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A multireference coupled-cluster approach to quantum dynamics
Volume 186, number 1
1 November 1991
CHEMICAL PHYSICS LETTERS
A multireference coupled-cluster approach to quantum dynamics Sruti Guha and Debashis Mukherjee ’ Theory Group,Department ofPhysical Chemistry,Indian Association& the Cultivationof Science. Calcutta 700-032,India Received I6 July 199I
A time-dependent coupled-cluster(TDCC) formalism with a multireference model space for studying quantum dynamics is presented. The evolution operator I/acting on the model space is factorized as U&, where I& evolves in the model space and r/,, brings in virtual functions. Both Uexand UMare written as normal ordered exponentials involving cluster operators. The evolution of UMis governed by an effective Hamiltonian that is size-extensive. An illustrative application on a nontrivial problem indicates the potentiality of the formalism.
1. Introduction Significant progress has been made over the last two decades in the development and application of various time-dependent methods for studying quantum dynamical processes. Recent works have paid special attention to providing compact descriptions of the wavefunctions, admitting of systematic truncation schemes, and generating viable strategies for systems with several degrees of freedom. Particular mention may be made of the methods of Gaussian wave-packet dynamics [ 11, the numerically oriented methods [ 2-5 3, the Lie-algebra inspired formulations [6-81 and the methods based on time-dependent variational principles [9-l 11. We present in this Letter another general method for solving time-dependent problems that combines the advantages of Lie-algebraic ideas and the coupled-cluster (CC) representation of the evolution operators to provide a nonperturbative computational strategy. The theory has the computational advantage of not getting rapidly unwieldy with the increase of the degrees of freedom of the system. Moreover, the CC approach ensures automatically a correct separation of the reactive species in a dynamics and hence should be well suited to study associative and fragmentation dynamical processes. Most of the earlier developments of the time-de’ To whom correspondence
84
should be addressed.
pendent CC methods have used a single reference function from which the evolution was monitored [ 12,131. This is a rather serious restriction as to the types of problems of real interest that could be studied, since quite often we require more than one reference function even for a qualitatively correct description. We shall generalize the concept of effecctive Humiltanians of stationary problems [ 141 to encompass nonstationary situations and develop a CC theory for the evolution process, starting from the reference functions defining a model space.
2. Theory of effective Hamiltonians for timedependent problems Let us consider a situation where we prepare a set nonstationary functions of time-dependent { pK(f)} at some initial time tu, built as superpositions of a set of quasi-degenerate, and generally strongly interacting, time-independent functions spanning a model space Wh)
= c GQ, I
*
(1)
Since the states pK are nonstationary, they will nontrivially evolve in time according to ‘Y,(r)=U(t, to) YIOK(t) >
(2)
with U satisfying 0009-2614/91/S 03.50 0 1991 Elsevier Science Publishers B.V. All rights reserved.
CHEMICAL PHYSICS LETTERS
Volume 186, number 1
iaU( t, to) =H(t)U(t, at
(3)
lo) 2
where we have assumed that H is time dependent in general. U mixes with ul”,sthe virtual functions and also induces a rotation within the model space, amounting to changing the coefficients of {ai}. From now on, we shall use P and Q to indicate projectors onto model and virtual functions. Our desire is to define a model evolution operator UM and a model time-dependent effective Hamiltonian H,, such that the evolution of the model space component { PK( t)} of (YK(t)} can be equivalently described via an equation of motion of the form %(t)=Gl(t,
(4)
to) %(to) ,
where U, satisfies
iWd4 to)
at
=Kfdt)
UM(t,to) *
(5)
Let us note that UMand &are defined in the model space only; also, we need to know only UP, rather than the full U. The key step in our formulation is the use of factorization Ansatz for U as follows: UP= we, U,P)
(6)
where U,, brings the admixture with the complementary space functions. Although U is unitary, we have a freedom in the choice of our model space projection of U,,, PU,,P. This adtomatically fixes U, via unitarity of U, which fixes the orientation of PYK in the model space with respect to the evolved model functions PK(f)_ Substituting eq. (6) into eq. (3) and using eqs. (4) and (5), we obtain iXJ L
at
= HU,, - U,, He,
(7)
From the PP component of eq. (7), we obtain He,= [ U:;] -I
[HUe,]fP-
F
.
(8)
The evolution of Ugp is governed by =
at
= [Huex]Qp-U~pHeff.
