for one dimension; for many dimensions the matrix is no longer tridiagonal

and special sparse-matrix techniques should be used.

An elegant and e¬cient way to solve (5.11) has been devised by de Raedt

2 Most textbooks on partial di¬erential equations will treat the various schemes to solve initial

value problems like this one. In the context of the Schr¨dinger equation Press et al. (1992)

o

and Vesely (2001) are useful references.

5.2 Quantum dynamics in a non-stationary potential 117

time

‘

t+„

t+„/2

t

’x/δ

n-1 n n+1

Figure 5.3 Space-time grid for the implicit, time-reversible Crank“Nicholson

scheme. A virtual point at xn , t + „ /2 is approached from t and from t + „ , yielding

U (’„ /2)Ψ(t + „ ) = U („ /2)Ψ(t).

(1987, 1996), using the split-operator technique (see Chapter 14). It is

possible to split the operator H in (5.9) into a sum of easily and exactly

solvable block-diagonal matrices, such as

H = H1 + H2 ,

with

⎛ ⎞

a0 b

⎜ b a1 /2 ⎟

⎜ ⎟

⎜ ⎟

a2 /2 b

H1 = ⎜ ⎟ (5.15)

⎜ ⎟

b a3 /2

⎝ ⎠

..

.

⎛ ⎞

0

⎜ ⎟

a1 /2 b

⎜ ⎟

⎜ ⎟

b a2 /2

⎜ ⎟

H2 = ⎜ ⎟ (5.16)

a3 /2 b

⎜ ⎟

⎜ ⎟

⎝ ⎠

b a4 /2

..

.

Each 2 — 2 exponential matrix can be solved exactly, and independently

of other blocks, by diagonalization (see Chapter 14, page 388), yielding a

2 — 2 matrix. Then the total exponential matrix can be applied using a

form of Trotter“Suzuki splitting, e.g., into the second-order product (see

118 Dynamics of mixed quantum/classical systems

Chapter 14, page 386)

e’iH„ = e’iH1 „ /2 e’iH2 „ e’iH1 „ /2 . (5.17)

The Hamiltonian matrix can also be split up into one diagonal and two block

matrices of the form

’i sin b„

0b cos b„

exp ’i„ = , (5.18)

’i sin b„

b0 cos b„

simplifying the solution of the exponential block matrices even more. For d

dimensions, the operator may be split into a sequence of operators in each

dimension.3

While the methods mentioned above are pure real-space solutions, one

ˆ

may also split the Hamiltonian operator into the kinetic energy part K and

ˆ

the potential energy part V . The latter is diagonal in real space and poses

no problem. The kinetic energy operator, however, is diagonal in reciprocal

space, obtained after Fourier transformation of the wave function. Thus the

exponential operator containing the kinetic energy operator can be applied

easily in Fourier space, and the real-space result is then obtained after an

inverse Fourier transformation.

An early example of the evolution of a wave function on a grid, using

this particular kind of splitting, is the study of Selloni et al. (1987) on

the evolution of a solvated electron in a molten salt (KCl). The electron

occupies local vacancies where a chloride ion is missing, similar to F-centers

in solid salts, but in a very irregular and mobile way. The technique they

use relies on the Trotter expansion of the exponential operator and uses

repeated Fourier transforms between real and reciprocal space.

For a small time step ”t the update in Ψ is approximated by

t+”t

i ˆ ˆ

Ψ(t + ”t) = exp ’ [K + V (t )] dt Ψ(r, t)

t

ˆ

ˆ iV (t + 1 ”t)”t

iK”t

≈ exp ’ exp ’ 2

2

ˆ

iK”t

— exp ’ Ψ(r, t). (5.19)

2

Here we have used in the last equation the Trotter split-operator approxi-

mation for exponential operators with a sum of non-commuting exponents

3 De Raedt (1987, 1996) employs an elegant notation using creation and annihilation operators to

index o¬-diagonal matrix elements, thus highlighting the correspondence with particle motion

on lattice sites, but for the reader unfamiliar with Fermion operator algebra this elegance is of

little help.

5.2 Quantum dynamics in a non-stationary potential 119

(see Chapter 14), which is of higher accuracy than the product of two ex-

ponential operators. Note that it does not matter what the origin of the

time-dependence of V actually is: V may depend parametrically on time-

dependent coordinates of classical particles, it may contain interactions with

time-dependent ¬elds (e.g., electromagnetic radiation) or it may contain

stochastic terms due to external ¬‚uctuations. The operator containing V

is straightforwardly applied to the spatial wave function, as it corresponds

simply to multiplying each grid value with a factor given by the potential,

ˆ

but the operator containing K involves a second derivative over the grid.

ˆ

The updating will be straightforward in reciprocal space, as K is simply pro-

portional to k 2 . Using fast Fourier transforms (FFT), the wave function can

ˆ

¬rst be transformed to reciprocal space, then the operator with K applied,

and ¬nally transformed back into real space to accomplish the evolution in

the kinetic energy operator. When the motion does not produce ¬‚uctua-

tions in the potential in the frequency range corresponding to transition to

the excited state, i.e., if the energy gap between ground and excited states

is large compared to kB T , the solvated electron behaves adiabatically and

remains in the ground Born“Oppenheimer state.

5.2.2 Time-independent basis set

Let us ¬rst consider stationary basis functions and a Hamiltonian that con-

tains a time-dependent perturbation:

ˆ ˆ ˆ

H = H 0 + H 1 (t). (5.20)

ˆ

Assume that the basis functions φn (r) are solutions of H0 :

ˆ