r)
Ir, ) .
e(5)
Eq. ( 5 ) demonstrates that the density operator j.?and the position operator E have the same eigenstates { I r) }. The set of position eigenstates is continuous and complete with
J jr)
dr(rl=l
.
(6)
Therefore, the eigenstates { Ip) = I r) } of p form a complete set and the density operator fi is an observable. The same conclusions are reached for a many-particle system. The density operator in this case is BED(~)= C a(?, -r) I
,
A particle in the state I r,) is localized at the point rw Since I ro) is an eigenstate of p, the eigenvalue is 6( r. - r ), which is also the wavefunction Y(r) in the case of complete localization. This follows from (8)
For a system described by the general state vector I !P), the wavefunction is Y(r)=
(rl Y) .
and thus the expectation value of the density operator is the particle density at the point r p(r) = Y*(r)
(9)
In this system the particle is not localized and one can determine only the average value, the expectation value, of the density operator. This is
Y(r) = (
W(r)
(12)
I Y> .
In considering the many-particle case, wherein one obtains the corresponding result, the spin properties of the system are included by replacing ri with Xi= rpi, where si is the spin coordinate to yield p(r)=
J
d.q...dx,
(Yylx,...x,)
X C6(ri-r)(X1...XnIY), p(r)= X
3. The charge density as the expectation value of fi
Jdr’ Y*(r’)6(r’-r)Y(r’)
(11)
(7)
with the complete set of eigenstates { ]p) = (r,.., ri... rn) } and eigenvalues C $( ri- r).
Y(r)=(rIro)=6(r-r,).
(Ylp(r)]Y)=
J
(13)
dx ,... dx, Y(x,...x,)
1 d(ri-r)Y(X,...X,)
,
(14)
and the integrations are performed over the space and spin coordinates. McWeeny and Mizumo [ 31 have shown that when the operator @is generalized to 6( rj - r)6( ri - r' ), its expectation value becomes the spinless first-order density matrix. For a system of electrons, p(r) is the electronic charge density. The measurable [ 4 ] electronic charge density is therefore, the expectation value of the charge density operator, a quantum-mechanical observable. The Heisenberg equation of motion for the density operator yields the quantum equation of continuity for the charge density. That is, dp/dt= (i/h) ( Y[fi,,P] I Y) = -V-j(r)
,
(15)
where the vector current density is given by j(r) = (fi/2mi){ Y*VY-
YVY*}
.
(16)
The diagonal element of the commutator appearing in eq. ( 15) is given by
55
Volume 19 I, number
1,2
CHEMICAL
PHYSICS
(r’I[A,p]lr’)=-(fz2/2m){V:6(r’-r) +2[V,.d(r’-r)]*V,.},
(17)
and when this is averaged over the state function obtains, term for term, the result
1992
Since the coordinate eigenstates form a complete set, the number operator fi is a quantum-mechanical observable whose expectation value is N, the total number of particles in the system N= ( Ylfil Y) = j dr J dx, ... d.x, Y(x, ... X, )
Y*(r)Y(r))
- (P/2m){V2(
-2[VY*(r)*VY(r)+ =-V-j(r)
one
27 March
LETTERS
X x6( rr -r)
I
Y*(r)V’Y(r)]}
.
Y(x, ... x, ) .
(23)
(18) 5. Observables for a quantum subsystem
4. The number operator
To determine the average number of particles in a given region of space, the density operator fi must be transformed into an operator that counts the number of particles. The number operator for Q, a region of lR3, is fi(Q)=
jdrp(r)= R
CX~(?~). I
(19)
Associated with the operator xn( i) is the step function xn( r) Xn(r) = 1 =0
rrzR, otherwise
.