(9)
1November 1991
Since He, depends only err U,, and not on UM,eqs. (8 ) and (9 ) imply that we have decoupled U,, and U,. UrL and its time-evolution are fixed by us, and CJgpis obtained from eq. (9). U, is then obtained by solving eq. (5) with H,. as given by eq. (8). We propose to call eq. (7) the time-dependent anaiogue of the Bloch equation, This was put forward earlier by one of us [ 151 in the context of deriving open-shell CC stationary equations for incomplete model spaces via adiabatic hypothesis. The treatment was, however, general which enables us to use the same ideas to derive CC equations for any model space for nonstationary problems.
3. Time-dependent CC equations for a general multireference model space Let us assume that H in the occupation number representation is given by H(t)=
1 hi(t)&, I
(10)
where h,s are the time-dependent matrix elements and 1,sare products of suitable creation-annihilation operators defined in a Fock space. lis always satisfy a Lie-algebra. Let us note that we have kept I,s as independent of time. In case the elements {II} of H do not satisfy a closed Lie-algebra, we may formally assume that the sum over i in eq. (10) runs over the rest of the elements needed for completion with the corresponding h,s as zero. This will be assumed always, even when the algebra is countably infinite. In exact analogy with our earlier works [ 15,161, we classify all the operators into two categories. An operator is closed, denoted as AC,,if it induces transitions only within the model space. An operator is exiernal, denoted as A,, if it can induce a transition outside the model space from at least oozemodel space function oP We shall write U,, and U, as cluster expansions of these two kinds of operators and generate an Hem which is closed and connected and thus respects size-extensivity [ 1516 1. We take a suitable physical vacuum, and rewrite H in normal order. For fermions, the choice of a hole-particle vacuum is often physically quite straightforward, but for bosons a nontrivial Bogilyubov transformation may be necessary. U, is 85
Volume 186, number I
CHEMICAL PHYSICS LETTERS
written as a normal ordered exponential of a closedcluster operator X=X,,: UM =iexp(x>)
.
(11)
For U,,, we do not have a unique choice. It should obviously involve external operators S,,, but it may additionally have closed operators S,, depending on how we want to fix [ CJe,] cl. For the minimal description, it is suflcient to have only S=S,, in Uex: G, ={exp(S)l . Other
choices,
(12)
e.g., the
isometry
condition (viz, giving Hermitian Hen). We do not consider them here, but quote our earlier works [ 15,161. The choice (12) leads to a non-Hermitian Heff, giving a nonunitary U,. This is understandable on physical grounds: with the passage of time, the system propagates in the virtual space induced via Ue, which “stretches” the function via S,,; U, not only rotates it within the model space but also “squeezes” it to restore normalization. UMcannot thus be normconserving by itself. Substituting CJei,, from eq. ( 12) into (7), and rewriting them in normal order, we have [WA Uex]cl= ICI, are sometimes convenient
iaS, ex = (HU.Z, Q, I- I we,,&J&J 1 at 1
3
(13)
where (HU,,} etc. denote the connected expressions like tH[exp(S) I} J‘oining H with powers of S from the exponential, omitting contractions among the S operators. Since we are working in the Fock space, the n-body components of U,, are linearly independent, and it follows then that if
={HCT,,]-{G&-r}
-
(14)
Taking the closed part of eq. ( 14), using [SIC,=@ we have ~ULL&f1=IHU,,L,
1
(15)
which defines Hen. With the external part, we likewise have i~={HU,,}..-(LI,,H,J,,.
(16)
We get S by solving eq. ( 16). X is solved next via an analogous CC equation 86
.ax -
1%
={&Al),
1November 1991
(17)
obtained by substituting eq. ( 11) into eq. (5). Eqs. ( 15)- ( 17) indicate that S, X and & are all connected. The TDCC theory thus generates a sizeextensive formulation - a very important requirement for treating large systems. To obtain survivalities in the model space, we need the matrix cl- < @,I CJe,UM (@ii>. It may appear that U,, does not contribute to rti since U,, has only external operators S,,. This is not generally so. For a complete model space, which is realizable generally for fermions only, powers of external operators remain external [ 1% 181. But for incomplete model spaces, where all possible occupancies of the partially filled orbitals are not exhausted (this is naturally the case for bosons because of no restriction in occupancy) S,, can lead to scattering within the model space, and powers thereof can even be closed [ 15-181. [ iYe,Jcl is thus not generally I,,, and this should be borne in mind while using expressions which potentially involve [ Ve,]c,. The situation is analogous to that in stationary problems [ 17,181. An attempt to restore [ U,, ] cI= 1clleads to disconnected H,,and a breakdown of the linked cluster expansion 1171.
