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Time correlation of localized excitations in extended molecular systems: A self-consistent field treatment
Volume 162, number 4,5
CHEMICAL PHYSICS LETTERS
20 October 1989
TIME CORRELATION OF LOCALIZED EXCITATIONS IN EXTENDED MOLECULAR SYSTEMS: A SELF-CONSISTENT FIELD TREATMENT * David A. MICHA and D. SRIVASTAVA’ Departments of Chemistry and Physics, UniversityofFlorida, Gainesville, FL. 32611, USA Received 22 May 1989; in final form 7 August 1989
A timedependent self-consistent field approach is developed for the dynamics of a local excitation coupled to its molecular environment, starting from the time-correlation function of transition operators. A variational functional is used to derive a factorized TCF with the correct normalization and phase factor. We also show what conditions are sufficient to justify the assumption of localized dynamics.
1. Introduction When a particle interacts with an extended molecular system, such as an adsorbate or a polymer, the system is usually excited in a localized region, made up of a set of bonds, or a group of atoms. During its excitation, the localized region evolves in time while strongly coupled to its environment, under the influence of effective forces that depend on the thermodynamical state of the system. Energy absorbed during the interaction leads to rearrangement or break-up in the localized region, and dissipates into its environment. Here we present a general formalism in the Liouville space of operators, to quantitatively characterize the localized region. We derive equations of motion for operators of the system, from which the localized dynamics of the system can be calculate& and consider what physical conditions justify a separate treatment of the localized dynamics. Extended systems can be conveniently described with time-correlation functions (TCFs) of operators for different processes, such as dipole TCFs for molecular spectra, atomic density fluctuation TCFs for neutron inelastic scattering, and flux TCFs for rate coefficients [l-4]. We base our treatment on time* Work partly supported by NSF Grant CHE-8615334. ’ Present address: Chemistry Department, Pennsylvania State University, University Park, PA 16802, USA.
376
correlation functions of transition operators, which contain many other TCFs as special cases [ 5 1. We also use the formalism of operators in Liouville space to obtain the TCFs under very general conditions [6]. This approach accounts for the thermodynamical state of the system in terms of the statistical averages implicit in the TCFs, and can be applied even when a system is not in a pure quantum state. We divide the system into a primary region, containing the localized part that is excited and undergoes rearrangement, and a remaining, secondary region which serves as the environment and is perturbed only through coupling with the primary region. Our treatment introduces a self-consistent field approximation, to include strong couplings between the primary and the secondary regions. Special care must however be taken within this approximation to obtain correct phase factors when constructing the corresponding TCFs. These factors make a difference when one takes the Fourier transforms of TCFs, that are related to measured properties. We derive the approximation from a general time-dependent variational functional for operators, that leads to the correct phases and normalization factors and can be used in applications with different trial operators. Along the way we find what conditions are sufficient to justify the assumption of localized dynamics. The time-dependent SCF approximation developed here has several aspects in common with recent
0 009-2614/89/$ 03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
20
CHEMICAL PHYSICS LETTERS
Volume 162, number 43
treatments of intramolecular dynamics [ 7-9 1. However, it is more general in that it is applicable to systems in mixed (statistical) states as well as systems in pure quanta1 states. It also includes special cases of scattering by surfaces treated in the literature [ lo].
October1989
obtain a self-consistent field (or mean field) approximation. We consider a system for which the primary degrees of freedom are designated by x, and the secondary ones by y. The Hamiltonian is given by H=H,+H,+H,,
2. Variational derivation of operatorequations We work with the TCF of the transition operator A = T,, where k and k’ are the wavevectors of the incoming and outgoing particles. This is an operator in the variables of the target, evolving in time in aG cordance with the target Hamiltonian H. We introduce the Liouville operator 2, where &‘A-HA-AH, and set fz= 1, so that A(t) satisfies the equation of motion for operators in the Heisenberg picture, -idA/dt=XA(t).
