- Email: [email protected]
Fock space construction of the quantum thermal flux operator
Journal of Molecular Structure: THEOCHEM 768 (2006) 159–162 www.elsevier.com/locate/theochem
Fock space construction of the quantum thermal flux operator M. Durga Prasad * School of Chemistry, University of Hyderabad, Hyderabad 500 046, India Received 1 May 2006; accepted 1 May 2006 Available online 23 May 2006
Abstract The construction of the flux operator in the Fock space of harmonic oscillator eigenfunctions is outlined. In one dimension, the flux operator is found to be FZ(iu/Op) exp[KaC2/O2] [j1O!0jKj0O!1j] exp[Ka2/2]. Two approaches for the temperature propagation are discussed, and the thermal flux eigenvalues for a double well potential are presented. q 2006 Elsevier B.V. All rights reserved. Keywords: Flux operator; Fock space construction
1. Introduction The symmetrized flux operator in one dimension is defined as FZ
½dðxÞ$p C p$dðxÞ ; 2m
(1.1)
where m is the mass of the particle, and p is momentum operator. As shown by Yamamoto [1] and Miller and coworkers [2,3] it is particularly useful in the direct evaluation of thermal rate constants through various correlation functions without going through a detailed, state-to-state, energy dependent, rate calculation. For example, in one popular form, the rate constant, k, is given by [3] ð kðTÞ Z Cff ðtÞdt=Qr ; (1.2) where Qr is the reactant partition function and Cff is the flux– flux autocorrelation function Cff ðtÞ Z Tr½Fð0ÞFðtÞr:
(1.3a)
Here rZexp(KbH) is the density operator, bZ1/kBT is the inverse temperature, and H is the Hamiltonian of the system. The time evolved flux operator, F(t) is given by: FðtÞ Z expðKiHtÞ$F$expðiHtÞ:
(1.3b)
* Tel.: C91 40 2756 4164. E-mail address: [email protected]
0166-1280/$ - see front matter q 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2006.05.020
Using the invariance of traces to cyclic permutations and the observation that the projection operator on to the product state space commutes with the Hamiltonian, Eq. (1.3) and its equivalent forms can be rewritten in several ways. One such form [4] is Cff ðtÞ Z Tr½Fb=2 ð0ÞFb=2 ðtÞ;
(1.4)
where the thermal flux operator Fb is defined by: Fb Z expðbH=2Þ$F$expðKbH=2Þ:
(1.5)
Park and Light [5] noted that the thermal flux operator has only two non-zero eigenvalues, equal in magnitude with opposite signs, and the absolute value of these eigenvalues approaches infinity as the temperature goes to zero. A proof of this observation was first provided by Seideman and Miller [6], and, later by Pollak [7], and Park [8]. Even in multidimensional systems, the flux operator is of low rank. In all practical applications so far, these properties were utilized numerically [9–16]. The goal of the present work is to construct the flux operator in the Fock space of the harmonic oscillator eigenfunctions. Though, the flux operator is singular, it turns out that it can be written in terms of projection operators constructed from infinitely squeezed ground and first excited states of the harmonic oscillator. This part of the work is presented in Section 2. In Section 3, we present the calculation of the thermal flux for some model systems. Section 4 summarizes our results. 2. The flux operator We need a reference system to define the harmonic oscillator basis. Let the Hamiltonian of the reference system be
