1.5 Gaussian Wave Packet based methods
1.5.2 Corrections to the classical path equations: Hermite corrections 37
field (SCF) approach to correct the Gaussian form of the Wave packet [14]. One such correction is the multitrajectory approach [15], where the interaction between the two subsystems, namely the quantum and the classical, demands more GWP’s as the coupling increases. However, these equations of motion for the GWP’s can be troublesome to work with in practical calculations. The GWP’s form a nonorthogonal basis set and the solution of the equations of motion involves the inversion of nearly singular matrices.
Another correction is the attempt to correct the separability approximation, namely, if the degrees of freedom are strongly coupled, we can not expect to get an accurate description with the simple product form of the wave function. This kind of correction is termed as the MCSCF (multiconfiguration self-consistent field) approach. If the coupling between the R and r degrees of freedom is weak we expect the separability approximation to be valid. Since GWP is the correct form of the asymptotic (the initial) wavefunction leading to the classical limit in a natural fashion, the corrected theory has the classical path theory as the limit. A convenient way to achieve this correction is based on the use of Hermite polynomial with GWP [15]. The Hermite correction is also known as the quantum correction to the classical path equation.
The quantum correction to the classical path equation is based upon the intro- duction of time-dependent basis function which in the limit gives the classical path equation. This basis set is introduced only in those degrees of freedom which are expected to behave classically. The method corrects both in the SCF and MCSCF direction, and is therefore ideal for the correction of the classical path equations. It also contains the options for an easy determination of the correlation between the classical and the quantum degree(s) of freedom and thereby, to determine whether the primitive classical path theory is expected to work or not.
1.5.3 The Hermite correction method The product type wavefunction is expressed as
Ψ(r, R, t) =X
nk
ank(t)φn(r)Φk(R, t) (1.5.9) The total Hamiltonian contains the following terms:
H(r, R) = ˆH0+TR+V(r, R) (1.5.10) where the function φn(r) are chosen to be the eigenfunction ofH0,
H0φn(r) =Enφn(r) (1.5.11) and TR is the kinetic energy for the R-motion and V(r, R) the potential which cou- ples the the two degrees of freedom. At t = t0, the system is represented by the wavefunction
Ψ(r, R, t0) =φl(r)ΦGW P(R, t0) (1.5.12) where ΦGW P(R, t0) is a Gaussian wave packet
ΦGW P(R, t0) = exp i
¯
h[γ(t0) +P0(R−R(t0)) +A(t0)(R−R(t0))2]
(1.5.13)
centered around R0 = R(t0) in coordinate and P0 in momentum space. The initial condition of the parameters can be taken to be
R0 large, (1.5.14)
P0 arbitrary, (1.5.15)
ReA(t0) = 0, (1.5.16)
ImA(t0) arbitrary, (1.5.17)
Reγ(t0) = 0, and (1.5.18)
Imγ(t0) =−¯h
4ln(2ImA(t0)/π¯h) (1.5.19) where the last equation follows from normalization. The initial value of the width parameter ImA(t0) and P0 are arbitrary in the sense that the final solution is inde- pendent of where in momentum space we place the wave packet and also of its width.
In practical calculation there will be a dependence on those parameters due to nu- merical inaccuracies but the wave packet can represent a finite energy range around the center momentum accurately.
Let the function Φk(R, t) at any time t is Φk(R, t) =π14 exp
i
¯
h γ(t) +P(t)[R−R(t)] +ReA(t)[R−R(t)]2
ξk(x)(1.5.20) where
x=
r2ImA(t)
¯
h [R−R(t)] (1.5.21)
and
ξk(x) = 1 pk!2k√
πexp−x2 2
Hk(x) (1.5.22)
are the Harmonic oscillator wave functions. The basis function has two important properties. First they form an orthonormal basis, i.e.,
Z
dRΦ∗k(R, t)φn(R, t) =δkn (1.5.23)
and second the ground state is the Gaussian wave packet given by Eq. (1.5.13). It is important to note that the normalization condition not only holds at t0 but at all time t.