(1)
(6)
with a corresponding decomposition for X. Next we specify conditions that lead to the time-dependent self-consistent field (SCF) equations. Firstly, the initial statistical state of the system is given by a density operator (7)
P=PxPy ,
where the two factors are determined self-consistently, and they are normalized by the trace relations tr,(p,) =tr,(p,) = 1. Hence the factor px satisfies the equation
The TCF of A is given by
F(t)=
(2)
(8)
where (( ( ) )) = tr [ ( )p], in terms of the trace and the density operator p of the target. To obtain approximate equations of motion for A, we construct the variational functional 12 -6; dt ((A(t)+ (id/dt+X)A(t))} +c.c. , U II >
and similarly for py’ Secondly we choose as a trial operator A the factored form
(3) where C.C.means the complex conjugate of the preceding term. This functional must be varied under the constraint U(r)+‘Jt(r)>)
=JlrU)=F(O)
I
(4)
which is a constant fixed at the initial time t, =O. This can be imposed introducing the Lagrange multiplier A(t) and the functional dtk(t)
(9)
where V and V are primary and secondary region operators respectively. This assumes that at the initial time t, the operator A can also be written in a factored form. In what follows we drop the subindices in U and V. The functional J depends on V, V, and their adjoints. Introducing the independent variations i!XJ+,6Vt, the condition &J=O for
6V=6V=O
(10)
gives the coupled equations (( Vt(id/dt+&+lZ)UV}),=O, (( Vt(id/dt+Rt2)UV)),=0,
12 /=S+s
A(r) = VA) V.(t) ,
J’-(t) ,
(5)
II
which must satisfy Sj=O for variations 64 (t) = Ea,B&I, I a> (81, where 1a) is an internal state, around the exact values of A(t). Variations 6A( t)f can be assumed to be independent of &4( t ). The procedure just described can be followed to
(11)
where the subindices indicate the degrees of freedom that are not averaged. Next we define the time-dependent operators in Liouville space
~:[V(t)l=~++(v+~~V})x/~v+V~
I
~:[V(t)l=JE”++u+~~~~,l~U+U}) 3
(12)
and the averages 377
CHEMICALPHYSICSLETTERS
Volume 162,number 4,5
20 October 1989
a(t)=((Ut(id/df+X,)U))/((UtU)),
u(t)=a(t)l(~a+a))Y
B(t)=((v+(idldt+~Y,)Y))/((V+V)), Y(t)=((Y’Ut~,UV))/(((V+V))cu+u))),
u(t)
3
=b(t)l> :“,
(19)
in terms of which we find from eqs. (2) and (17), and with tl = 0, (13)
and also make the transformations
t
I U(t)=a(t)
exp i U
xexp dt’ [A(t’)+/I(t’)]
t1
,
>
y(t)
t
V(t)=b(t)exp
i
(I ‘I
dt’ [A(t’)+cu(t’)]
, )
(14)
to obtain from eq. ( 11) the self-consistently coupled differential equations of motion ida/dt+Y1[b]a(t)=O, idb/dt+YY[u] b(t) =O.
(15)
To impose the normalization condition we note that
=
{{
(I
i
dt’Rey(t’)
u+u+xvuu))
, )
0
I .
(20)
This expression provides the correct normalization and phase factor for a TCF in the self-consistent field approximation. The two double brackets in eq. (20) represent reduced primary and secondary TCFs. The operators in eq. ( 12 ) are in general non-Hermitian in Liouville space, from which it follows that the norms of a(l) and b(t) can change with time. Furthermore, in some calculations it is convenient to introduce complex times; hence the denominators in eq. ( 19) are not necessarily constant, and must be included in the equations.
A(t)=a(t)b(t)
(’
Xexp 1 jdi. 11
Izn(l~)+ol(l’)+B(t’)1).
(16)
To simplify the exponent here, we multiply the first eq. ( 11) times pJJt on the left and take the trace, do the same with p,,Vt in the second equation, and add the results to obtain a+P+y+L=O.
