Formulation of the TDDVR method for multi - dimensional multi -

2.3 Formulation of the TDDVR method for multi

The kinetic energy operator, ˆT_{s^k_} =−¯h² 2µ

h ∂²

∂s¹²+ ∂²

∂s²²+···+ ∂²

∂s^p2

i, and the diagonal elements of the matrix in Eq. (2.3.3) represent potential energy operators for different surfaces where off-diagonal elements define the coupling among them. The lthelement of the wavefunction matrix in Eq. (2.3.4) is the wavefunction for the lth surface and we can expand the time-dependent wavefunction (Ψl({s^k}, t)) as below,

Ψl({s^k}, t) = X

i1i2...ip

ci1i2....ip,l(t) Yp k=1

ψik(s^k, t), (2.3.5) where ψik(s^k, t) is the TDDVR basis function for the kth mode which can again be expanded in terms of DVR basis multiplied by plane wave to represent the coordinate, s^k, as a function of time, t,

ψik(s^k, t) = φ(s^k, t)

Nk

X

n=0

ξ_n^⋆(x^k_ik)ξn(x^k) (2.3.6)

=

Nk

X

n=0

ξ_n^⋆(x^k_ik)Φn(s^k, t) (2.3.7) φ(s^k, t) = π^1/4expi

¯ h

ps^k_c(t)[s^k−s^k_c(t)] , (2.3.8)

with harmonic oscillator eigenfunctions as the primitive ones to construct the DVR basis,

ξ_n(x^k) = 2ImA^k π¯h

1/4 1 pn!2ⁿ√

πexpn

− (x^k)² 2

oH_n(x^k) (2.3.9)

and

x^k =

r2ImA^k

¯

h (s^k−s^k_c(t)), (2.3.10) x^k_ik =

r2ImA^k

¯

h (s^k_ik(t)−s^k_c(t)). (2.3.11) Alternatively, we can say{Φn(s^k, t)}form the G-H basis set. Among the parame- ters in the G-H basis (Eqs. (2.3.5) - (2.3.11)) we choose the center of the wavepacket ({s^k_c}) and its’ momentum ({p_s^k_c}) as time-dependent variables, imaginary part of

the width ({ImA^k}) as time-independent parameter and real part of{γ^k}and width ({ReA^k}) are zero for all the time.

The index, ik, in Eqs. (2.3.5) and (2.3.6) counts the DVR basis functions for the kth mode where Nk is the maximum number of Hermite basis functions included in Eq. (2.3.6) to expand a particular DVR basis for the same mode, p and M are the maximum number of modes and surfaces, respectively involved in a system. A DVR grid-point,s^k_ik, will be determined by Eq. (2.3.11) using the root,x^k_ik, ofNkthHermite polynomial,HNk(x^k) [17] Though the roots (x^k_iks) of the polynomial are fixed values, the position of the DVR grid-points (s^k_iks) will change as a function of time due to the time-dependent variables, s^k_c(t),

s^k_ik(t) =s^k_c(t) +

r ¯h

2ImA^kx^k_ik (2.3.12)

The Gauss-Hermite basis, Φ_n(s^k, t), for the kth mode as introduced in Eq. (2.3.7) have the following two important properties. Firstly, they form an orthonormal basis,

Z

ds^kΦ^⋆_m(s^k, t)Φn(s^k, t) = δ^k_mn (2.3.13) and secondly, the ground state of the G-H basis is the Gaussian Wave Packet.

The DVR basis functions, ψiks, in Eq. (2.3.6) for the kth mode constitute an orthogonal but not normalized set,

Z

dskψ_i^⋆_k(s^k, t)ψi^′_k(s^k, t) =δiki^′_kA^k_iki′

k, (2.3.14)

where A^k is the normalization factor.