(20)
The number operator, like fi, is a real dynamical variable with a complete set of eigenstates. That m(Q) is real is readily established using the completeness relation for the position eigenstates,
=
tXn(e@l y> >
(21)
since the function xn( r), obtained after the action of its corresponding operator, can be moved within the brackets. When R equals R’, the operator is denoted simply by fl. In this case, fi has as its eigenstates the coordinate states with eigenvalues N, the number of particles. That is, Rlr l...rN)=
~~d~~)lr~...r~) (22)
56
It is again to Dirac [ 1] that one turns for the final step in the quantum definition of an atomic population, the definition of a quantum subsystem. Dirac demonstrated that the mathematical foundation of the analogy between classical and quantum mechanics resides in the correspondence of the infinitesimal contact transformations in the former mechanics with infinitesimal unitary transformations in the latter. Such infinitesimal transformations can be used to generate all possible changes in a system, classical or quantal, and it is on this basis that Schwinger formulated his principle of stationary action [ 5 1. Schwinger’s principle is obtained through a generalized variation of the action integral operator in which the quantum observables are introduced in the form of generators of infinitesimal unitary transformations which act at the time end-points. The principle states that s*,=e(t+e(t,,)
(24)
where the generator e is an infinitesimal times an observable, &=& This single dynamical principle yields not only the equation of motion, as is obtained as the sole result in Hamilton’s principle wherein S?$Z =O, but it defines in addition the observables, their equations of motion and the commutation relations. The corresponding statement of the principle for an infinitesimal time interval is given in terms of the Lagrange function operator [ 51, 9 t&=
(ie/fi) [fi, E] ,
(25)
which is a variational statement of the Heisenberg equation of motion for the observable l? Schwinger’s principle has been shown to apply to regions of lR3 which satisfy a particular boundary
Volume
191, number
CHEMICAL
1,2
PHYSICS
condition [ 6,7]. This condition demands that each region R be bounded by a surface S(r) through which there is a local zero flux in the gradient vector field of the electronic charge density, i.e. Vp(r)*n(r)=O,
Vr&(r)
.
(26)
The regional statement of the principle of stationary action given in eq. (25), expressed in the Schriidinger representation, is
+complex
conjugate)
.
(27)
The variation in d;p[ Y, Q] is caused by the action of the generator & on the wavefunction. The manner in which the commutator is averaged over the subsystem, or indeed the subsystem average of any operator, is determined by eq. (27 ). This equation applies to any region of space which satisfies the boundary condition given in eq. (26 ). This includes the total system as well as its subsystems. Thus eq. (27 ) extends the quantum description of system to its subsystems and it will be referred to as the extended principle of stationary action. The subsystem expectation value of an observable F, as defined by the extended principle, exhibits the property of additivity, i.e. (28) and the sum of the subsystem expectation values equals the expectation value for the total system. This result obtains for all properties, including those induced by the response of a system to applied electric or magnetic fields [ 8 1.
6. An electron population as the expectation value of an observable The definition of N(R), the average population for a subsystem, as the integral of p over the region Q, together with its equation of motion are obtained from the principle of stationary action with the observable fl set equal to the number operator fi. In this case the commutator [fi, fi] vanishes and one obtains the result that the variation in L!‘[ Y, Q] induced by the generator ( -ie/fi)# vanishes to yield
[61
27 March
LETTERS
SJZ[fiY, Q] =dN(Ll)/dt+ -
$ dS G(r).n(r)
(Wr)lWp(r)l=O,
which is an integrated of continuity, dN(Q)/dt=-
1992
(29) form of eq. ( 16 ), the equation
fdSD.(r)*n(r)-(%(r)/at)p(r)}. (30)
The temporal change in a subsystem population is equal to the flux in the vector current density through its surface, together with a contribution arising from the change in its surface with time. The definition of the average population and its equation of motion apply equally to the total system or to a quantum subsystem, as is true for all observables. The electronic charge density and its gradient vector field are measurable properties of a system [ 4,9]. A quantum subsystem and its average population are therefore, defined in terms of measurable expectation values of the observables p and I?.
7. Atoms in molecules The boundary condition for a quantum subsystem eq. (26), leads to an exhaustive and disjoint partitioning of the real space of a system. In the vast majority of systems, each of the resulting subsystems contains a single nucleus and they are identified with the chemical atom. Eq. (28 ) provides the basis for this identification for it recovers the operational basis of chemistry, that atoms and functional groupings of atoms contribute characteristic and measurable sets of properties to every system in which they occur. When the form of a group in real space - its distribution of charge - is transferable without change amongst systems, then its properties are transferable, as well as being additive. In such cases theory predicts the existence of group additivity schemes and it is at this limit of near perfect transferability that one can compare the predicted properties of atoms with the experimental measured values. It has been shown that theory recovers the observed additivity and transferability of molar volumes, moments, heats of formation and the related strain energies, and group polarizabilities of hydrocarbon molecules 57
Volume I9 1, number I,2
CHEMICAL PHYSICS LETTERS
[ 10,111. The recovery of the measured properties of atoms in molecules justifies the identification of the atoms of theory with the atoms of chemistry #I. The theory of atoms in molecules enables one to apply the theorems and principles governing the behaviour of quantum-mechanical observables to measurable objects in real space. Included amongst these observables are the density and number operators whose expectation values are the charge density and atomic population, respectively. Acknowledgement
The authors wish to thank Dr. P.L.A. Popelier and Mr. T.A. Keith for useful discussions concerning this work.