(17)
The normalization condition of eq. (4) leads then to {>l{(btb>>,
3. Discussion The content of the previous equations can be understood within a simple model in which the primary region is described by a general Hamiltonian H, and the secondary region contains a single degree of freedom y oscillating harmonically around its equilibrium value y,, with mass and force constants chosen equal to unity. To lowest order in the displacement q=y-ye, the full Hamiitonian with a linear coupling is (21)
H=H,+f($+q*)-flX)q, xexp
- dt’2[Iml(t’)-Imy( (I II
>
=F(O>, (18)
where F( 0) = {( i?U)),((
V l~‘)}~_ This expression, together with the choice Rel=O, allows us to eliminate the Lagrange multiplier from the following expressions. To obtain the TCF, we define the normalized operators 378
where p is the momentum conjugated to q, and f(n ) is a general force. From eq. (8) we find K=H,-
<4)lf(x)
~Y=1w+~2b-uw
, (22)
The density operator px at temperature T is given by Px=exp( -FJkJ)/tr[exp( -F,/kJ’)], with a similar expression for pY;hence pYis the density OPerator of a linearly driven harmonic oscillator and
CHEMICAL
Volume 162, number 43
{(4)) is not zero. For the operator A we consider the dipole D(x, y) of the system, which to lowest order in q can be chosen as D(-%y)=&V,,
l$=1,+4/1,
<(u(t)+u(t)f(x)>?
t) =QI(X,0 V(Y,t) exp[ia(t)1 ,
%&Y,
1989
(25)
where 4 and w satisfy TDSCF equations [ 8,9 ] with effective Hamiltonians F,(t) and F,(t) and
(23)
where D, is the dipole operator of the primary region, 1, is the unit operator of the secondary region and I is a constant. Eq. ( 15 ) must be solved for a( t ) and b(t) with the initial conditions a(O) =O, and b(0) = V,. The equation for b(t) can be written in terms of the Hamiltonian for a linearly driven harmonic oscillator, for which the time evolution operator has a simple expression, and contains a nonHermitian potential. For the phase in eq. (20) one finds
+<(v(t)‘v(t)q>
20 October
PHYSICS LETTERS
(24)
where we have used that HXY=-f(x)q. The real part of this r(t) is not zero. Hence F(t) is not just the product of the correlation function of the dipole in the primary region times a secondary region correlation function, but it includes in addition a phase factor that will affect its Fourier transform. This example illustrates the sufficient conditions under which the dynamics of the system can be assumed to occur only in the localized region. These conditions are given by ( 1) a factorizable initial density operator, eq. (7 ), and (2) a transition operator A which at the initial and later times is of the form A= U,V,, with V, close to the unit operator. These conditions provide a quantitative definition of the localized dynamics. In a process characterized by a transition operator A, given for example by a dipole or a density fluctuation, one can analyze the initial form of A to determine what variables does it depend upon, such as certain bond coordinates or atomic positions. The primary region must then contain at least those variables. It follows from this that the primary region must be defined differently for different probes of the whole system. We can next analyze the connection with time-dependent SCF approximations, which can be made for wavefunctions or for transition amplitudes. Factoring only the time-dependent wavefunction Y’(x,y, t) of the system into components for the two regions gives
a(t)=
I
dl’ W(t’),
w~)=
II
(26) The TCF, now given by (27)
F(r)= { yo IA(t)+A(O) I %v,>>
with !?‘ou,= Y( t= 0), does not factorize except if one assumes in addition that A(O)= U,(O) V,(O) and HZZHsCF= F,+ F,,,- W. Furthermore, the TCF does not display the correct phase factor coming from the normalization constraint. An alternative is to define a time-dependent amplitude given by a product of the excitation operator times the wavefunction, r (t) =A( t) PO.Then provided the eigenenergy E, of the initial state is known, one can write -iX/&=(H-&)T(t)
(28)
and can introduce the TDSCF approximation, as done in some of the treatments with wavepackets [ 111 and path integrals [ 121, by writing rscF(x, Y, t) =5(x, t) rt(y, 0 exp!iX(t) I. This leads to the desired factorization of the TCF since F(t) = Wt)
