Essentials of Computational Chemistry

326 9 CHARGE DISTRIBUTION AND SPECTROSCOPIC PROPERTIES
cubic contribution to the energy change can be measured (although technically it becomes
increasingly challenging to fit the data reliably) and this change can be used to define the
first hyperpolarizability, β (now a third-rank tensor).
It is possible to generalize this discussion in a useful way. Spectral measurements invariably
assess how a molecular system changes in energy in response to some sort of external
perturbation. The example presently under discussion involves application of an external
electric field. If we write the energy as a Taylor expansion in some generalized vector
perturbation X, wehave
E(X) = E(0) + ∂E
∂X
X=0
· X + 1
2!
∂2
E
∂X2
X=0
· X 2 + 1
3!
∂3
E
∂X3
X=0
· X 3 +··· (9.32)
Thus, Eq. (9.32) makes more clear the measurement of the Stark effect, for instance. At
low electric field strengths, the only expansion term having significant magnitude involves
the first derivative, and it defines the permanent dipole moment. At higher field strengths, the
second derivative term begins to be noticeable, and it contributes to the energy quadratically
and defines the polarizability. Finally, we see naturally how additional terms in the Taylor
expansion can be used to define the first hyperpolarizability, the second hyperpolarizability
γ , etc. (Note that conventions differ somewhat on whether the 1/n! term preceding the
corresponding nth derivative term is included in the value of the physical constant or not, so
that care should be exercised in comparing values reported from different sources to ensure
consistency in this regard.)
Analogous quantities to the electric moments can be defined when the external perturbation
takes the form of a magnetic field. In this instance the first derivative defines the permanent
magnetic moment (always zero for non-degenerate electronic states), the second derivative
the magnetizability or magnetic susceptibility, etc.
Equation (9.32) is also useful to the extent it suggests the general way in which various
spectral properties may be computed. The energy of a system represented by a wave function
is computed as the expectation value of the Hamiltonian operator. So, differentiation of the
energy with respect to a perturbation is equivalent to differentiation of the expectation value
of the Hamiltonian. In the case of first derivatives, if the energy of the system is minimized
with respect to the coefficients defining the wave function, the Hellmann–Feynman theorem
of quantum mechanics allows us to write
∂
〈|H|〉 = |
∂X ∂H
∂X |
(9.33)
Note that H here is the complete Hamiltonian, that is, it presumably includes new terms
dependent on the nature of X. It is occasionally the case that the integral on the r.h.s. of
Eq. (9.33) can be readily evaluated. Indeed, it is choice of X = E that leads to the definition
of the dipole moment operator presented in Eq. (9.1).
However, even when it is not convenient to solve the integral on the r.h.s. of Eq. (9.33)
analytically, or when Eq. (9.33) does not hold because the wave function is not variationally
optimized, it is certainly always possible to carry out the differentiation numerically. That