#’ In certain systems one finds quantum subsystems without nuclear attractors. This occurs when there is mobile, loosely bound electronic charge density, as found in first-row metals, for example [ 121. Such non-nuclear attractors are found to link the Si atoms in crystalline silicon, a semiconductor, but not the carbon atoms in the diamond, an insulator with the same crystal structure [ 131. The observance of non-nuclear attractors does not constitute an exception to the theory. Theory predicts that the properties of any system are determined by the sum of its subsystem contributions, eq. (28). It is to be anticipated that the observation of unusual properties will, in certain instances, be a consequence of the presence of unusual topological properties in the charge distribution, relationships between form and properties that are brought to the fore by theory.
58
27 March 1992
References [ I] P.A.M. Dirac, The principles of quantum mechanics (Oxford Univ. Press, Oxford, 1958). [2] H.C. Longuet-Higgins, Proc. Roy. Sot. A 235 (1956) 537. [ 31 R. McWeeny and Y. Mizuno, Proc. Roy. Sot. A 259 ( 196 1) 554. [4] P. Coppens, J. Phys. Chem. 93 (1989) 7979; R. Destro, R. Bianchi, C. Gatti and F. Merati, Chem. Phys. Letters 186 ( 1991) 47; C. Gatti, R. Bianchi, R. Destro and F. Merati, J. Mol. Struct. (THEOCHEM) (1991), toappear; M. Kappkhan, V.G. Tsirel’son and R.P. Ozerov, Dokl. Phys. Chem. 303 (1989) 1025; R.F. Stewart, in: The application of charge density research to chemistry and drug design, eds. G.A. Jeffrey and J.F. Piniella (Plenum Press, New York, 199 1) 63; W.T. Klooster, S. Swaminathan, R. Nanni and B.M. Craven, Acta. Cryst. ( 199 1) , to appear. [5] J. Schwinger, Phys. Rev. 82 (1951) 914. [6] R.F.W. Bader and T.T. Nguyen-Dang, Advan. Quantum Chem. 14 (1981) 63. [7] R.F.W. Bader, Atoms in molecules: a quantum theory (Oxford Univ. Press, Oxford, 1990); Chem. Rev. 9 1 ( 1991) 893. [S] R.F.W. Bader, J. Chem. Phys. 91 (1989) 6989. (91 R.F. Stewart, Chem. Phys. Letters 65 (1979) 335. [lO]K.B.Wiberg,R.F.W.BaderandC.D.H.Lau,J.Am.Chem. Sot. 109 (1987) 1001; R.F.W. Bader, A. Larouche, C. Gatti, M.T. Carroll, P.J. MacDougall and K.B. Wiberg, J. Chem. Phys. 87 ( 1987) 1142; R.F.W. Bader, M.T. Carroll, J.R. Cheeseman and C. Chang, J. Am. Chem. Sot. 109 (1987) 7968. [ll]R.F.W. Bader, T.A. Keith, K.M. Gough and K.E. Laidig, Mol. Phys. ( 1991), to appear. [ 121 C. Gatti, P. Fantucci and C. Pacchioni, Theoret. Chim. Acta 72 (1987) 433; W.L. Cao, C. Gatti, P.J. MacDougall and R.F.W. Bader, Chem. Phys. Letters 141 (1987) 380. [ 131 P.F. Zou and R.F.W. Bader, unpublished work.
I,2
CHEMICAL
PHYSICS
LETTERS
27 March
1992
An atomic population as the expectation value of a quantum observable R.F.W.