Ino) > 5% (T(Oliv)>
(v(t)lr(O)>
x
exp [ix(i)] where the initial state has been approximated by a time-independent SCF wavefunction. The present Liouville formalism is however more general, because it does not require knowledge of the eigenstates and eigenenergies of the extended system at the initial time, that must usually be approximated. An additional advantage of the present formalism is that it consistently incorporates the temperature in the TDSCF dynamics. This is to be compared with the situation arising at finite temperatures with the TDSCF approximation for wavefunctions. Here ground and excited states are separately approximated, averages are calculated and then weighted by the statistical distributions of initial states. This can lead to unphysical temperature dependences if the errors in the SCF approximation are different for low and high excited states. The present formalism instead starts with the correct statistical density and 379
Volume 162, number 4,5
CHEMICAL PHYSICS LETTERS
the temperatures in the effective Liouville operators in eq. (12) so that the SCF approximation maintains the temperature balance. incorporates
References [ 1 ] L. Van Hove, Phys. Rev. 95 (1954) 249. [ 21 R.G. Gordon, Advan. Magn. Resort. 3 ( 1968) I. [ 31 B.J. Beme, in: Physical chemistry. An advanced treatise, Vol. 8B, ed. D. Henderson (Academic Press, New York, I971 ) ch. 9.
380
20 October 1989
[4] W.H. Miller, SD. Schwartz and J.W. Tromp, I. Chem. Phys. 79 (1983) 4889. [ 51 D.A. Micha, Intern. J. Quantum Chem. Symp. 19 (1986) 443. [6 J D.A. Micha, Chem. Phys. Letters 8 I ( 1981) 517. [7] J.M. Bowman, Accounts Chem. Res. 19 (1986) 202 [8 ] R.B. Gerber, V. Buch and M.A. Ratner, J. Chem. Phys. 77 (1982) 3022. [9J R.J. Kosloff, J. Phys. Chem. 92 (1988) 2087. [IO] D. Kumamoto and R. Silbey, J. Chem. Phys. 75 (198 I) 5164. [ll]S.Y. LceandE.J. Heller, J. Chem.Phys. 76 (1982) 3035. [ 121D. Srivastava and D.A. Micha, Chem. Phys. Letters 162 (1989) 381.
CHEMICAL PHYSICS LETTERS
20 October 1989
TIME CORRELATION OF LOCALIZED EXCITATIONS IN EXTENDED MOLECULAR SYSTEMS: A SELF-CONSISTENT FIELD TREATMENT * David A. MICHA and D. SRIVASTAVA’ Departments of Chemistry and Physics, UniversityofFlorida, Gainesville, FL. 32611, USA Received 22 May 1989; in final form 7 August 1989
A timedependent self-consistent field approach is developed for the dynamics of a local excitation coupled to its molecular environment, starting from the time-correlation function of transition operators. A variational functional is used to derive a factorized TCF with the correct normalization and phase factor. We also show what conditions are sufficient to justify the assumption of localized dynamics.
1. Introduction When a particle interacts with an extended molecular system, such as an adsorbate or a polymer, the system is usually excited in a localized region, made up of a set of bonds, or a group of atoms. During its excitation, the localized region evolves in time while strongly coupled to its environment, under the influence of effective forces that depend on the thermodynamical state of the system. Energy absorbed during the interaction leads to rearrangement or break-up in the localized region, and dissipates into its environment. Here we present a general formalism in the Liouville space of operators, to quantitatively characterize the localized region. We derive equations of motion for operators of the system, from which the localized dynamics of the system can be calculate& and consider what physical conditions justify a separate treatment of the localized dynamics. Extended systems can be conveniently described with time-correlation functions (TCFs) of operators for different processes, such as dipole TCFs for molecular spectra, atomic density fluctuation TCFs for neutron inelastic scattering, and flux TCFs for rate coefficients [l-4]. We base our treatment on time* Work partly supported by NSF Grant CHE-8615334. ’ Present address: Chemistry Department, Pennsylvania State University, University Park, PA 16802, USA.