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(in atomic units): Href ZKð1=2mÞd2 =dx2 C mu2 x2 =2:
(2.1)
The dimensionless coordinate q is defined in the usual manner as q Z xOmu;
(2.2)
be reproduced by an appropriate normalization constant, Eq. (3.8) can be rewritten as: 1 F Z ðiu=OpÞexp½K2v$Us ½j1O!0jKj0O!1jUK s :
(2.11)
It can be readily verified that the two non-zero eigenvalues and the associated eigenfunctions of F are:
and the harmonic oscillator ladder operators are given by:
fG ZGðu=OpÞexp½K2v/N;
(2.12a)
a Z ðq C d=dqÞ=O2
(2.3a)
4G Z Us ½j0OGij1O:
(2.12b)
aC Z ðqKd=dqÞ=O2
(2.3b)
We now turn to the Fock space representation of the d-function. One of the representations of the d-function is the limit: dðxÞ Z Lta/Nð2mua=pÞ1=2 exp½K2muax2 :
(2.4)
Substitution of Eq. (2.3a,b) into Eq. (2.4) leads to: dðxÞ Z Lta/Nð2mua=pÞ1=2 exp½Kaða C aCÞ2 :
(2.5)
3. Thermal propagation of the flux operator We now turn to the calculation of the flux eigenvalues at finite temperature. The first system we consider is the analytically soluble parabolic barrier potential. The Hamiltonian for such a system is: Hpb ZKð1=2mÞd2 =dx2 Kmu2 x2 =2:
(3.1a)
Disentangling the operators in the exponential, we finally arrive at
We take the frequency of the reference Hamiltonian as the barrier frequency. With this choice:
dðxÞ Z Lta/Nð2mua=pÞ1=2 exp½uaC2 exp½KvaCaexp½ua2 ;
Hpb ZKðu=2ÞðaC2 C a2 Þ:
(3.1b)
(2.6a)
Since the Hamiltonian belongs to the harmonic oscillator algebra, Fb can be parametrized as
u ZKa=ð2a C 1Þ;
(2.6b)
Fb Z ðiu=OpÞD½j1O!0jKj0O!1jD;
v Z lnð2a C 1Þ:
(2.6c)
where
with:
(3.2)
Note that the pre-exponential factor and v go to N as a/N, but u remains finite. Inserting the resolution of identity, IZ SnjnO!nj, on either side of the middle exponential and retaining only the non-zero terms after taking the limit, we arrive at the Fock space representation of the delta function:
D Z exp½uaC2 $exp½vðaCa C 1=2Þ$exp½wa2 ;
dðxÞ Z ðmu=pÞ1=2 exp½KaC2 =2j0O!0jexp½Ka2 =2;
The working equations for the parameters u, v, and w are obtained by premultiplying Eq. (3.4) with DK1, and invoking the Hausdorff expansion. Note that due to the Lie-algebraic decoupling conditions [17], the equation for u is decoupled from v and w, while the equation for v is decoupled from w. On integration, we obtain
(2.7)
Substituting Eq. (2.7) for the delta function and the momentum operator from Eq. (2.3a,b) into Eq. (1.1), we finally arrive at the flux operator F Z ðiu=OpÞexp½uaC2 ½j1O!0jKj0O!1jexp½ua2 ;
(2.8)
with uZK1/2. Eq. (2.8) is the central result of the present work. Note that exp [uaC2] is the first term in the disentangled form of the squeezing operator Us Us Z exp½sðaC2 Ka2 Þ Z exp½uaC2 $exp½vðaCa C 1=2Þ$exp½Kua2 ;
(3.3)
with the initial conditions u(0)ZK1/2, v(0)Zw(0)Z0. It satisfies: dD=db ZKHpb D:
(3.4)
u Z tanðbu=2Kp=4Þ=2;
(3.5a)
v Z ln½ð1 C 4u2 Þ=2=2:
(3.5b)
w is not needed, since its action produces zero on the two states of interest. With this the thermal flux operator turns out to be: (2.9)
Fb Z ðiu$sec2 ðub=2Kp=4Þ=OpÞ$exp½uaC2 =2½j1O!0j
with the relation u Z tanhð2sÞ=2;
(2.10a)
v Z ln½sechð2sÞ:
(2.10b)
Since the operator a2 annihilates both the ground state and the first excited state, and the effect of the number operator can
Kj0O!1jexp½ua2 =2;
(3.6)
Its eigenvalues are given by f ðbÞ ZGu=ð2 sinðubÞO2pÞ; as they should [7].