Bader
and P.F. Zou
Department of Chemistry, McMaster University, Hamilton, Ontario, Canada L8S 4Ml Received
30 October
199 1; in final form 18 December
199 1
Dirac defines an observable to be a real dynamical variable with a complete set of eigenstates. It is shown that the density operatorb= Z, 6( i$ - r), is a quantum-mechanical observable whose expectation value is the particle density and that the integral form of this operator, the number operator fi, is also a quantum-mechanical observable whose expectation value is the average number of particles. The principle of stationary action defines the expectation value and the equation of motion for every observable. Using this principle it is demonstrated that an atomic population is the expectation value of the observable fi when p is the electron density operator. An atom and its population are defined in terms of experimentally measurable expectation values of the observables b and @.
1. Introduction One frequently encounters the statement that “an atomic population is not a quantum-mechanical observable”. This statement certainly applies to populations based upon a partitioning of the basis functions in the Hilbert space of the orbitals, as found in the Mulliken and other orbitally based definitions of atomic populations. It does not however, apply to populations determined by the integration of the electronic charge density over certain bounded regions of real space. To prove this assertion one must first show that the measurable electronic charge density is the expectation value of a quantum mechanical observable, the density operator. This is followed by the demonstration that an atomic population is in turn obtained as the expectation value over a quantum mechanical subsystem of the corresponding integral operator, the number operator, which is also a quantum observable.
a real dynamical variable whose eigenstates form a complete set. This definition is preceded by a discussion of how the result of the measurement of a real dynamical variable is one of its eigenvalues. Dirac states that while not all observables are measurable in practice, any quantity that can be measured is an observable. We show that the density operator is an observable whose expectation value for a given state is the measurable charge density, the expectation value being the average value obtained by the measurement of the observable a large number of times [ 11. The density of a classical particle can be expressed as )
p(r)&(r’-r)
(1)
where r’ is the position of the particle. Corresponding to this definition one has the quantum density operator [ 2 ] jkD(r)=d(i-r)
.
(2)
2. The density operator as a quantum-mechanical observable
The operator p is linear and is clearly Hermitian, as it is a real function of the position operator. To complete Dirac’s definition of an observable, it must be shown that (p), the eigenstates of p
Dirac [ 1 ] terms a linear Hermitian operator a real dynamical variable. He defines an observable to be
BIP) =PlP)
54
0009-2614/92/$
T
05.00 0 1992 Elsevier Science Publishers
(3) B.V. All rights reserved.
Volume 19 1, number I,2
form a complete set. This one can do by showing that the set { ]p)} satisfies the completeness relation for any state ) !P),
IV=
J
27 March 1992
CHEMICAL PHYSICS LETTERS
=
IP)~P(PIw,
(4)
J
dr’ (Ylr’)
(r’(d(r’-r)l
Y)
.
(10)
In terms of the wavefunction One sees from the definition of j? that the coordinate state I r, ) is one of its eigenstates and the corresponding eigenvalue is 6( rl - r ), j31rl )=6(+-r)
Ir, ) =6(r,
-r)
Ir, ) .
e(5)
Eq. ( 5 ) demonstrates that the density operator j.?and the position operator E have the same eigenstates { I r) }. The set of position eigenstates is continuous and complete with
J jr)
dr(rl=l
.
(6)
Therefore, the eigenstates { Ip) = I r) } of p form a complete set and the density operator fi is an observable. The same conclusions are reached for a many-particle system. The density operator in this case is BED(~)= C a(?, -r) I
,
A particle in the state I r,) is localized at the point rw Since I ro) is an eigenstate of p, the eigenvalue is 6( r. - r ), which is also the wavefunction Y(r) in the case of complete localization. This follows from (8)
For a system described by the general state vector I !P), the wavefunction is Y(r)=
(rl Y) .
and thus the expectation value of the density operator is the particle density at the point r p(r) = Y*(r)
(9)
In this system the particle is not localized and one can determine only the average value, the expectation value, of the density operator. This is
Y(r) = (
W(r)
(12)
I Y> .
In considering the many-particle case, wherein one obtains the corresponding result, the spin properties of the system are included by replacing ri with Xi= rpi, where si is the spin coordinate to yield p(r)=
J
d.q...dx,
(Yylx,...x,)
X C6(ri-r)(X1...XnIY), p(r)= X
3. The charge density as the expectation value of fi
Jdr’ Y*(r’)6(r’-r)Y(r’)
(11)
(7)
with the complete set of eigenstates { ]p) = (r,.., ri... rn) } and eigenvalues C $( ri- r).