376
correlation functions of transition operators, which contain many other TCFs as special cases [ 5 1. We also use the formalism of operators in Liouville space to obtain the TCFs under very general conditions [6]. This approach accounts for the thermodynamical state of the system in terms of the statistical averages implicit in the TCFs, and can be applied even when a system is not in a pure quantum state. We divide the system into a primary region, containing the localized part that is excited and undergoes rearrangement, and a remaining, secondary region which serves as the environment and is perturbed only through coupling with the primary region. Our treatment introduces a self-consistent field approximation, to include strong couplings between the primary and the secondary regions. Special care must however be taken within this approximation to obtain correct phase factors when constructing the corresponding TCFs. These factors make a difference when one takes the Fourier transforms of TCFs, that are related to measured properties. We derive the approximation from a general time-dependent variational functional for operators, that leads to the correct phases and normalization factors and can be used in applications with different trial operators. Along the way we find what conditions are sufficient to justify the assumption of localized dynamics. The time-dependent SCF approximation developed here has several aspects in common with recent
0 009-2614/89/$ 03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
20
CHEMICAL PHYSICS LETTERS
Volume 162, number 43
treatments of intramolecular dynamics [ 7-9 1. However, it is more general in that it is applicable to systems in mixed (statistical) states as well as systems in pure quanta1 states. It also includes special cases of scattering by surfaces treated in the literature [ lo].
October1989
obtain a self-consistent field (or mean field) approximation. We consider a system for which the primary degrees of freedom are designated by x, and the secondary ones by y. The Hamiltonian is given by H=H,+H,+H,,
2. Variational derivation of operatorequations We work with the TCF of the transition operator A = T,, where k and k’ are the wavevectors of the incoming and outgoing particles. This is an operator in the variables of the target, evolving in time in aG cordance with the target Hamiltonian H. We introduce the Liouville operator 2, where &‘A-HA-AH, and set fz= 1, so that A(t) satisfies the equation of motion for operators in the Heisenberg picture, -idA/dt=XA(t).
(1)
(6)
with a corresponding decomposition for X. Next we specify conditions that lead to the time-dependent self-consistent field (SCF) equations. Firstly, the initial statistical state of the system is given by a density operator (7)
P=PxPy ,
where the two factors are determined self-consistently, and they are normalized by the trace relations tr,(p,) =tr,(p,) = 1. Hence the factor px satisfies the equation
The TCF of A is given by
F(t)=
(2)
(8)
where (( ( ) )) = tr [ ( )p], in terms of the trace and the density operator p of the target. To obtain approximate equations of motion for A, we construct the variational functional 12 -6; dt ((A(t)+ (id/dt+X)A(t))} +c.c. , U II >
and similarly for py’ Secondly we choose as a trial operator A the factored form
(3) where C.C.means the complex conjugate of the preceding term. This functional must be varied under the constraint U(r)+‘Jt(r)>)
=JlrU)=F(O)
I
(4)
which is a constant fixed at the initial time t, =O. This can be imposed introducing the Lagrange multiplier A(t) and the functional dtk(t)
(9)
