26
Y. Zhang et al.
for simulation of vibronic excitation of N 2
+ [22, 23, 33] and by including the rotational levels to construct complete wave functions, we can simulate time-dependent
rotational excitation in N 2
+ to clarify the rotational effect in air-lasing, so that we will
be able to simulate theoretically the role of the electronic, vibrational, and rotational
dynamics of N 2
+ in the air lasing [39].
2.2 Theoretical Model
A nitrogen molecular ion, N 2
+ can be treated as a multi-level system composed
only of several vibrational levels and their rotational levels in the respective three
electronic states, X
2
g
+ , A
2
u , and B
2
u
+ . The time-dependent population transfer
can be obtained by solving the time-dependent Schrödinger equation,
i
∂
∂t
(r, t) = H (t))(r, t),
(2.6)
where is Planck’s constant divided by 2π , and H is the Hamiltonian operator.
The general solution can be expressed as
(r, t) =
α=X,A + ,A − ,B
V max
v=0
K max
K =0
K
m=−K
c αvK m (t)ψ αvK m (r),
(2.7)
where |r| is the internuclear distance, P αvK m = |c αvK m (t)|
2 represents the probability
of finding the system in the rotational state with quantum numbers K , m in the vth
vibrational state in the electronic state α. In (2.7), α denotes one of the four lowestenergy electronic states, X
2
g
+ , the doubly degenerate A
2
u , labeled as A + and A − ,
and B
2
u
+ , v is the vibrational quantum number, K is the total angular momentum
in which the electron spin angular momenta are excluded, m is the projection of the
total angular momentum onto the z-axis, and r is the internuclear separation vector.
In the simulation, the maximum vibrational quantum number and the maximum
rotational quantum number included in each electronic state are V max and K max ,
respectively. ψ αvK m (r) represents the complete molecular basis wave function of the
K th rotational state with magnetic quantum number m in the vth vibrational state in
the electronic α state.
2.2.1 Vibrational Basis
The field-free basis wavefunction of N 2
+ is calculated by the finite difference method
with the potential curves of X
2
g
+ , A
2
u and B
2
u
+ given in [30, 31]. The vibrational basis set is obtained as the eigen functions of Hamiltonian H 0 with no electronic
Y. Zhang et al.
for simulation of vibronic excitation of N 2
+ [22, 23, 33] and by including the rotational levels to construct complete wave functions, we can simulate time-dependent
rotational excitation in N 2
+ to clarify the rotational effect in air-lasing, so that we will
be able to simulate theoretically the role of the electronic, vibrational, and rotational
dynamics of N 2
+ in the air lasing [39].
2.2 Theoretical Model
A nitrogen molecular ion, N 2
+ can be treated as a multi-level system composed
only of several vibrational levels and their rotational levels in the respective three
electronic states, X
2
g
+ , A
2
u , and B
2
u
+ . The time-dependent population transfer
can be obtained by solving the time-dependent Schrödinger equation,
i
∂
∂t
(r, t) = H (t))(r, t),
(2.6)
where is Planck’s constant divided by 2π , and H is the Hamiltonian operator.
The general solution can be expressed as
(r, t) =
α=X,A + ,A − ,B
V max
v=0
K max
K =0
K
m=−K
c αvK m (t)ψ αvK m (r),
(2.7)
where |r| is the internuclear distance, P αvK m = |c αvK m (t)|
2 represents the probability
of finding the system in the rotational state with quantum numbers K , m in the vth
vibrational state in the electronic state α. In (2.7), α denotes one of the four lowestenergy electronic states, X
2
g
+ , the doubly degenerate A
2
u , labeled as A + and A − ,
and B
2
u
+ , v is the vibrational quantum number, K is the total angular momentum
in which the electron spin angular momenta are excluded, m is the projection of the
total angular momentum onto the z-axis, and r is the internuclear separation vector.
In the simulation, the maximum vibrational quantum number and the maximum
rotational quantum number included in each electronic state are V max and K max ,
respectively. ψ αvK m (r) represents the complete molecular basis wave function of the
K th rotational state with magnetic quantum number m in the vth vibrational state in
the electronic α state.
2.2.1 Vibrational Basis
The field-free basis wavefunction of N 2
+ is calculated by the finite difference method
with the potential curves of X
2
g
+ , A
2
u and B
2
u
+ given in [30, 31]. The vibrational basis set is obtained as the eigen functions of Hamiltonian H 0 with no electronic