Eqs. ( 16) and ( 17) are the principal working equations of our time-dependent CC formalism using any arbitrary model space. A very appealing feature of the TDCC equations is the systematic hierarchical decoupling of the equations for the various n-particle model space sectors. Owing to the normal ordering, the cluster operators cannot contract among themselves. As a result, for the zero-valence problem (i.e. the vacuum as the model space), the equations will have only those S operators having no destruction operators. Operators with destruction operators will trivially give zero by their action on the model space. For the one-valence problem (one quasi-particle on the vacuum), we would need additional operators having one destruction operator. Since the zero-valence cluster operators are already known, they will simply appear as time-dependent parameters in the equations for one-valence operators. This same pattern is repeated for the higher valence problems. The situation is entirely analogous to what one encounters in the sta-
Volume 186,number I
I November 1991
CHEMICAL PHYSICS LETTERS
tionary case, and may be termed as the time-dependent counterpart of the “subsystem embedding condition“ [ 191. Another interesting aspect of our development is that, when our choice of S and X exhausts the Liealgebra, the solution U= &,U, is also the global solution, independent of the choice of u particular type of model space.
We also note here that for any given n-valence sector, the TDCC equations will contain only finite powers of the n-valence cluster amplitudes; these equations, unlike the Magnus [ 201 or Wie-Norman expansions [ 2 11, have thus terminating series.
operators with destruction components for the modes 1 and 2: S, bta, and S,b+a,. There are now new closed operators scattering only within the model space: ata, with i,j= 1,2. Let us note that we have included the operators alb and btaz in our description, although they do not figure in H. We have thus exhausted the algebra, and can even write the global solution U as a product of CJ,,and U,, which are given by U~,={exp(s,a]btszaQ ts, !J+a,+S2b+a*)} ,
(19)
(20) 4. Numerical applications We shall illustrate our formalism by applying it to a rather nontrivial problem of reasonable complexity. The problem is prototypical in the sense that it displays the generic features of time-dependent problems and involves several degrees of freedom. The problem has two oscillators of frequencies w and w+& We switch on, at time to=0, both a coupling between them and a coupling of one with a quantized field of frequency w@ The Hamiltonian for the problem is
The hierarchical decoupling of the first model space (zero-valence problem) and the second one (onevalence problem) is clearly illustrated by the set of equations numbered respectively as (2 1) and (22):
.a$,
1%
_
tys~-s,wo-crs:,
as, = (w+d)Sz
-cflSlS2-S2Wo,
(21a (21b
ix
tp,
ax, lx =wotas,
;
(21c )
-4xd:-ys2-ws,,
Wa)
. as,
+ar(a~b+b+a,)+y(a~a,+a~a,)
=o!tws,
lx
=atw,s,
i%
=woS,-S,S*cu-S;y-S2(wtB),
(22b)
i$
=G,t
(22c)
(18)
The Lie-algebra for the Hamiltonian is not complete. We also need to include operators a$b and btu* for its completion (vide infra). We want to study the survivalities of the ground states of the two oscillators and the states when either of them are excited to the respective first mode. We take the photon to be in an eigenstate I n) at t = 0. The model space for our first problem is 10, 0, n} in an obvious notation. This is the zero-valence problem for the modes. For the second problem, the model space is spanned by ] 1, 0, n> and ]0, 1, n) . For the first model space, the external cluster operators can be written as s,a]b and szaJ b, and the closed operator is x,b+b. There are no mode destruction operators. For the second model space, which are one-valence oscillator cases, the equations for s, , s2and &, remain unchanged, but there are now new
c GiKxxJ, K
where G is a matrix, given by GE
wtas;
(
Y
ytaF2 wt6
)
.
(23)
Eqs. (2 1) and (22 ) can be solved by the RungeKutta followed by Adams-Moulton predictor-corrector method. We show in fig. 1 the survivality of the model state 10, 0, n) and the transition probabilities to the “dark” mode 2, i.e. to ]0, 1, n- 1.) and to the “lighted” mode 1, i.e. to I 1, 0,n - 1). For the second, 2,x2 model space, we compute the survivality of the states ] I, 0, n > and ]0,1, n) and also the 87
Volume 186, number I
CHEMICAL PHYSICS LETTERS
1 November 1991
transition probability from mode I to mode 2. It should be noted that mode 1 gets excited by the light and this gets transferred to the dark mode 2 via the oscillator coupling. These quantities are shown in fig. 2. For these computations, we have taken 6= 0 (no detuning) and followed the quantities as o. is varied from off-resonance to near-resonance conditions. The survivalities PoO,P,. and PO, are given as poo= lexp(-M
I*,
(24)
P,0=I(l+nsl~,)exp(~bn)(1+~II)12,
(25)
Po,=~(l+nsz~,)exp(~hn)(1+x22)~2.