(3.7)
M.D. Prasad / Journal of Molecular Structure: THEOCHEM 768 (2006) 159–162
161
The second system that we study is a quartic double well potential: Hdw ZKð1=2mÞd2 =dx2 Kmu2 x2 =2 C a4 x4 :
(3.8)
The barrier height Eb is related to the quartic coefficient a4 through the relation: Eb Z m2 u4 =16a4 :
(3.9)
For a general anharmonic surface such as this, the thermal flux operator is no longer limited to two boson operators as in the ansatz (3.3). Instead, a more general coupled cluster ansatz must be used if the propagation in temperature is to be carried out via Eq. (3.4). It is possible to approximate the density operator with only two boson operators, and use the Hausdorff expansion as before. The equations of motion for u, v, and w would not be decoupled, however. Here, we explore an alternative approach by constructing an effective harmonic oscillator (EHO) [18–20] to describe the system and use it to construct the density matrix. The method is based on the GibbsBogoliubov inequality F% Fo C!H KHo O ;
(3.10)
where, Fo and Ho are the free energy and the Hamiltonian of the EHO, respectively. The last term is the thermal average of the residual perturbation with respect to the effective harmonic oscillator Hamiltonian. The frequency of the EHO is obtained by minimizing the right hand side of Eq. (3.10). The resulting Hamiltonian of the EHO is used both as the reference Hamiltonian and to construct the density matrix r in the harmonic approximation
in Eq. (2.10a,b) is now given in terms of ub of Eq. (3.12b). The eigenfunctions are now square integrable. The flux eigenvalues obtained by this procedure for mZuZ1.0, and EbZ4 are presented in Fig. 1. As can be seen, the flux eigenvalues decrease monotonically from N with increasing b, and the results do not show the unphysical behavior of the parabolic barrier. 4. Concluding remarks
(3.12a)
In this work, we have constructed the symmetrized flux operator in the Fock space of the harmonic oscillator eigenfunctions. It has a simple structure in this representation, being given in terms of the projection operators constructed from the (squeezed) ground and first excited states. At infinite temperature, the squeezing leads to unnormalizable eigenfunctions and infinite eigenvalues. We have considered one possible approach to construct the flux operator at finite temperature based on the Gibbs-Bogoliubov inequality, and have shown it to be a workable approach. For practical applications, the approach must be extended to multi-dimensional spaces. The flux operator in D-dimensional space is given by the product of the flux operator in one dimension and the identity operator in the remaining degrees freedom
(3.12b)
FD Z Fr IDK1 :
Ho Zuo ðaCa C 1=2Þ;
(3.11a)
r Z exp½Kbuo ðaCa C ð1=2ÞÞ;
(3.11b)
where uo is the variationally optimized frequency of the EHO. With this, Fb is obtained from its definition (1.5) directly: Fb Z ðiu=OpÞexp½ub aC2 ½j1O!0jKj0O!1jexp½ub a2 ;
ub ZKexpðKbuo Þ=2:
Fig. 1. Variation of the flux eigenvalue with b for the double well barrier.
