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| k (t + T ) = | k (t),
(4.5)
and
( ˆ
H d (t) − i∂ t )| k (t) = ˆ
H F (t)| k (t) = ε k | k (t).
(4.6)
As can be verified using (4.6), if ε k is a quasienergy, then ε k + mω 1 is also a
quasienergy with a corresponding Floquet state e
imω 1 t
| k (t), where m is an integer.
As (4.4) demonstrates, such a shifted quasienergy does not represent a new physical
state, because ε k + mω 1 with e
imω 1 t
| k (t) gives the same contribution to the wave
function as ε k with | k (t).
Because | k (t) are periodic in time, they can be expanded as a Fourier series,
| k (t) =
n
|ϕ kn e
inω 1 t
.
(4.7)
Substituting (4.7) into (4.6) and assuming ˆ
W 1 (t) = −E 1 ˆ
μcos(ω 1 t) = −
1
2
E 1 ˆ
μ(e
iω 1 t
+ e
−iω 1 t
), (4.6) becomes
( ˆ
H mol + nω 1 )
n
|ϕ kn e
inω 1 t
−
1
2
E 1 ˆ
μ
n
|ϕ kn (e
i(n+1)ω 1 t
+ e
i(n−1)ω 1 t
)
= ε k
n
|ϕ kn e
inω 1 t
.
(4.8)
Multiplying (4.8) with
1
T
e
−imω 1 t and integrating over the time period T leads to
( ˆ
H mol + mω 1 )|ϕ km −
1
2
E 1 ˆ
μ
|ϕ k,m−1 + |ϕ k,m+1
= ε k |ϕ km .
(4.9)
The Fourier components |ϕ km can be further expressed as a linear combination of
field-free rovibronic molecular eigenstates
|ϕ km =
α,v,J
C
(k)
m,αv J |αv J ,
(4.10)
where α, v, and J represent electronic, vibrational, and rotational quantum numbers, respectively. Using the expansion of (4.10), (4.9) can be turned into the matrix
eigenvalue problem
n,α,v,J
(H F ) mα v J ,nαv J C
(k)
n,αv J = ε k C
(k)
m,α v J ,
(4.11)