(26)
The photon number it has been taken equal to 5 in all computations. We again note that products like S;Ji(afa,btb) from V,, can contribute to survivalities, since they are closed operators. These factors become important when n is rather large, which can be interpreted as “laser enhancement factors”.
5. Concluding remarks
Fig. I. For c1=0.8, o= 2.0, y=O.4,6=0.0: (a) Survivality of the ground state of the system, (b) Transition probability from the ground to the first excited state of mode 1. (c) Transition probability from the ground state to the first excited state of mode 2.
88
We have shown in this Letter a way to formulate a time-dependent CC approach to quantum dynamics via a time-dependent H,, starting from a multireference model space. The central idea is to factorize the evolution operator U as U= U,,U, where U, is a model evolution operator generated by H,, and U,, brings in admixture with virtual functions. By defining suitable “external” and “closed” cluster operators appearing in U,, and U,, very general compact solutions may be found. The equations are terminatingin the series of the cluster amplitudes, owing to the normal ordering imposed on U,, and WM.If we classify the model spaces in terms of n-valence sectors with increasing n, where n-valence sectors have n valence occupancies on top of the vacuum, one can generate the cluster amplitudes for the n-valence problems with increasing n in a hierarchical manner. Cluster amplitudes for the lower valence sectors appear in the equations for the higher valence sectors as known time-dependent quantities. Since the quantities of interest in U,, and U, are the cluster operators, the operator H,, remains extensive, thus not deteriorating in quality for large number of particles. The method should thus prove use-
Volume 186. number I
CHEMICAL PHYSICS LETTERS
I November 1991
ful for studying processes with several degrees of freedom.
Acknowledgement The authors thank G. Dedieu for her kind help in preparing the typescript. Thanks are also due to the DST (New Delhi ) and CNRS (France ) for financial support.
References [I ] E.J. Heller, J. Chem. Phys. 62 ( 1975) 154;Accounts Chem. Res. 14 ( 198I ) 368, and references therein. [ 21D. Kosloff and R. Kosloff, J. Comput. Phys. 52 ( 1983) 35;
Fig. 2. For (r=O.8, 0=2.0, y=O.4, r&0.0: (a) Survivality of the first excited state of mode I. (b) Survivality of the first excited state of mode 2. (c) Transition probability of mode 1 to mode 2 for the model spaces with excited states.
R. Kosloff, J. Phys. Chem. 92 (1988) 2087: R.B. Gerber, R. Kosloff and M. Berman, Computer Phys. Rept. 5 (1986) 59. [ 31 E.A. McCulloughand R.A. Wyatt, J. Chem. Phys. 54 ( I97 I ) 3578; A. Nauts and R.E. Wyatt, Phys. Rev. Letters 51 ( 1983) 2238. f [ 41 G.V. Voth and R.A. Marcus, J. Chem. Phys. 84 ( 1986) 2254. [ 51G.Jolicard and G.D. Billing, Chem. Phys. 104 (1986) 357; 149 (1991) 261. [6] M. Matti Maricq, Phys. Rev. B 25 ( 1982) 6622; W.R. Salzman, Chem. Phys. Letters 124 (1986) 531; I. Chem. Phys. 85 (1986) 4605; G. Campolieti and B.C. Sanctuary, ‘J. Chem. Phys. 87 (1987) 4673; J.Z.H. Zhang, J. Chem. Phys. 92 (1990) 324. [7] F. Wolfand H.J. Korsch, Phys. Rev. A 37 (1988) 1934. [S] B. Grazdy and D.A. Micha, J. Chem. Phys. 82 (1985) 4926; EM. Femandez, Phys. Rev. A 40 (1989) 41; J. Math. Phys. 30 (1989) 1522; J. Echave, P.M. Hemandez and D.A. Micha, J. Chem. Phys. 94(1991)3537. [9] P. Kramer and M. Saraceno, Geometry of the timedependent variational principles (Springer, Berlin, I98 I )_ [IO] F. Arichx, J. Broeckhove, W. Coene and P. van Leuven, Intern. J. Quantum Chem. Symp. 20 (1986) 471. [ I1 ] J. Kucar, H.D. Meyer and L.S. Cederbaum, Chem. Phys. Letters 140 (1987) 525; J. Math. Phys. 29 (1988) 1417; H.D. Meyer, V. Manthe and LX Cederbaum, Chem. Phys. Letters 165 (1990) 73. [ 121H.J. Monkhorst, Intern. J. Quantum Chem. Symp. I I (1977) 421; P. Hoodbhoyand J.W. Negele,Phys. Rev. C 18 ( 1978) 2380; K. Schiinhammer and 0. Gunnarsson, Phys. Rev. C I8 (1978) 660. [ 131K.L. Sebastian, Phys. Rev. B 31 ( 1985) 6976; M.D. Prasad, J. Chem. Phys. 88 (1988) 7005.