Note that the thermal flux operator, in this approximation, is still given in terms of the squeezed states and has only two nonzero eigenvalues. It is not singular for any non-zero value of b. As is well known, the eigenvalues of the thermal flux operator diverge for the parabolic barrier because of the divergent nature of the potential in the asymptotic region. Since no realistic barrier potential has such divergent behavior, the flux eigenvalues do not show such behavior. It is thus important that the approximate density operator does not lead to such unphysical divergences. As the present results show, the approximate density operator obtained from the variational principle reflects the bounded nature of the potential correctly, though, we have used only the harmonic algebra to construct it. The flux eigenvalues and eigenfunctions are once again given by Eq. (2.12a,b). However, the squeezing parameter s defined
(4.1)
here, Fr is the flux operator along the reactive coordinate, and is given by Eq. (1.1) or equivalently by Eq. (2.8), and IDK1 is identity operator in remaining degrees of freedom. Unlike the flux operator, the identity operator cannot be written in bra-ket form. However, using the thermo field dynamical convention [21], the identity operator can written in ket form in an extended Hilbert space with double the number of degrees of freedom: X X IZ jnO!nj Z jn;n 0O Z expðaCa 0CÞj0;0 0 O : (4.2) n n Here, the bra part of the projection operator is rewritten in a ket form based on the observation that the action of the annihilation operators to the left is identical to the action of the creation operators to the right. Using this convention, it is possible to write down the flux operator in D-dimensional
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space as:
hX
C2 i C 0C 0C2 FD Zðiu=OpÞexp a a K a C a =2 r r b b b j1r ;0b ;0r0 0b0 OKj0r ;0b ;1r0 0b0O
(4.3)
This operator would have to be propagated both in temperature to obtain the thermal flux operator. Acknowledgements It is a pleasure and privilege to dedicate this paper to Prof. Debashis Mukherjee on the occasion of his sixtieth birth day. Financial support from DST is gratefully acknowledged. References
[5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
[1] [2] [3] [4]
T. Yamamoto, J. Chem. Phys. 33 (1960) 281. W.H. Miller, J. Chem. Phys. 61 (1974) 1823. W.H. Miller, S.D. Schwartz, W. Tromp, J. Chem. Phys. 79 (1983) 4889. D. Brown, C. Light, J. Chem. Phys. 97 (1992) 5465.
[21]
T.J. Park, C. Light, J. Chem. Phys. 91 (1989) 974. T. Seideman, H. Miller, J. Chem. Phys. 95 (1991) 1768. E. Pollak, J. Chem. Phys. 107 (1997) 64. T.J. Park, J. Chem. Phys. 107 (1997) 671. W.H. Thompson, H. Miller, J. Chem. Phys. 102 (1995) 7409. D.H. Zhang, C. Light, J. Chem. Phys. 104 (1996) 6184. D.H. Zhang, C. Light, J. Chem. Phys. 106 (1997) 551. W.H. Thompson, H. Miller, J. Chem. Phys. 106 (1997) 142. U. Manthe, H. Miller, J. Chem. Phys. 99 (1993) 3411. F. Matzkies, U. Manthe, J. Chem. Phys. 112 (2000) 130. D.M. Medvedeev, K. Gray, J. Chem. Phys. 120 (2004) 9060. A. Chakraborty, G. Truhlar, Proc. Natl Acad. Sci. USA 102 (2005) 6744. G.M. Sastry, M. Durga Prasad, Theor. Chim. Acta 89 (1994) 193. V.P. Spiridonov, B.S. Butayev, A.Ya. Nasarenko, O.N. But, A.G. Gershivkov, Chem. Phys. Lett. 103 (1984) 363. V.P. Spiridonov, V.S. Lyutsarev, S. Butayev, Chem. Phys. Lett. 110 (1984) 32. K.V. Ermakov, B.S. Butayev, P. Spiridonov, Chem. Phys. Lett. 144 (1988) 497. H. Umezawa, H. Masumoto, M. Tachiki, Thermo-fleld Dynamics and Condensed States, North-Holland, Amsterdam, 1982; H. Umezawa, Y. Yamanaka, Adv. Phys. 37531.