If we include only one basis function and replace an0 with an we obtain Ψ(r, R, t) = X
n
an(t)φn(r)φ0(R, t) (1.5.24) When this expansion is inserted in the TDSE, the classical path equations appears as
i¯ha˙n(t) = Enan(t) +X
m
Vnm(R(t))am(t), (1.5.25) P˙(t) = −X
nm
a∗n(t)am(t)∂Vnm(R)
∂R
R=R(t), (1.5.26)
R˙ = P(t)
µ , (1.5.27)
A(t) =˙ −2
µA(t)2−1 2
X
nm
a∗n(t)am(t)∂2Vnm
∂R2
R=R(t), (1.5.28)
˙
γ(t) = P(t)2
µ +i¯hA(t)
µ , (1.5.29)
where the matrix element Vnm is defined by
Vnm(R) =hφn|V(R, r)|φmi (1.5.30) At this junction, we can formulate rigorous classical path equations of motion within the MCSCF framework by using the following equations
d
dtR(t) = P(t)
µ (1.5.31)
d
dtImA(t) = −4
µReA(t)ImA(t) (1.5.32)
d
dtImγ(t) = ¯h
µReA(t) (1.5.33)
The first equation introduce the classical path picture (the center of the wave packet follows a classical equation of motion) and the two other equations ensures that the
basis set is normalized at all time t. If we insert the Eq. (1.5.9) in the TDSE and use Eqs. (1.5.26) - (1.5.28), the coupled equations for the expansion coefficients aml are as below,
i¯ha˙ml = Emaml+X
nk
ankVmnlk(t) +aml
2l¯hImA(t)
µ + P(t)2
2µ − ¯h(l− 12) 4ImA(t)Vef f′′
− 1
2Vef f′ p
¯
h/ImA(t)√
lam,l−1+√
l+ 1am,l+1
− 1 8
¯ h
ImA(t)Vef f′′ hp
l(l−1)am,l−2+p
(l+ 1)(l+ 2)am,l+2
i (1.5.34)
where
Vmnlk(t) = Z
dRΦ∗l(R, t)Vmn(R)Φk(R, t) (1.5.35) but Vef f′ and Vef f′′ are not specified yet. These quantities enter into the equations of motion forP(t) and ReA(t),
P˙(t) = −Vef f′ (1.5.36)
d
dtReA(t) = −2
µ(ReA(t)2−ImA(t)2)− 1
2Vef f′′ (1.5.37) Comparing Eqs. (1.5.21) and (1.5.23) we can write
Vef f′ = X
ml;nk
a∗ml(t)ank(t) Z
dR×Φ∗l(R, t) d
dRVnm(R)|R=R(t)Φk(R, t) (1.5.38) Vef f′′ = X
ml;nk
a∗ml(t)ank(t) Z
dR×Φ∗l(R, t) d dR
2
Vnm(R)|R=R(t)Φk(R, t)(1.5.39) However it is possible to derive rigorous expressions by using the Dirac - Frenkel variational principle by minimizing the following integral
Z
drdR
−i¯h∂Ψ(r, R, t)∗
∂t −HΨ(r, R, t)∗
×
i¯h∂
∂tΨ(r, R, t)−HΨ(r, R, t) (1.5.40) The equations of motion are obtained by differentiation with respect to ˙P and d
dtReA(t)
and after some manipulations(!) we finally get
Vef f′ =
Q1 S(3)−X12
Q2 S(4)−X22
S(2)−X11 S(3)−X12
S(3)−X21 S(4)−X22
(1.5.41)
and
1
2Vef f′′ =
S(2)−X11 Q1
S(3)−X21 Q2
S(2)−X11 S(3)−X12
S(3)−X21 S(4)−X22
(1.5.42)
where
Q1 = X
nk;ml
a∗nkaml Vml;nk(1) −X
p
Skp(1)Vnp;ml(0)
!
(1.5.43)
Q2 = X
nk;ml
a∗nkaml Vml;nk(2) −X
p
Skp(2)Vnp;ml(0)
!
(1.5.44) Xij = X
nk;np
X
l
a∗nkanpSkl(i)Slp(j) (1.5.45) Vml;nk(j) =
Z
dRΦ∗l(R, t)Vmn(R)(R−R(t))jΦk(R, t) (1.5.46) Slk(j) =
Z
dRΦ∗l(R, t)(R−R(t))jΦk(R, t) (1.5.47) S(j) = X
n
X
lk
a∗nkanlSkl(j) (1.5.48)
One can introduce the projection operatorQ=I−P, whereI is the identity operator and
P = XN m=0
|ΦmihΦm| (1.5.49)
andN is the highest Hermite polynomial used in the expansion. Thus we can express
the quantities involved in calculating the potential derivatives as Qi = X
mnkl
a∗nkamlhΦk|△RiQVmn(R)|Φli (1.5.50) S(i+j)−Xij = X
nkl
a∗nkanlhΦk|△RiQ△Rj|Φli (1.5.51) Though in the limit P → I we have Q → 0, i.e., both numerator and denominator approaches zero, still a meaningful ratio can be obtained.
1.5.4 Time-dependent discrete variable representation method (TDDVR)