89
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CHEMICAL PHYSICS LETTERS
[ 141I. Lindgren and J. Morrison, Atomic many body theory (Springer, Berlin, 1986); P. Durand and J.P. Malrieu, Advan. Chem. Phys. 67 ( 1987) 321. [ 151D. Mukerjee, Intern. J. Quantum Chem. Symp. 20 (1986) 409. [ 161R. Chaudhuri, S. Guha, D. Sinha and D. Mukherjee, in: Lecture notes in chemistry, ed. U. Kaldor (Springer, Berlin, 1989); S. Guha, R. Chaudhuri and D. Mukherjee, in: Condensed matter theory, Vol. 4, ed. J. Keller (Plenum Press, New York, 1989).
90
I November 199I
[ 171D. Mukherjee, Chem. Phys. Letters 125 (1986) 207. [IS] D. Mukherjee and S. Pal, Advan. Quantum Chem. 20 (1989)
291.
[ 191 A. Haque and D. Mukherjee, J. Chem. Phys. 80 (1984) 5058. [20] W. Magnus, Comm. Pure Appl. Math. 7 (1954) 649; P. Pechukas and J.C. Light, J. Chem. Phys. 44 ( 1966) 3897; R.M. Wilcox, J. Math. Phys. 8 (1967) 962. [ 2 I ] J. Wei and E. Norman, J. Math. Phys. 4 ( 1963) 575; Proc. Ann. Math. Sot. 15 (1964) 327.
1 November 1991
CHEMICAL PHYSICS LETTERS
A multireference coupled-cluster approach to quantum dynamics Sruti Guha and Debashis Mukherjee ’ Theory Group,Department ofPhysical Chemistry,Indian Association& the Cultivationof Science. Calcutta 700-032,India Received I6 July 199I
A time-dependent coupled-cluster(TDCC) formalism with a multireference model space for studying quantum dynamics is presented. The evolution operator I/acting on the model space is factorized as U&, where I& evolves in the model space and r/,, brings in virtual functions. Both Uexand UMare written as normal ordered exponentials involving cluster operators. The evolution of UMis governed by an effective Hamiltonian that is size-extensive. An illustrative application on a nontrivial problem indicates the potentiality of the formalism.
1. Introduction Significant progress has been made over the last two decades in the development and application of various time-dependent methods for studying quantum dynamical processes. Recent works have paid special attention to providing compact descriptions of the wavefunctions, admitting of systematic truncation schemes, and generating viable strategies for systems with several degrees of freedom. Particular mention may be made of the methods of Gaussian wave-packet dynamics [ 11, the numerically oriented methods [ 2-5 3, the Lie-algebra inspired formulations [6-81 and the methods based on time-dependent variational principles [9-l 11. We present in this Letter another general method for solving time-dependent problems that combines the advantages of Lie-algebraic ideas and the coupled-cluster (CC) representation of the evolution operators to provide a nonperturbative computational strategy. The theory has the computational advantage of not getting rapidly unwieldy with the increase of the degrees of freedom of the system. Moreover, the CC approach ensures automatically a correct separation of the reactive species in a dynamics and hence should be well suited to study associative and fragmentation dynamical processes. Most of the earlier developments of the time-de’ To whom correspondence
84
should be addressed.
pendent CC methods have used a single reference function from which the evolution was monitored [ 12,131. This is a rather serious restriction as to the types of problems of real interest that could be studied, since quite often we require more than one reference function even for a qualitatively correct description. We shall generalize the concept of effecctive Humiltanians of stationary problems [ 141 to encompass nonstationary situations and develop a CC theory for the evolution process, starting from the reference functions defining a model space.
