76
2 Molecular States
Franck–Condon transition energies. Loosely speaking, this is due to the fact that
normally, at the ground-state geometry, the ionic configurations are more important
in the description of singlet excited states than for S 0 . Now, in CASSCF the orbitals
are optimized in an averaged field corresponding to a neutral charge distribution,
and therefore are more suited to covalent than to ionic configurations [33, 34], thus
introducing a bias in favor of S 0 . This phenomenon may be especially important
for π → π
∗ states (see Sect. 2.6.4), less for n → π
∗ states. To obtain quantitative
agreement with experimental spectroscopic data, the CASSCF results can be refined
either variationally, by multireference CI (MRCI) or, more commonly, perturbatively
(using CASPT2 [35] or NEVPT2 [36]). However, due to the high computational cost
of these multireference methods, their applicability is limited to small molecules
(say, ∼20 second row atoms).
An alternative to ab initio methods is provided by density functional theory (DFT),
a single-reference scheme for which excited state properties are available from the
“time-dependent” approach (TD-DFT). Due to its low computational cost, TD-DFT
is usually the method of choice for large systems, and in many cases, at geometries
close to the ground-state minimum, gives results in good agreement with the best
ab initio methods. At distorted geometries it suffers from the problems of singlereference methods mentioned above.
Problems
2.1 Consider a one dimensional system described by the following Hamiltonian
ˆ
H (x) = −
2
2m
d
2
dx 2 − F x
(F > 0)
appropriate, for example, for a particle of mass m and charge q in the presence of a
constant electric field E 0 (in that case F = q E 0 ). Working in the momentum representation (use x = id/d p), evaluate the time evolution of a Gaussian wavepacket,
i.e., find Ψ (p, t) given Ψ (p, 0) = (2α/π)
−1/4 exp(−αp
2
), with α > 0.
2.2 The potential energy of a Morse oscillator is V (x) = D(1 − e
−a(x−x 0 )
)
2 , where
D is the well depth. The corresponding energy eigenvalues are
E v = ω(v + 1/2) +
2
ω
2
4D
(v + 1/2)
2
ω = a
2D/m
where v = 0, 1, . . . and m is the mass of the oscillator. Evaluate the density of
states for a single Morse oscillator and use it to obtain a classical expression (i.e.,
without taking into account the quantization of energy) for the density of states of
two noninteracting identical Morse oscillators.
2 Molecular States
Franck–Condon transition energies. Loosely speaking, this is due to the fact that
normally, at the ground-state geometry, the ionic configurations are more important
in the description of singlet excited states than for S 0 . Now, in CASSCF the orbitals
are optimized in an averaged field corresponding to a neutral charge distribution,
and therefore are more suited to covalent than to ionic configurations [33, 34], thus
introducing a bias in favor of S 0 . This phenomenon may be especially important
for π → π
∗ states (see Sect. 2.6.4), less for n → π
∗ states. To obtain quantitative
agreement with experimental spectroscopic data, the CASSCF results can be refined
either variationally, by multireference CI (MRCI) or, more commonly, perturbatively
(using CASPT2 [35] or NEVPT2 [36]). However, due to the high computational cost
of these multireference methods, their applicability is limited to small molecules
(say, ∼20 second row atoms).
An alternative to ab initio methods is provided by density functional theory (DFT),
a single-reference scheme for which excited state properties are available from the
“time-dependent” approach (TD-DFT). Due to its low computational cost, TD-DFT
is usually the method of choice for large systems, and in many cases, at geometries
close to the ground-state minimum, gives results in good agreement with the best
ab initio methods. At distorted geometries it suffers from the problems of singlereference methods mentioned above.
Problems
2.1 Consider a one dimensional system described by the following Hamiltonian
ˆ
H (x) = −
2
2m
d
2
dx 2 − F x
(F > 0)
appropriate, for example, for a particle of mass m and charge q in the presence of a
constant electric field E 0 (in that case F = q E 0 ). Working in the momentum representation (use x = id/d p), evaluate the time evolution of a Gaussian wavepacket,
i.e., find Ψ (p, t) given Ψ (p, 0) = (2α/π)
−1/4 exp(−αp
2
), with α > 0.
2.2 The potential energy of a Morse oscillator is V (x) = D(1 − e
−a(x−x 0 )
)
2 , where
D is the well depth. The corresponding energy eigenvalues are
E v = ω(v + 1/2) +
2
ω
2
4D
(v + 1/2)
2
ω = a
2D/m
where v = 0, 1, . . . and m is the mass of the oscillator. Evaluate the density of
states for a single Morse oscillator and use it to obtain a classical expression (i.e.,
without taking into account the quantization of energy) for the density of states of
two noninteracting identical Morse oscillators.