We substitute Eqs. (2.3.2) - (2.3.5) in the time-dependent Schroedinger equation (Eq. (2.3.1)) and due to the special choice of the form of the TDDVR basis functions as defined in Eq. (2.3.6), the classical path picture appears naturally along with the DVR matrix equation. The general form of differential equation for any surface, l, (

l, l^′ ∈ {1, M}) turns out to be, i¯h X

i1i2...ip

˙

ci¹i²....ip,l(t) Yp k=1

nX^Nk

n=0

ξ_n^⋆(x^k_ik)Φn(s^k, t)o

+ i¯hX

k

X

i¹i²...ip

ci¹i²....ip,l(t)nX^Nk

n=0

ξ^⋆_n(x^k_ik)∂

∂tΦn(s^k, t)oY^p

k^′6=k

nX^Nk′

n=0

ξ_n^⋆(x^k_ik′^′ )Φn(s^k′, t)o

= X

k

X

i¹i²...ip

ci¹i²....ip,l(t)nX^Nk

n=0

ξ_n^⋆(x^k_ik) ˆT_s^kΦn(s^k, t)o Y^p

k^′6=k

nX^Nk′

n=0

ξ_n^⋆(x^k_i^′

k′)Φn(s^k′, t)o

+ X

i¹i²...ip

c_i1i²....ip,l(t)V_ll({s^k_ik}) Yp k=1

nX^Nk

n=0

ξ^⋆_n(x^k_i)Φ_n(x^k)o

+ X

l^′6=l

X

i1i2...ip

ci1i2....ip,l^′(t)Vll^′({s^k_ik}) Yp k=1

nX^Nk

n=0

ξ_n^⋆(x^k_i)Φn(x^k)o

(2.3.15)

When we evaluate the terms of _∂t^∂Φn(s^k, t) and ˆT_s^kΦn(s^k, t) for the kth mode and substitute back in Eq. (2.3.15), some of these terms cancel each other. Collecting the remaining terms, along with simple manipulation we arrive at the evolution equation for quantum motion on the lth surface in a compact matrix form,

i¯hA ˙C^l = H^tllC^l+AX

l′6=l

V^ll′C^l′ (2.3.16) which can be reorganized into a more convenient form by a similarity transformation,

i¯hD˙^l(t) =A^−1/2H^tllA^−1/2D^l+X

l′6=l

V^ll′D^l′ (2.3.17) where D^l =A^1/2C^l.

The explicit expression of an element of the TDDVR coefficients, di1i2....ip,l is, i¯hd˙i¹i²....ip,l = 1

2 n X

k

˙ p_s^k_c

r ¯h

ImA^kX¯_i^k_kiko

di¹i²....ip,l+n X

k

µ( ˙s^k_c)² 2

o

di¹i²....ip,l

+ Vll(i1i2....ip)di¹i²....ip,l+X

l^′6=l

Vll^′(i1i1....ip)di¹i²....ip,l^′

+ X

k

n¯hImA^k 2µ

X

i^′1i^′2....i^′_p

2 ¯Y_i^k_ki′

k −Z¯_i^k_ki′

k di^′1i^′2...i^′_p,l

Yp k^′6=k

δi_k′i^′

k′

o

(2.3.18)

where X¯_i^k_k,i′

k = X_i^k

k,i^′_k

qA^k_ik,ikA^k_i′ k,i^′_k

, Y¯_i^k_k,i′

k = Y_i^k

k,i^′_k

qA^k_ik,ikA^k_i′ k,i^′_k

, and ¯Z_i^k_k,i′

k = Z_i^k

k,i^′_k

qA^k_ik,ikA^k_i′ k,i^′_k

(2.3.19)

d_i1i²....ip,l = c_i1i²....ip,l

Yp k=1

(A^k_ik,ik)¹², (2.3.20)

A^k_ik,i′

k =

N^k

X

n=0

ξ_n^⋆(x^k_ik)ξn(x^k_i′

k), (2.3.21)

X_i^k_k,i′

k =

NX^k−1 n=0

ξ_n+1^⋆ (x^k_ik)√

n+ 1ξn(x^k_i′ k) +

N^k

X

n=1

ξ_n−1^⋆ (x^k_ik)√

nξn(x^k_i′

k), (2.3.22) Y_i^k_k,i′

k =

N^k

X

n=0

ξ_n^⋆(x^k_ik)2nξn(x^k_i′

k), (2.3.23)

Z_i^k_k,i′

k =

NX^k−2 n=0

ξ_n+2^⋆ (x^k_ik)p

(n+ 1)(n+ 2)ξn(x^k_i′ k) +

N^k

X

n=2

ξ_n−2^⋆ (x^k_ik)p

n(n−1)ξn(x^k_i′ k) +

N^k

X

n=0

ξ_n^⋆(x^k_ik)(2n+ 1)ξ_n(x^k_i′

k) (2.3.24)

