6.6 Computational Note: Diabatic States for ET and CT Studies
209
The use of localized molecular orbitals (LMO) is necessary to define group functions and greatly facilitates any diabatization procedure, such as the one just sketched.
Nonorthogonal LMOs can be produced by applying quantum chemistry methods
(DFT, CASSCF, etc.) separately to single molecules or fragments that compose the
whole system, as in the divide and conquer strategies seen above. Orthogonal LMOs
can be produced by many localization procedures that are usually applied separately
to different subsets of MOs. For instance, the many-electron wavefunction can be
kept invariant by applying orthogonal localizing transformations to the occupied and
virtual MOs obtained by HF or DFT calculations, and the same can be done with
CASSCF by further separating the active MOs. Some localization algorithms were
proposed with the aim of improving the efficiency of the configuration interaction
method, as for instance the Foster–Boys procedure [31] that minimizes the second
moments of the electron densities |φ i |
2 associated with the LMOs, or the Edmiston–
Ruedenberg one [32], that maximize the self-repulsion. These two criteria make no
explicit reference to the molecular structure. Other methods tend instead to place the
LMOs in particular regions, i.e., near bonds, atoms, or groups of atoms [33–39]. Of
particular interest in this context are the procedures put forward by de Silva et al.
[40] and by Zhang and Li [38], which allow to choose the groups of atoms in which
the user wants to bring the LMOs.
As a last remark, we feel that energy transfer and charge transfer, among other
subjects of this textbook, are one of the fields in which the broadest variety of theoretical models and computational methods is being proposed and tested in the last years,
probably because of the challenging problems posed by the attempt of understanding
fundamental biological processes and of contributing to technological advancements.
Problems
6.1 In a study of the quenching of excited anthraquinone (C 14 H 8 O 2 ) by electron
transfer to amines in gas phase [41], the bimolecular rate constant with pyridine
(C 5 H 5 N) was found to be 2.3 · 10
−19 s
−1 molc
−1 m
3 , at T = 433 K. Compute the
cross section and compare it with the geometrical cross section, obtained by considering two rigid spheres with volumes equal to the molecular volumes. To evaluate the
volumes, use the density of solid anthraquinone (1.308 g/cm
3 ) and of liquid pyridine
(0.982 g/cm
3 ).
6.2 Compute the average time between two gas-phase collisions and between two
encounters in solution, for a given molecule. Make the following “standard” assumptions: hard spheres with radii = 4 Å and molecular masses = 100 a.m.u., T = 300
K, P = 1 atm, concentration of the other solute 1 mol/L, viscosity of the solvent
10
−3 kg·m
−1 s
−1 .
209
The use of localized molecular orbitals (LMO) is necessary to define group functions and greatly facilitates any diabatization procedure, such as the one just sketched.
Nonorthogonal LMOs can be produced by applying quantum chemistry methods
(DFT, CASSCF, etc.) separately to single molecules or fragments that compose the
whole system, as in the divide and conquer strategies seen above. Orthogonal LMOs
can be produced by many localization procedures that are usually applied separately
to different subsets of MOs. For instance, the many-electron wavefunction can be
kept invariant by applying orthogonal localizing transformations to the occupied and
virtual MOs obtained by HF or DFT calculations, and the same can be done with
CASSCF by further separating the active MOs. Some localization algorithms were
proposed with the aim of improving the efficiency of the configuration interaction
method, as for instance the Foster–Boys procedure [31] that minimizes the second
moments of the electron densities |φ i |
2 associated with the LMOs, or the Edmiston–
Ruedenberg one [32], that maximize the self-repulsion. These two criteria make no
explicit reference to the molecular structure. Other methods tend instead to place the
LMOs in particular regions, i.e., near bonds, atoms, or groups of atoms [33–39]. Of
particular interest in this context are the procedures put forward by de Silva et al.
[40] and by Zhang and Li [38], which allow to choose the groups of atoms in which
the user wants to bring the LMOs.
As a last remark, we feel that energy transfer and charge transfer, among other
subjects of this textbook, are one of the fields in which the broadest variety of theoretical models and computational methods is being proposed and tested in the last years,
probably because of the challenging problems posed by the attempt of understanding
fundamental biological processes and of contributing to technological advancements.
Problems
6.1 In a study of the quenching of excited anthraquinone (C 14 H 8 O 2 ) by electron
transfer to amines in gas phase [41], the bimolecular rate constant with pyridine
(C 5 H 5 N) was found to be 2.3 · 10
−19 s
−1 molc
−1 m
3 , at T = 433 K. Compute the
cross section and compare it with the geometrical cross section, obtained by considering two rigid spheres with volumes equal to the molecular volumes. To evaluate the
volumes, use the density of solid anthraquinone (1.308 g/cm
3 ) and of liquid pyridine
(0.982 g/cm
3 ).
6.2 Compute the average time between two gas-phase collisions and between two
encounters in solution, for a given molecule. Make the following “standard” assumptions: hard spheres with radii = 4 Å and molecular masses = 100 a.m.u., T = 300
K, P = 1 atm, concentration of the other solute 1 mol/L, viscosity of the solvent
10
−3 kg·m
−1 s
−1 .