2. Theory of effective Hamiltonians for timedependent problems Let us consider a situation where we prepare a set nonstationary functions of time-dependent { pK(f)} at some initial time tu, built as superpositions of a set of quasi-degenerate, and generally strongly interacting, time-independent functions spanning a model space Wh)
= c GQ, I
*
(1)
Since the states pK are nonstationary, they will nontrivially evolve in time according to ‘Y,(r)=U(t, to) YIOK(t) >
(2)
with U satisfying 0009-2614/91/S 03.50 0 1991 Elsevier Science Publishers B.V. All rights reserved.
CHEMICAL PHYSICS LETTERS
Volume 186, number 1
iaU( t, to) =H(t)U(t, at
(3)
lo) 2
where we have assumed that H is time dependent in general. U mixes with ul”,sthe virtual functions and also induces a rotation within the model space, amounting to changing the coefficients of {ai}. From now on, we shall use P and Q to indicate projectors onto model and virtual functions. Our desire is to define a model evolution operator UM and a model time-dependent effective Hamiltonian H,, such that the evolution of the model space component { PK( t)} of (YK(t)} can be equivalently described via an equation of motion of the form %(t)=Gl(t,
(4)
to) %(to) ,
where U, satisfies
iWd4 to)
at
=Kfdt)
UM(t,to) *
(5)
Let us note that UMand &are defined in the model space only; also, we need to know only UP, rather than the full U. The key step in our formulation is the use of factorization Ansatz for U as follows: UP= we, U,P)
(6)
where U,, brings the admixture with the complementary space functions. Although U is unitary, we have a freedom in the choice of our model space projection of U,,, PU,,P. This adtomatically fixes U, via unitarity of U, which fixes the orientation of PYK in the model space with respect to the evolved model functions PK(f)_ Substituting eq. (6) into eq. (3) and using eqs. (4) and (5), we obtain iXJ L
at
= HU,, - U,, He,
(7)
From the PP component of eq. (7), we obtain He,= [ U:;] -I
[HUe,]fP-
F
.
(8)
The evolution of Ugp is governed by =
at
= [Huex]Qp-U~pHeff.
(9)
1November 1991
Since He, depends only err U,, and not on UM,eqs. (8 ) and (9 ) imply that we have decoupled U,, and U,. UrL and its time-evolution are fixed by us, and CJgpis obtained from eq. (9). U, is then obtained by solving eq. (5) with H,. as given by eq. (8). We propose to call eq. (7) the time-dependent anaiogue of the Bloch equation, This was put forward earlier by one of us [ 151 in the context of deriving open-shell CC stationary equations for incomplete model spaces via adiabatic hypothesis. The treatment was, however, general which enables us to use the same ideas to derive CC equations for any model space for nonstationary problems.
3. Time-dependent CC equations for a general multireference model space Let us assume that H in the occupation number representation is given by H(t)=
1 hi(t)&, I
(10)
where h,s are the time-dependent matrix elements and 1,sare products of suitable creation-annihilation operators defined in a Fock space. lis always satisfy a Lie-algebra. Let us note that we have kept I,s as independent of time. In case the elements {II} of H do not satisfy a closed Lie-algebra, we may formally assume that the sum over i in eq. (10) runs over the rest of the elements needed for completion with the corresponding h,s as zero. This will be assumed always, even when the algebra is countably infinite. In exact analogy with our earlier works [ 15,161, we classify all the operators into two categories. An operator is closed, denoted as AC,,if it induces transitions only within the model space. An operator is exiernal, denoted as A,, if it can induce a transition outside the model space from at least oozemodel space function oP We shall write U,, and U, as cluster expansions of these two kinds of operators and generate an Hem which is closed and connected and thus respects size-extensivity [ 1516 1. We take a suitable physical vacuum, and rewrite H in normal order. For fermions, the choice of a hole-particle vacuum is often physically quite straightforward, but for bosons a nontrivial Bogilyubov transformation may be necessary. U, is 85
Volume 186, number I
CHEMICAL PHYSICS LETTERS
written as a normal ordered exponential of a closedcluster operator X=X,,: UM =iexp(x>)
.