Component matrices ({A^k},{X^k},{Y^k}, {Z^k}) of the TDDVR Hamiltonian matrix Eq. (2.3.18) can infer very important significance on the quantum-classical nature of this approach. {A^k}, {X^k}, {Y^k} and {Z^k} matrices (Tables 2.1 - 2.4) are time- independent and need to be evaluated once for all. As{X^k}matrices are diagonal and multiplied with the “classical” variables{p˙_s^k_c(t)} respectively, the quantum equations of motion is free from any non-linearity due to “classical” motion, i.e., “classical”

mechanics can only affect the convergence of the quantum dynamics. Again, as couplings among the grid-points come through the full matrices {Y^k}and {Z^k} and dominantly shape the quantum dynamics, we wish them to be free from any non- linearity due to “classical” propagation by choosing{ImA^k}as the time-independent variable. Moreover, both the matrices {Y^k} and {Z^k} are diagonally dominant, but the average value of the off-diagonal elements of {Z^k} matrix is much larger than these of the {Y^k}matrix. Hence the {Z^k}matrix unlike{Y^k}can not facilitate the

converged solution of quantum equations of motion if {ImA^k}is considered as time- dependent. On the other hand, in order to formulate a rigorous equation of motion, one needs to include {Z^k} matrix and also treat {ImA^k} as a time-independent variable. Last but not the least, Eq. (2.3.18) clearly indicates that {Y^k} and {Z^k} matrices couple grid-points or basis function of the kth mode only, hence numerical handling of differential equation for each mode can be done by parallel computation, reducing computational cost and making large dimensional calculations possible.

Similarly, the classical path equations for the kth mode can be written as s˙^k_c(t) = p_s^k_c(t)

µ (2.3.25)

˙

ps^k_c(t) = −D

Ξ({s^k}, t)

dVˆ({s^k}) ds^k

s^k=s^k_c(t)

Ξ({s^k}, t)E

(2.3.26) where the classical force is evaluated by substituting the matrices, Ξ({s^k}, t) and Vˆ({s^k}) (Eqs. (2.3.3) and (2.3.4)), respectively.

Using Dirac-Frenkel variational principle [37], we now arrive at a rigorous expres- sion of ˙p_s^k_c for multi-dimensional multi-surface systems by minimizing the following integral,

I = Z

−i¯h∂Ξ^⋆({s^k}, t)

∂t −H({p^k_s},{s^k})Ξ^⋆({s^k}, t)

×

i¯h∂Ξ({s^k}, t)

∂t −H({p^k_s},{s^k})Ξ({s^k}, t)Y^p

k=1

ds^k, (2.3.27) with respect to {p˙_s^k_c}. When this derivation (!!!) is done, {p˙_s^k_c} takes the following

