Handbook of Solvents - George Wypych - ChemTech - Ventech!

11.1 Theoretical treatment of solvent effects 657
From equation [11.1.47], the following Schrödinger equation is obtained for the electronic
state |ψ > of the solute molecule
H� � � |
0 − Γψμ ψμ = W ψ
[11.1.48]
tot el
W plays the role of the quantum mechanical motif that is directly obtained from the
Schrödinger equation as follows
ψ � Γ ψμ� ψ μ� ψ ψ � ψ Γ ψμ� ψ ψμ� ψ [11.1.49]
0 0
W = H − tot el = H − tot el
By adding the part for the interaction of the nuclear component of the solute dipole with the
total reaction field and assuming
ψ � 0 0
H ψ ψ H�
0 0
≈ ψ
[11.1.50]
where ψ 0 is the wavefunction of the solute molecule, unperturbed by the reaction field, one
obtains that
2
0
0 0 0
E = E − E = ψ � �
ψ − ψμ� ψ + μ − ψ �
�
H Γ H ψ ≈ −Γ
μ
( )
2
el rf 0
el nuc tot
[11.1.51]
The comparison of the last equation with the starting equation [11.1.44] reveals a difference
by the factor of two in the final result. Of course, the approximation [11.1.50] brings
up a certain error and it has been therefore proposed 25,27 to correct the last formula by the addition
of the “solvent cost”, i.e., the additional work required to reorganize the solvent due
to the electrostatic field of the solute
1
2
E el = E rf − E 0 + tot
2 Γ � μ [11.1.52]
Alternatively, the electrostatic solvation energy can be derived proceeding from the following
variational functional 27
� �
( � � 2 � nuc nuc)
( | 1)
0 1
2
L = E − Γ ψμψ ψμψ + μ ψμψ + μ −W ψ ψ −
4
[11.1.53]
This leads to a Schrödinger equation which eigenvalue is directly related to the total
electrostatic (dipolar) solvation energy, E el,
� ( )
H � � | |
0 ⎛ 1
⎞
e
⎜ − Γψμ tot ψμ el ⎟ ψ = E ψ
[11.1.54]
el
⎝ 2
⎠
( e)
1 � �
E el = E − Γμ E
rf
nucμ tot − 0
[11.1.55]
2