where V and V are primary and secondary region operators respectively. This assumes that at the initial time t, the operator A can also be written in a factored form. In what follows we drop the subindices in U and V. The functional J depends on V, V, and their adjoints. Introducing the independent variations i!XJ+,6Vt, the condition &J=O for
6V=6V=O
(10)
gives the coupled equations (( Vt(id/dt+&+lZ)UV}),=O, (( Vt(id/dt+Rt2)UV)),=0,
12 /=S+s
A(r) = VA) V.(t) ,
J’-(t) ,
(5)
II
which must satisfy Sj=O for variations 64 (t) = Ea,B&I, I a> (81, where 1a) is an internal state, around the exact values of A(t). Variations 6A( t)f can be assumed to be independent of &4( t ). The procedure just described can be followed to
(11)
where the subindices indicate the degrees of freedom that are not averaged. Next we define the time-dependent operators in Liouville space
~:[V(t)l=~++(v+~~V})x/~v+V~
I
~:[V(t)l=JE”++u+~~~~,l~U+U}) 3
(12)
and the averages 377
CHEMICALPHYSICSLETTERS
Volume 162,number 4,5
20 October 1989
a(t)=((Ut(id/df+X,)U))/((UtU)),
u(t)=a(t)l(~a+a))Y
B(t)=((v+(idldt+~Y,)Y))/((V+V)), Y(t)=((Y’Ut~,UV))/(((V+V))cu+u))),
u(t)
3
=b(t)l
(19)
in terms of which we find from eqs. (2) and (17), and with tl = 0, (13)
and also make the transformations
t
I U(t)=a(t)
exp i U
xexp dt’ [A(t’)+/I(t’)]
t1
,
>
y(t)
t
V(t)=b(t)exp
i
(I ‘I
dt’ [A(t’)+cu(t’)]
, )
(14)
to obtain from eq. ( 11) the self-consistently coupled differential equations of motion ida/dt+Y1[b]a(t)=O, idb/dt+YY[u] b(t) =O.
(15)
To impose the normalization condition we note that
=
{{
(I
i
dt’Rey(t’)
u+u+xvuu))
, )
0
I .
(20)
This expression provides the correct normalization and phase factor for a TCF in the self-consistent field approximation. The two double brackets in eq. (20) represent reduced primary and secondary TCFs. The operators in eq. ( 12 ) are in general non-Hermitian in Liouville space, from which it follows that the norms of a(l) and b(t) can change with time. Furthermore, in some calculations it is convenient to introduce complex times; hence the denominators in eq. ( 19) are not necessarily constant, and must be included in the equations.
A(t)=a(t)b(t)
(’
Xexp 1 jdi. 11
Izn(l~)+ol(l’)+B(t’)1).
(16)
To simplify the exponent here, we multiply the first eq. ( 11) times pJJt on the left and take the trace, do the same with p,,Vt in the second equation, and add the results to obtain a+P+y+L=O.
(17)
The normalization condition of eq. (4) leads then to {
3. Discussion The content of the previous equations can be understood within a simple model in which the primary region is described by a general Hamiltonian H, and the secondary region contains a single degree of freedom y oscillating harmonically around its equilibrium value y,, with mass and force constants chosen equal to unity. To lowest order in the displacement q=y-ye, the full Hamiitonian with a linear coupling is (21)
H=H,+f($+q*)-flX)q, xexp
- dt’2[Iml(t’)-Imy( (I II
>
=F(O>, (18)
where F( 0) = {( i?U)),((
V l~‘)}~_ This expression, together with the choice Rel=O, allows us to eliminate the Lagrange multiplier from the following expressions. To obtain the TCF, we define the normalized operators 378
where p is the momentum conjugated to q, and f(n ) is a general force. From eq. (8) we find K=H,-
<4)lf(x)
~Y=1w+~2b-uw
, (22)
The density operator px at temperature T is given by Px=exp( -FJkJ)/tr[exp( -F,/kJ’)], with a similar expression for pY;hence pYis the density OPerator of a linearly driven harmonic oscillator and
CHEMICAL
Volume 162, number 43
{(4)) is not zero. For the operator A we consider the dipole D(x, y) of the system, which to lowest order in q can be chosen as D(-%y)=&V,,
l$=1,+4/1,
<(u(t)+u(t)f(x)>?