form,

˙

p_s^k_c(t) =

2(ImA^k)² µ

n X

l

X

iki^′_k

c^⋆_i1i²..ik..ip,l(t) P

i^′′_k ci1i2..i^′′_k..ip,l(t)S_i⁽²⁾′′

ki^′_k

A_i^′

ki^′_k

S^⋆(1)_i

ki^′_k

+ X

l

X

iki^′_k

P

i^′′c^⋆_i1i2..i′′

k..ip,l(t)S^⋆(2)_i′′

kik

A_ikik c_i1i²..i^′_k..ip,l(t)S^⋆(1)_i

ki^′_k

− 2X

l

X

iki^′_k

c^⋆_i1i²..ik..ip,l(t)ci1i2..i^′_k..ip,l(t)S^⋆(3)_i

ki^′_k

o

− hImA¯ ^k µ

n X

l

X

iki^′_k

c^⋆_i1i²..ik..ip,l(t) P

i^′′_kc_i1i²..i^′′_k..ip,l(t)R_i^′′

ki^′_k

Ai^′_ki^′_k

S^⋆(1)_i

ki^′_k

+ X

l

X

iki^′_k

P

i^′′_kc^⋆_i1i2..i′′

k..ip,l(t)R^⋆_i′′

kik

Aikik

ci1i2..i^′_k..ip,l(t)S^⋆(1)_i

ki^′_k

− 2X

l

X

ikik′

c^⋆_i1i²..ik..ip,l(t)ci1i2..i^′_k..ip,l(t)T_i^⋆_ki′ k

o

X

l

X

iki^′_k

c^⋆_i1i²..ik..ip,l(t) P

i^′′_kci1i2..i^′′_k..ip(t)S_i⁽¹⁾′′

ki^′_k

A_i^′

ki^′_k

S^⋆(1)_i

ki^′_k

+ X

l

X

iki^′_k

P

i^′′_kc^⋆_i1i2..i′′

k..ip,l(t)S^⋆(1)_i′′

kik

A_ikik ci1i2..i^′_k..ip,lS^⋆(1)_i

ki^′_k

− 2X

l

X

iki^′_k

c^⋆_{i1i2..ik..ip,l}(t)ci1i2..i^′_k..ip,l(t)S^⋆(2)_i

ki^′_k

(2.3.28)

where

Riki^′_k = X

p

ξ^⋆_p(x^k_ik)ξp(x^k_i′ k)2p S_i⁽ⁿ⁾

ki^′_k = X

pq

ξ^⋆_p(x^k_ik)ξq(x^k_i′ k)

Z

Φ^⋆q(s^k, t)(s^k−s^k_c(t))ⁿΦp(s^k, t)ds^k T_iki^′

k = X

pq

ξp(x^k_ik)ξ^⋆_q(x^k_i′ k)2p

Z

Φ^⋆p(s^k, t)(s^k−s^k_c(t))Φq(s^k, t)ds^k (2.3.29)

with Z

Φ^⋆p(s^k, t)(s^k−s^k_c(t))Φq(s^k, t)ds^k = 1 2

r ¯h ImA^k

npp+ 1δp+1,q+√pδp−1,q

o Z

Φ^⋆p(s^k, t)(s^k−s^k_c(t))²Φq(s^k, t)ds^k = ¯h 4ImA^k

np(p+ 1)(p+ 2)δp+2,q

+ (2p+ 1)δp,q+p

p(p−1)δp−2,q

o Z

Φ^⋆p(s^k, t)(s^k−s^k_c(t))³Φq(s^k, t)ds^k = 1 8

¯h ImA^k

3/2np

(p+ 1)(p+ 2)(p+ 3)δp+3,q

+ 3(p+ 1)p

p+ 1δp+1,q + 3p√pδp−1,q

+ p

p(p−1)(p−2)δp−3,q

o

(2.3.30) Matrices R, S⁽ⁿ⁾, T and A involved in Eq. (2.3.28) are time-independent and can be calculated once for all the time while the time-dependence of ˙p_s^k_c arises from the time-dependent coefficient {ci1i2..ik..ip,l(t)}. At this point, it is important to note that

˙

p_s^k_c, k= 1, . . . , p does not depend explicitly on the potential of the system unlike its form in ref [27]. The compact form of ˙p_s^k_c in Eq. (2.3.28) can further be simplified when we replace S_i⁽¹⁾

ki^′_k by S_i⁽¹⁾

ki^′_kδiki^′_k as the first order term (s^k−s^k_c(t)) is diagonal in DVR (Table 2.2),

˙

p_s^k_c(t) = X

l

X

iki^′_k

c^⋆_iii2..ik..ip,l(t)ci1i2..i^′_k..ip,l(t)

× n2(ImA^k)² µ

hS_i⁽²⁾

ki^′_k

S^(1)⋆_ikik Aikik

−S_i⁽³⁾

ki^′_k

i−hImA¯ ^k µ

hR_iki^′

k

S^(1)⋆_ikik Aikik

−T_i^⋆_ki′ k

io

/ h X

l

X

ik

c^⋆_{i1i2..ik..ip,l}(t)c_i1i²..ik..ip,l(t)S^(1)⋆_ikikS_i⁽¹⁾_kik Aikik

− X

i

X

iki^′_k

c^⋆_i1i²..ik..ip,l(t)ci1i2..i^′_k..ip,l(t)S^⋆(2)_i

ki^′_k

i

(2.3.31)