Fock space construction of the quantum thermal flux operator M. Durga Prasad * School of Chemistry, University of Hyderabad, Hyderabad 500 046, India Received 1 May 2006; accepted 1 May 2006 Available online 23 May 2006
Abstract The construction of the flux operator in the Fock space of harmonic oscillator eigenfunctions is outlined. In one dimension, the flux operator is found to be FZ(iu/Op) exp[KaC2/O2] [j1O!0jKj0O!1j] exp[Ka2/2]. Two approaches for the temperature propagation are discussed, and the thermal flux eigenvalues for a double well potential are presented. q 2006 Elsevier B.V. All rights reserved. Keywords: Flux operator; Fock space construction
1. Introduction The symmetrized flux operator in one dimension is defined as FZ
½dðxÞ$p C p$dðxÞ ; 2m
(1.1)
where m is the mass of the particle, and p is momentum operator. As shown by Yamamoto [1] and Miller and coworkers [2,3] it is particularly useful in the direct evaluation of thermal rate constants through various correlation functions without going through a detailed, state-to-state, energy dependent, rate calculation. For example, in one popular form, the rate constant, k, is given by [3] ð kðTÞ Z Cff ðtÞdt=Qr ; (1.2) where Qr is the reactant partition function and Cff is the flux– flux autocorrelation function Cff ðtÞ Z Tr½Fð0ÞFðtÞr:
(1.3a)
Here rZexp(KbH) is the density operator, bZ1/kBT is the inverse temperature, and H is the Hamiltonian of the system. The time evolved flux operator, F(t) is given by: FðtÞ Z expðKiHtÞ$F$expðiHtÞ:
(1.3b)
* Tel.: C91 40 2756 4164. E-mail address: [email protected]
0166-1280/$ - see front matter q 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2006.05.020
Using the invariance of traces to cyclic permutations and the observation that the projection operator on to the product state space commutes with the Hamiltonian, Eq. (1.3) and its equivalent forms can be rewritten in several ways. One such form [4] is Cff ðtÞ Z Tr½Fb=2 ð0ÞFb=2 ðtÞ;
(1.4)
where the thermal flux operator Fb is defined by: Fb Z expðbH=2Þ$F$expðKbH=2Þ:
(1.5)
Park and Light [5] noted that the thermal flux operator has only two non-zero eigenvalues, equal in magnitude with opposite signs, and the absolute value of these eigenvalues approaches infinity as the temperature goes to zero. A proof of this observation was first provided by Seideman and Miller [6], and, later by Pollak [7], and Park [8]. Even in multidimensional systems, the flux operator is of low rank. In all practical applications so far, these properties were utilized numerically [9–16]. The goal of the present work is to construct the flux operator in the Fock space of the harmonic oscillator eigenfunctions. Though, the flux operator is singular, it turns out that it can be written in terms of projection operators constructed from infinitely squeezed ground and first excited states of the harmonic oscillator. This part of the work is presented in Section 2. In Section 3, we present the calculation of the thermal flux for some model systems. Section 4 summarizes our results. 2. The flux operator We need a reference system to define the harmonic oscillator basis. Let the Hamiltonian of the reference system be
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M.D. Prasad / Journal of Molecular Structure: THEOCHEM 768 (2006) 159–162
