<<
ω
η , the first term in the equation (2.66) will dominate. The second and the third terms in the equation (2.66) involved in the derivatives of the electrical contribution to the polarizability and of the electrical contribution to the first hyperpolarizability are the lattice relaxation expression for vibrational contributions to the second
hyperpolarizability. Moreover, there is no pure vibrational contribution to the second hyperpolarizability. That is to say, the contribution of the order of
∆
1 to the second hyperpolarizability comes from the coupled motion of the electronic excitation and the nuclear vibration within the adiabatic approximation. Besides, the other terms in the equation (2.66) and the lattice relaxation expression for vibrational contributions to the second hyperpolarizability are the same order.
III. Multi-level system with multiple vibrational modes
The method in section II is also applicable to multi-level system with multiple vibrational modes. In the same way, the vibrational levels of the ground and excited electronic states will be modeled by harmonic oscillators (mechanical harmonicity approximation). Furthermore, for simplicity, the oscillators of the same mode are assumed to have identical frequencies ω, but the minima for the ground electronic state and for the excited electronic state of the same mode are displaced.
A. Polarizability
From sum-over-state method, the polarizability α can be expressed as
∑
≠ −′=
=
0 0
0 0
! 2
1
m Em E
m
m µ
α µ
α (3.1) Therefore, the electrical contribution to the polaizability turns into
∑
∑
∞=
≠ = ∆
= −
′
1 2
0 0 i i
ge
i i
i
i i
E E
g e e
g µ µ µ
α (3.2)
where ∆i = Ei −E0. E and 0 E refer to the ground and the i-th excited electronic i state energy respectively. If we include multiple vibrational modes with multi-level electronic states, we have
{ }
∑ ∑
≠ +∑ ∑
∞= { }= ∆ +∑
′=
1 0
2
0 0 0
2
0 0
0
0 i m
a
a i a i i
a i i
m a
a a
a
a i
a m
m e g m
gm g
ω µ
ω α µ
η
η (3.3)
In the equation (3.3),
{ }
m0a refers to(
m1o,mo2,mo3,Λ)
and m denotes the vibrational 10 quantum number of the first mode in the ground electronic state, m denotes the 02 vibrational quantum number of the second mode in the ground electronic state and so on. Similarly,{ }
mia refers to(
m1i,mi2,mi3,Λ)
and m denotes the vibrational i1 quantum number of the first mode in the i-th electronic state, m denotes the i2vibrational quantum number of the second mode in the i-th electronic state and so on.
Within the Born-Oppenheimer approximation,
( )
agg
a Q m
gm
g0µ 0 = 0 µ 0 (3.4) For a collection of oscillators, we have =
∏ ( )
a
g0a Qa
0 χ and =
∏ ( )
a gm a
a Q
m0 χ a
where χgma denotes the harmonic oscillator eigenfunction of the a-th mode with vibrational quantum number m in the ground electronic state. In the same way, we a can write the dipole moment of the ground electronic state as a function of the vibrational coordinate and expand it in a Taylor series about the minimum of the ground electronic state. For simplicity, we set the minimum for the ground electronic state to be zero. Furthermore, the electronic transition moment is only expanded to the first derivative term (electrical harmonicity approximation). Therefore,
( ) ( ) ∑
∂ + ∂
=
b
b b gg gg
gg Q
Q Q
0
0 µ
µ
µ (3.5) Substituting the equation (3.5) into the equation (3.4), we obtain
( )
b ab b
a gg gg
a
gg Q m
m Q
m 0
0 0
0 0 0 0
0
∑
∂ + ∂
= µ
µ
µ (3.6)
Since
{ }
m0a ≠(
0,0,0,Λ)
for the first summation in the equation (3.3), 0 m0a =0. 00Qb m0a = unless a=b, m0a =1a0 and the vibrational quantum numbers of other modes are equal to zero. Consequently,
a a
a a gg
gg Q
Q 0 0
0 0 1
1
0
∂
= ∂µ
µ (3.7)
where a
a a
Qa
0
0 2
1
0 µ ω
= η . Therefore, the first term in the equation (3.3) then becomes
∑
∂
∂
a
a a a
gg
a Q
Q
2 0 2
0 0
1 1 µ 0
ω
η (3.8) By neglecting the second term in the equation (3.3) that has ∆i in the denominator, we can express the polarizability as
0
The equation (3.9) is the lattice relaxation expression for vibrational contribution to the polarizability.
Similar to the result in section II, the result we have obtained is that if the
vibrational frequencies are much smaller than the electronic frequencies (ηω0a <<∆i), we can find that the dominant vibrational contribution to the polarizability α is mainly from the pure vibrational motion.