Y(r)=(rIro)=6(r-r,).
(Ylp(r)]Y)=
J
(13)
dx ,... dx, Y(x,...x,)
1 d(ri-r)Y(X,...X,)
,
(14)
and the integrations are performed over the space and spin coordinates. McWeeny and Mizumo [ 31 have shown that when the operator @is generalized to 6( rj - r)6( ri - r' ), its expectation value becomes the spinless first-order density matrix. For a system of electrons, p(r) is the electronic charge density. The measurable [ 4 ] electronic charge density is therefore, the expectation value of the charge density operator, a quantum-mechanical observable. The Heisenberg equation of motion for the density operator yields the quantum equation of continuity for the charge density. That is, dp/dt= (i/h) ( Y[fi,,P] I Y) = -V-j(r)
,
(15)
where the vector current density is given by j(r) = (fi/2mi){ Y*VY-
YVY*}
.
(16)
The diagonal element of the commutator appearing in eq. ( 15) is given by
55
Volume 19 I, number
1,2
CHEMICAL
PHYSICS
(r’I[A,p]lr’)=-(fz2/2m){V:6(r’-r) +2[V,.d(r’-r)]*V,.},
(17)
and when this is averaged over the state function obtains, term for term, the result
1992
Since the coordinate eigenstates form a complete set, the number operator fi is a quantum-mechanical observable whose expectation value is N, the total number of particles in the system N= ( Ylfil Y) = j dr J dx, ... d.x, Y(x, ... X, )
Y*(r)Y(r))
- (P/2m){V2(
-2[VY*(r)*VY(r)+ =-V-j(r)
one
27 March
LETTERS
X x6( rr -r)
I
Y*(r)V’Y(r)]}
.
Y(x, ... x, ) .
(23)
(18) 5. Observables for a quantum subsystem
4. The number operator
To determine the average number of particles in a given region of space, the density operator fi must be transformed into an operator that counts the number of particles. The number operator for Q, a region of lR3, is fi(Q)=
jdrp(r)= R
CX~(?~). I
(19)
Associated with the operator xn( i) is the step function xn( r) Xn(r) = 1 =0
rrzR, otherwise
.
(20)
The number operator, like fi, is a real dynamical variable with a complete set of eigenstates. That m(Q) is real is readily established using the completeness relation for the position eigenstates,
=
tXn(e@l y> >
(21)
since the function xn( r), obtained after the action of its corresponding operator, can be moved within the brackets. When R equals R’, the operator is denoted simply by fl. In this case, fi has as its eigenstates the coordinate states with eigenvalues N, the number of particles. That is, Rlr l...rN)=
~~d~~)lr~...r~) (22)
56
It is again to Dirac [ 1] that one turns for the final step in the quantum definition of an atomic population, the definition of a quantum subsystem. Dirac demonstrated that the mathematical foundation of the analogy between classical and quantum mechanics resides in the correspondence of the infinitesimal contact transformations in the former mechanics with infinitesimal unitary transformations in the latter. Such infinitesimal transformations can be used to generate all possible changes in a system, classical or quantal, and it is on this basis that Schwinger formulated his principle of stationary action [ 5 1. Schwinger’s principle is obtained through a generalized variation of the action integral operator in which the quantum observables are introduced in the form of generators of infinitesimal unitary transformations which act at the time end-points. The principle states that s*,=e(t+e(t,,)
(24)
where the generator e is an infinitesimal times an observable, &=& This single dynamical principle yields not only the equation of motion, as is obtained as the sole result in Hamilton’s principle wherein S?$Z =O, but it defines in addition the observables, their equations of motion and the commutation relations. The corresponding statement of the principle for an infinitesimal time interval is given in terms of the Lagrange function operator [ 51, 9 t&=
(ie/fi) [fi, E] ,
(25)
which is a variational statement of the Heisenberg equation of motion for the observable l? Schwinger’s principle has been shown to apply to regions of lR3 which satisfy a particular boundary
Volume
191, number
CHEMICAL
1,2
PHYSICS
condition [ 6,7]. This condition demands that each region R be bounded by a surface S(r) through which there is a local zero flux in the gradient vector field of the electronic charge density, i.e. Vp(r)*n(r)=O,
Vr&(r)
.