(in atomic units): Href ZKð1=2mÞd2 =dx2 C mu2 x2 =2:
(2.1)
The dimensionless coordinate q is defined in the usual manner as q Z xOmu;
(2.2)
be reproduced by an appropriate normalization constant, Eq. (3.8) can be rewritten as: 1 F Z ðiu=OpÞexp½K2v$Us ½j1O!0jKj0O!1jUK s :
(2.11)
It can be readily verified that the two non-zero eigenvalues and the associated eigenfunctions of F are:
and the harmonic oscillator ladder operators are given by:
fG ZGðu=OpÞexp½K2v/N;
(2.12a)
a Z ðq C d=dqÞ=O2
(2.3a)
4G Z Us ½j0OGij1O:
(2.12b)
aC Z ðqKd=dqÞ=O2
(2.3b)
We now turn to the Fock space representation of the d-function. One of the representations of the d-function is the limit: dðxÞ Z Lta/Nð2mua=pÞ1=2 exp½K2muax2 :
(2.4)
Substitution of Eq. (2.3a,b) into Eq. (2.4) leads to: dðxÞ Z Lta/Nð2mua=pÞ1=2 exp½Kaða C aCÞ2 :
(2.5)
3. Thermal propagation of the flux operator We now turn to the calculation of the flux eigenvalues at finite temperature. The first system we consider is the analytically soluble parabolic barrier potential. The Hamiltonian for such a system is: Hpb ZKð1=2mÞd2 =dx2 Kmu2 x2 =2:
(3.1a)
Disentangling the operators in the exponential, we finally arrive at
We take the frequency of the reference Hamiltonian as the barrier frequency. With this choice:
dðxÞ Z Lta/Nð2mua=pÞ1=2 exp½uaC2 exp½KvaCaexp½ua2 ;
Hpb ZKðu=2ÞðaC2 C a2 Þ:
(3.1b)
(2.6a)
Since the Hamiltonian belongs to the harmonic oscillator algebra, Fb can be parametrized as
u ZKa=ð2a C 1Þ;
(2.6b)
Fb Z ðiu=OpÞD½j1O!0jKj0O!1jD;
v Z lnð2a C 1Þ:
(2.6c)
where
with:
(3.2)
Note that the pre-exponential factor and v go to N as a/N, but u remains finite. Inserting the resolution of identity, IZ SnjnO!nj, on either side of the middle exponential and retaining only the non-zero terms after taking the limit, we arrive at the Fock space representation of the delta function:
D Z exp½uaC2 $exp½vðaCa C 1=2Þ$exp½wa2 ;
dðxÞ Z ðmu=pÞ1=2 exp½KaC2 =2j0O!0jexp½Ka2 =2;
The working equations for the parameters u, v, and w are obtained by premultiplying Eq. (3.4) with DK1, and invoking the Hausdorff expansion. Note that due to the Lie-algebraic decoupling conditions [17], the equation for u is decoupled from v and w, while the equation for v is decoupled from w. On integration, we obtain
(2.7)
Substituting Eq. (2.7) for the delta function and the momentum operator from Eq. (2.3a,b) into Eq. (1.1), we finally arrive at the flux operator F Z ðiu=OpÞexp½uaC2 ½j1O!0jKj0O!1jexp½ua2 ;
(2.8)
with uZK1/2. Eq. (2.8) is the central result of the present work. Note that exp [uaC2] is the first term in the disentangled form of the squeezing operator Us Us Z exp½sðaC2 Ka2 Þ Z exp½uaC2 $exp½vðaCa C 1=2Þ$exp½Kua2 ;
(3.3)
with the initial conditions u(0)ZK1/2, v(0)Zw(0)Z0. It satisfies: dD=db ZKHpb D:
(3.4)
u Z tanðbu=2Kp=4Þ=2;
(3.5a)
v Z ln½ð1 C 4u2 Þ=2=2:
(3.5b)
w is not needed, since its action produces zero on the two states of interest. With this the thermal flux operator turns out to be: (2.9)
Fb Z ðiu$sec2 ðub=2Kp=4Þ=OpÞ$exp½uaC2 =2½j1O!0j
with the relation u Z tanhð2sÞ=2;
(2.10a)
v Z ln½sechð2sÞ:
(2.10b)
Since the operator a2 annihilates both the ground state and the first excited state, and the effect of the number operator can
Kj0O!1jexp½ua2 =2;
(3.6)
Its eigenvalues are given by f ðbÞ ZGu=ð2 sinðubÞO2pÞ; as they should [7].