B. First hyperpolarizability
From sum-over-state method, the first hyperpolarizability β can be expressed as
( )( ) ∑ ( )
At first, the terms involved in pure vibrational motion are
{ }
Through the same approximation and the equations (3.5), (3.6) and (3.7), the equation (3.11) then turns into
( ) ( ) ( ) ( )
0Hence, there is no pure vibrational contribution to the first hyperpolarizability β. Subsequently, the terms in the equation (3.10) that have only one pure vibrational frequency in the denominators are
{ }
equation (3.13) can be simplified
( )
The first summation in the equation (3.14) can be expanded in terms of
( )
i
Keeping the equation (3.15) to the first term, we can obtain
( )
transition moment as a function of the vibrational coordinate and expand it in a Taylor series about the minimum of the ground electronic state.( ) ( ) ∑
Substituting the equation (3.17) into the equation (3.16), we can obtain( )
In the same way, we can also obtain
( )
Substituting the equations (3.18) and (3.19) into the equation (3.14) and using the equation (3.7), we can obtain
∑ ( )
On the other hand, the first derivative of the electrical contribution to the
polarizability α with respect to the a-th vibrational mode at the minimum of the ground electronic state through the equation (3.2) is
∑
∞( )
=
∂
∂
= ∆
∂
∂
1 0
0
0 1 4
i a
ge ge
i a
e
Q Q
i i
µ µ
α (3.21)
Therefore, substituting the equation (3.21) into the equation (3.20), we can then express the first hyperpolarizability β as
0 0
2 0
1
∂
∂
∂
≅ ∂
′
∑
a e
a a
gg
a Q Q
k µ α
β (3.22) where k0a is the force constant of the a-th mode and 0
( )
0 2a a
ka =µ ω has been used.
The equation (3.9) is the lattice relaxation expression for vibrational contribution to the first hyperpolarizability.
Similar to the result in section II, the result we have obtained is that if the
vibrational frequencies are much smaller than the electronic frequencies (ηω0a <<∆i), we can find that the dominant vibrational contribution to the hyperpolarizability β is mainly from the term in the equation (3.22).
IV. Conclusion
In this paper, we first applied the exact sum-over-state (SOS) formulas for the
(hyper)polarizabilities expressed in terms of vibronic states to a two-level system with a single vibrational mode. Next, the same method was also applied to multi-level system with multiple vibrational modes. In these two systems, the lattice relaxation expression for vibrational contributions to (hyper)polarizabilities can be obtained. For the two-level system with a single vibrational mode we considered, the contributions of the next higher order terms can also be obtained by making using of the formula developed by Ting. Additionally, there were some extra terms not contained in the formulas obtained by using Placzek’s approximation.
If the vibrational frequencies are much smaller than the electronic frequencies, the lattice relaxation expression for vibrational contributions to the polarizability will dominate. Moreover, the term is a pure vibrational contribution to the polarizability.
On the same condition that vibrational frequencies are much smaller than the electronic frequencies, there are no pure vibrational contributions to the first and to the second hyperpolarizability. Similarly, the lattice relaxation expression for
vibrational contributions to the first hyperpolarizability is dominant. However, for the second hyperpolarizability, the contribution of another term of the order of
∆ 1 will dominate. Besides, there also are other terms the same order as the lattice relaxation expression for vibrational contributions to the second hyperpolarizability. On the other hand, if the ground electronic state dipole moment of symmetric modes is equal to zero, there are only two contributions of the same order to the second
hyperpolarizability. One term is the lattice relaxation expression for vibrational
contributions to the second hyperpolarizability and the other term is involved in the derivative of the electronic transition moment with respect to the vibrational mode.
We haved derived the vibrational contributions to (hyper)polarizabilities from the different method. Subsequently, in order to gauge the importance of the various contributions to (hyper)polarizabilities, it is significant to consider whether any qualitatively new effects will arise in polymer systems and polyacetylene will be examined using the tight binding approximation with harmonic vibrations.
References
[1] D. S. Chemla and J. Zyss, eds., Nonlinear optical properties of organic molecules and crystals (Academic Press, New York, 1978).
[2] C. Castiglioni, M. Gussoni, M. Del Zoppo and G. Zerbi,Relaxation contribution to hyperpolarizability. A semoclassical model. Solid State Communications, Vol 82, No 1, pp. 13-17 (1992).
[3] D. Yaron and R. J. Silbey, Vibrational contributions to third-order nonlinear optical susceptibilities. J. Chem. Phys., 95, 563 (1991).
[4] D. M. Bishop and B. Kirtman, A perturbation method for calculating vibrational dynamic dipole polarizabilities and hyper polarizabilities. J. Chem. Phys., 95, 563 (1991).
[5] C. H. Ting, Spectrochim. Acta A 24, 1177 (1968).