t) =QI(X,0 V(Y,t) exp[ia(t)1 ,
%&Y,
1989
(25)
where 4 and w satisfy TDSCF equations [ 8,9 ] with effective Hamiltonians F,(t) and F,(t) and
(23)
where D, is the dipole operator of the primary region, 1, is the unit operator of the secondary region and I is a constant. Eq. ( 15 ) must be solved for a( t ) and b(t) with the initial conditions a(O) =O, and b(0) = V,. The equation for b(t) can be written in terms of the Hamiltonian for a linearly driven harmonic oscillator, for which the time evolution operator has a simple expression, and contains a nonHermitian potential. For the phase in eq. (20) one finds
+<(v(t)‘v(t)q>
20 October
PHYSICS LETTERS
(24)
where we have used that HXY=-f(x)q. The real part of this r(t) is not zero. Hence F(t) is not just the product of the correlation function of the dipole in the primary region times a secondary region correlation function, but it includes in addition a phase factor that will affect its Fourier transform. This example illustrates the sufficient conditions under which the dynamics of the system can be assumed to occur only in the localized region. These conditions are given by ( 1) a factorizable initial density operator, eq. (7 ), and (2) a transition operator A which at the initial and later times is of the form A= U,V,, with V, close to the unit operator. These conditions provide a quantitative definition of the localized dynamics. In a process characterized by a transition operator A, given for example by a dipole or a density fluctuation, one can analyze the initial form of A to determine what variables does it depend upon, such as certain bond coordinates or atomic positions. The primary region must then contain at least those variables. It follows from this that the primary region must be defined differently for different probes of the whole system. We can next analyze the connection with time-dependent SCF approximations, which can be made for wavefunctions or for transition amplitudes. Factoring only the time-dependent wavefunction Y’(x,y, t) of the system into components for the two regions gives
a(t)=
I
dl’ W(t’),
w~)=
II
(26) The TCF, now given by (27)
F(r)= { yo IA(t)+A(O) I %v,>>
with !?‘ou,= Y( t= 0), does not factorize except if one assumes in addition that A(O)= U,(O) V,(O) and HZZHsCF= F,+ F,,,- W. Furthermore, the TCF does not display the correct phase factor coming from the normalization constraint. An alternative is to define a time-dependent amplitude given by a product of the excitation operator times the wavefunction, r (t) =A( t) PO.Then provided the eigenenergy E, of the initial state is known, one can write -iX/&=(H-&)T(t)
(28)
and can introduce the TDSCF approximation, as done in some of the treatments with wavepackets [ 111 and path integrals [ 121, by writing rscF(x, Y, t) =5(x, t) rt(y, 0 exp!iX(t) I. This leads to the desired factorization of the TCF since F(t) = Wt)
Ino) > 5% (T(Oliv)>
(v(t)lr(O)>
x
exp [ix(i)] where the initial state has been approximated by a time-independent SCF wavefunction. The present Liouville formalism is however more general, because it does not require knowledge of the eigenstates and eigenenergies of the extended system at the initial time, that must usually be approximated. An additional advantage of the present formalism is that it consistently incorporates the temperature in the TDSCF dynamics. This is to be compared with the situation arising at finite temperatures with the TDSCF approximation for wavefunctions. Here ground and excited states are separately approximated, averages are calculated and then weighted by the statistical distributions of initial states. This can lead to unphysical temperature dependences if the errors in the SCF approximation are different for low and high excited states. The present formalism instead starts with the correct statistical density and 379
Volume 162, number 4,5
CHEMICAL PHYSICS LETTERS
the temperatures in the effective Liouville operators in eq. (12) so that the SCF approximation maintains the temperature balance. incorporates
References [ 1 ] L. Van Hove, Phys. Rev. 95 (1954) 249. [ 21 R.G. Gordon, Advan. Magn. Resort. 3 ( 1968) I. [ 31 B.J. Beme, in: Physical chemistry. An advanced treatise, Vol. 8B, ed. D. Henderson (Academic Press, New York, I971 ) ch. 9.
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