(3.7)
M.D. Prasad / Journal of Molecular Structure: THEOCHEM 768 (2006) 159–162
161
The second system that we study is a quartic double well potential: Hdw ZKð1=2mÞd2 =dx2 Kmu2 x2 =2 C a4 x4 :
(3.8)
The barrier height Eb is related to the quartic coefficient a4 through the relation: Eb Z m2 u4 =16a4 :
(3.9)
For a general anharmonic surface such as this, the thermal flux operator is no longer limited to two boson operators as in the ansatz (3.3). Instead, a more general coupled cluster ansatz must be used if the propagation in temperature is to be carried out via Eq. (3.4). It is possible to approximate the density operator with only two boson operators, and use the Hausdorff expansion as before. The equations of motion for u, v, and w would not be decoupled, however. Here, we explore an alternative approach by constructing an effective harmonic oscillator (EHO) [18–20] to describe the system and use it to construct the density matrix. The method is based on the GibbsBogoliubov inequality F% Fo C!H KHo O ;
(3.10)
where, Fo and Ho are the free energy and the Hamiltonian of the EHO, respectively. The last term is the thermal average of the residual perturbation with respect to the effective harmonic oscillator Hamiltonian. The frequency of the EHO is obtained by minimizing the right hand side of Eq. (3.10). The resulting Hamiltonian of the EHO is used both as the reference Hamiltonian and to construct the density matrix r in the harmonic approximation
in Eq. (2.10a,b) is now given in terms of ub of Eq. (3.12b). The eigenfunctions are now square integrable. The flux eigenvalues obtained by this procedure for mZuZ1.0, and EbZ4 are presented in Fig. 1. As can be seen, the flux eigenvalues decrease monotonically from N with increasing b, and the results do not show the unphysical behavior of the parabolic barrier. 4. Concluding remarks
(3.12a)
In this work, we have constructed the symmetrized flux operator in the Fock space of the harmonic oscillator eigenfunctions. It has a simple structure in this representation, being given in terms of the projection operators constructed from the (squeezed) ground and first excited states. At infinite temperature, the squeezing leads to unnormalizable eigenfunctions and infinite eigenvalues. We have considered one possible approach to construct the flux operator at finite temperature based on the Gibbs-Bogoliubov inequality, and have shown it to be a workable approach. For practical applications, the approach must be extended to multi-dimensional spaces. The flux operator in D-dimensional space is given by the product of the flux operator in one dimension and the identity operator in the remaining degrees freedom
(3.12b)
FD Z Fr IDK1 :
Ho Zuo ðaCa C 1=2Þ;
(3.11a)
r Z exp½Kbuo ðaCa C ð1=2ÞÞ;
(3.11b)
where uo is the variationally optimized frequency of the EHO. With this, Fb is obtained from its definition (1.5) directly: Fb Z ðiu=OpÞexp½ub aC2 ½j1O!0jKj0O!1jexp½ub a2 ;
ub ZKexpðKbuo Þ=2:
Fig. 1. Variation of the flux eigenvalue with b for the double well barrier.
Note that the thermal flux operator, in this approximation, is still given in terms of the squeezed states and has only two nonzero eigenvalues. It is not singular for any non-zero value of b. As is well known, the eigenvalues of the thermal flux operator diverge for the parabolic barrier because of the divergent nature of the potential in the asymptotic region. Since no realistic barrier potential has such divergent behavior, the flux eigenvalues do not show such behavior. It is thus important that the approximate density operator does not lead to such unphysical divergences. As the present results show, the approximate density operator obtained from the variational principle reflects the bounded nature of the potential correctly, though, we have used only the harmonic algebra to construct it. The flux eigenvalues and eigenfunctions are once again given by Eq. (2.12a,b). However, the squeezing parameter s defined
(4.1)
here, Fr is the flux operator along the reactive coordinate, and is given by Eq. (1.1) or equivalently by Eq. (2.8), and IDK1 is identity operator in remaining degrees of freedom. Unlike the flux operator, the identity operator cannot be written in bra-ket form. However, using the thermo field dynamical convention [21], the identity operator can written in ket form in an extended Hilbert space with double the number of degrees of freedom: X X IZ jnO!nj Z jn;n 0O Z expðaCa 0CÞj0;0 0 O : (4.2) n n Here, the bra part of the projection operator is rewritten in a ket form based on the observation that the action of the annihilation operators to the left is identical to the action of the creation operators to the right. Using this convention, it is possible to write down the flux operator in D-dimensional
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M.D. Prasad / Journal of Molecular Structure: THEOCHEM 768 (2006) 159–162
space as:
hX
C2 i C 0C 0C2 FD Zðiu=OpÞexp a a K a C a =2 r r b b b j1r ;0b ;0r0 0b0 OKj0r ;0b ;1r0 0b0O
(4.3)
This operator would have to be propagated both in temperature to obtain the thermal flux operator. Acknowledgements It is a pleasure and privilege to dedicate this paper to Prof. Debashis Mukherjee on the occasion of his sixtieth birth day. Financial support from DST is gratefully acknowledged. References
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