(11)
For U,,, we do not have a unique choice. It should obviously involve external operators S,,, but it may additionally have closed operators S,, depending on how we want to fix [ CJe,] cl. For the minimal description, it is suflcient to have only S=S,, in Uex: G, ={exp(S)l . Other
choices,
(12)
e.g., the
isometry
condition (viz, giving Hermitian Hen). We do not consider them here, but quote our earlier works [ 15,161. The choice (12) leads to a non-Hermitian Heff, giving a nonunitary U,. This is understandable on physical grounds: with the passage of time, the system propagates in the virtual space induced via Ue, which “stretches” the function via S,,; U, not only rotates it within the model space but also “squeezes” it to restore normalization. UMcannot thus be normconserving by itself. Substituting CJei,, from eq. ( 12) into (7), and rewriting them in normal order, we have [WA Uex]cl= ICI, are sometimes convenient
iaS, ex = (HU.Z, Q, I- I we,,&J&J 1 at 1
3
(13)
where (HU,,} etc. denote the connected expressions like tH[exp(S) I} J‘oining H with powers of S from the exponential, omitting contractions among the S operators. Since we are working in the Fock space, the n-body components of U,, are linearly independent, and it follows then that if
={HCT,,]-{G&-r}
-
(14)
Taking the closed part of eq. ( 14), using [SIC,=@ we have ~ULL&f1=IHU,,L,
1
(15)
which defines Hen. With the external part, we likewise have i~={HU,,}..-(LI,,H,J,,.
(16)
We get S by solving eq. ( 16). X is solved next via an analogous CC equation 86
.ax -
1%
={&Al),
1November 1991
(17)
obtained by substituting eq. ( 11) into eq. (5). Eqs. ( 15)- ( 17) indicate that S, X and & are all connected. The TDCC theory thus generates a sizeextensive formulation - a very important requirement for treating large systems. To obtain survivalities in the model space, we need the matrix cl- < @,I CJe,UM (@ii>. It may appear that U,, does not contribute to rti since U,, has only external operators S,,. This is not generally so. For a complete model space, which is realizable generally for fermions only, powers of external operators remain external [ 1% 181. But for incomplete model spaces, where all possible occupancies of the partially filled orbitals are not exhausted (this is naturally the case for bosons because of no restriction in occupancy) S,, can lead to scattering within the model space, and powers thereof can even be closed [ 15-181. [ iYe,Jcl is thus not generally I,,, and this should be borne in mind while using expressions which potentially involve [ Ve,]c,. The situation is analogous to that in stationary problems [ 17,181. An attempt to restore [ U,, ] cI= 1clleads to disconnected H,,and a breakdown of the linked cluster expansion 1171.
Eqs. ( 16) and ( 17) are the principal working equations of our time-dependent CC formalism using any arbitrary model space. A very appealing feature of the TDCC equations is the systematic hierarchical decoupling of the equations for the various n-particle model space sectors. Owing to the normal ordering, the cluster operators cannot contract among themselves. As a result, for the zero-valence problem (i.e. the vacuum as the model space), the equations will have only those S operators having no destruction operators. Operators with destruction operators will trivially give zero by their action on the model space. For the one-valence problem (one quasi-particle on the vacuum), we would need additional operators having one destruction operator. Since the zero-valence cluster operators are already known, they will simply appear as time-dependent parameters in the equations for one-valence operators. This same pattern is repeated for the higher valence problems. The situation is entirely analogous to what one encounters in the sta-
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tionary case, and may be termed as the time-dependent counterpart of the “subsystem embedding condition“ [ 191. Another interesting aspect of our development is that, when our choice of S and X exhausts the Liealgebra, the solution U= &,U, is also the global solution, independent of the choice of u particular type of model space.
We also note here that for any given n-valence sector, the TDCC equations will contain only finite powers of the n-valence cluster amplitudes; these equations, unlike the Magnus [ 201 or Wie-Norman expansions [ 2 11, have thus terminating series.