(26)
The regional statement of the principle of stationary action given in eq. (25), expressed in the Schriidinger representation, is
+complex
conjugate)
.
(27)
The variation in d;p[ Y, Q] is caused by the action of the generator & on the wavefunction. The manner in which the commutator is averaged over the subsystem, or indeed the subsystem average of any operator, is determined by eq. (27 ). This equation applies to any region of space which satisfies the boundary condition given in eq. (26 ). This includes the total system as well as its subsystems. Thus eq. (27 ) extends the quantum description of system to its subsystems and it will be referred to as the extended principle of stationary action. The subsystem expectation value of an observable F, as defined by the extended principle, exhibits the property of additivity, i.e. (28) and the sum of the subsystem expectation values equals the expectation value for the total system. This result obtains for all properties, including those induced by the response of a system to applied electric or magnetic fields [ 8 1.
6. An electron population as the expectation value of an observable The definition of N(R), the average population for a subsystem, as the integral of p over the region Q, together with its equation of motion are obtained from the principle of stationary action with the observable fl set equal to the number operator fi. In this case the commutator [fi, fi] vanishes and one obtains the result that the variation in L!‘[ Y, Q] induced by the generator ( -ie/fi)# vanishes to yield
[61
27 March
LETTERS
SJZ[fiY, Q] =dN(Ll)/dt+ -
$ dS G(r).n(r)
(Wr)lWp(r)l=O,
which is an integrated of continuity, dN(Q)/dt=-
1992
(29) form of eq. ( 16 ), the equation
fdSD.(r)*n(r)-(%(r)/at)p(r)}. (30)
The temporal change in a subsystem population is equal to the flux in the vector current density through its surface, together with a contribution arising from the change in its surface with time. The definition of the average population and its equation of motion apply equally to the total system or to a quantum subsystem, as is true for all observables. The electronic charge density and its gradient vector field are measurable properties of a system [ 4,9]. A quantum subsystem and its average population are therefore, defined in terms of measurable expectation values of the observables p and I?.
7. Atoms in molecules The boundary condition for a quantum subsystem eq. (26), leads to an exhaustive and disjoint partitioning of the real space of a system. In the vast majority of systems, each of the resulting subsystems contains a single nucleus and they are identified with the chemical atom. Eq. (28 ) provides the basis for this identification for it recovers the operational basis of chemistry, that atoms and functional groupings of atoms contribute characteristic and measurable sets of properties to every system in which they occur. When the form of a group in real space - its distribution of charge - is transferable without change amongst systems, then its properties are transferable, as well as being additive. In such cases theory predicts the existence of group additivity schemes and it is at this limit of near perfect transferability that one can compare the predicted properties of atoms with the experimental measured values. It has been shown that theory recovers the observed additivity and transferability of molar volumes, moments, heats of formation and the related strain energies, and group polarizabilities of hydrocarbon molecules 57
Volume I9 1, number I,2
CHEMICAL PHYSICS LETTERS
[ 10,111. The recovery of the measured properties of atoms in molecules justifies the identification of the atoms of theory with the atoms of chemistry #I. The theory of atoms in molecules enables one to apply the theorems and principles governing the behaviour of quantum-mechanical observables to measurable objects in real space. Included amongst these observables are the density and number operators whose expectation values are the charge density and atomic population, respectively. Acknowledgement
The authors wish to thank Dr. P.L.A. Popelier and Mr. T.A. Keith for useful discussions concerning this work.
#’ In certain systems one finds quantum subsystems without nuclear attractors. This occurs when there is mobile, loosely bound electronic charge density, as found in first-row metals, for example [ 121. Such non-nuclear attractors are found to link the Si atoms in crystalline silicon, a semiconductor, but not the carbon atoms in the diamond, an insulator with the same crystal structure [ 131. The observance of non-nuclear attractors does not constitute an exception to the theory. Theory predicts that the properties of any system are determined by the sum of its subsystem contributions, eq. (28). It is to be anticipated that the observation of unusual properties will, in certain instances, be a consequence of the presence of unusual topological properties in the charge distribution, relationships between form and properties that are brought to the fore by theory.
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