operators with destruction components for the modes 1 and 2: S, bta, and S,b+a,. There are now new closed operators scattering only within the model space: ata, with i,j= 1,2. Let us note that we have included the operators alb and btaz in our description, although they do not figure in H. We have thus exhausted the algebra, and can even write the global solution U as a product of CJ,,and U,, which are given by U~,={exp(s,a]btszaQ ts, !J+a,+S2b+a*)} ,
(19)
(20) 4. Numerical applications We shall illustrate our formalism by applying it to a rather nontrivial problem of reasonable complexity. The problem is prototypical in the sense that it displays the generic features of time-dependent problems and involves several degrees of freedom. The problem has two oscillators of frequencies w and w+& We switch on, at time to=0, both a coupling between them and a coupling of one with a quantized field of frequency w@ The Hamiltonian for the problem is
The hierarchical decoupling of the first model space (zero-valence problem) and the second one (onevalence problem) is clearly illustrated by the set of equations numbered respectively as (2 1) and (22):
.a$,
1%
_
tys~-s,wo-crs:,
as, = (w+d)Sz
-cflSlS2-S2Wo,
(21a (21b
ix
tp,
ax, lx =wotas,
;
(21c )
-4xd:-ys2-ws,,
Wa)
. as,
+ar(a~b+b+a,)+y(a~a,+a~a,)
=o!tws,
lx
=atw,s,
i%
=woS,-S,S*cu-S;y-S2(wtB),
(22b)
i$
=G,t
(22c)
(18)
The Lie-algebra for the Hamiltonian is not complete. We also need to include operators a$b and btu* for its completion (vide infra). We want to study the survivalities of the ground states of the two oscillators and the states when either of them are excited to the respective first mode. We take the photon to be in an eigenstate I n) at t = 0. The model space for our first problem is 10, 0, n} in an obvious notation. This is the zero-valence problem for the modes. For the second problem, the model space is spanned by ] 1, 0, n> and ]0, 1, n) . For the first model space, the external cluster operators can be written as s,a]b and szaJ b, and the closed operator is x,b+b. There are no mode destruction operators. For the second model space, which are one-valence oscillator cases, the equations for s, , s2and &, remain unchanged, but there are now new
c GiKxxJ, K
where G is a matrix, given by GE
wtas;
(
Y
ytaF2 wt6
)
.
(23)
Eqs. (2 1) and (22 ) can be solved by the RungeKutta followed by Adams-Moulton predictor-corrector method. We show in fig. 1 the survivality of the model state 10, 0, n) and the transition probabilities to the “dark” mode 2, i.e. to ]0, 1, n- 1.) and to the “lighted” mode 1, i.e. to I 1, 0,n - 1). For the second, 2,x2 model space, we compute the survivality of the states ] I, 0, n > and ]0,1, n) and also the 87
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transition probability from mode I to mode 2. It should be noted that mode 1 gets excited by the light and this gets transferred to the dark mode 2 via the oscillator coupling. These quantities are shown in fig. 2. For these computations, we have taken 6= 0 (no detuning) and followed the quantities as o. is varied from off-resonance to near-resonance conditions. The survivalities PoO,P,. and PO, are given as poo= lexp(-M
I*,
(24)
P,0=I(l+nsl~,)exp(~bn)(1+~II)12,
(25)
Po,=~(l+nsz~,)exp(~hn)(1+x22)~2.
(26)
The photon number it has been taken equal to 5 in all computations. We again note that products like S;Ji(afa,btb) from V,, can contribute to survivalities, since they are closed operators. These factors become important when n is rather large, which can be interpreted as “laser enhancement factors”.
5. Concluding remarks
Fig. I. For c1=0.8, o= 2.0, y=O.4,6=0.0: (a) Survivality of the ground state of the system, (b) Transition probability from the ground to the first excited state of mode 1. (c) Transition probability from the ground state to the first excited state of mode 2.
88
We have shown in this Letter a way to formulate a time-dependent CC approach to quantum dynamics via a time-dependent H,, starting from a multireference model space. The central idea is to factorize the evolution operator U as U= U,,U, where U, is a model evolution operator generated by H,, and U,, brings in admixture with virtual functions. By defining suitable “external” and “closed” cluster operators appearing in U,, and U,, very general compact solutions may be found. The equations are terminatingin the series of the cluster amplitudes, owing to the normal ordering imposed on U,, and WM.If we classify the model spaces in terms of n-valence sectors with increasing n, where n-valence sectors have n valence occupancies on top of the vacuum, one can generate the cluster amplitudes for the n-valence problems with increasing n in a hierarchical manner. Cluster amplitudes for the lower valence sectors appear in the equations for the higher valence sectors as known time-dependent quantities. Since the quantities of interest in U,, and U, are the cluster operators, the operator H,, remains extensive, thus not deteriorating in quality for large number of particles. The method should thus prove use-
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ful for studying processes with several degrees of freedom.
Acknowledgement The authors thank G. Dedieu for her kind help in preparing the typescript. Thanks are also due to the DST (New Delhi ) and CNRS (France ) for financial support.
References [I ] E.J. Heller, J. Chem. Phys. 62 ( 1975) 154;Accounts Chem. Res. 14 ( 198I ) 368, and references therein. [ 21D. Kosloff and R. Kosloff, J. Comput. Phys. 52 ( 1983) 35;
Fig. 2. For (r=O.8, 0=2.0, y=O.4, r&0.0: (a) Survivality of the first excited state of mode I. (b) Survivality of the first excited state of mode 2. (c) Transition probability of mode 1 to mode 2 for the model spaces with excited states.
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