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Part 1
Improving Self-consistent Field Convergence
T
occ occ = N
C SC I , (1.23)
where F 0 is typically obtained as a weighted sum of the previous Fock matrices such as F in Eq.
(1.15). Since Eq. (1.22) represents a crude model of the true Hartree-Fock energy (with the same
first-order term, but different zero- and second-order terms), it has a rather small trust radius. A
global minimization of E RH (D), as accomplished by the solution of the Roothaan–Hall eigenvalue
problem Eq. (1.6), may therefore easily lead to steps that are longer than the trust radius and hence
unreliable. To avoid such steps, we shall impose on the optimization of Eq. (1.22) the constraint that
the new density matrix D does not differ much from the old D 0 , that is, the S-norm of the density
difference should be equal to a small number ∆
2
2
D− D0 S
= Tr ( D−D0 ) S( D− D0 ) S = − 2Tr D0SDS + N = ∆, (1.24)
where N is the number of electrons – see Eq. (1.2) – and the S-norm used throughout this thesis is
defined as
2
S
A = Tr ASAS (1.25)
for symmetric A. The optimization of Eq. (1.22) subject to the constraints Eq. (1.23) and Eq. (1.24)
may be carried out by introducing the Lagrangian
1
T
L = 2TrFD 0 −2µ
( TrDSDS 0 − ( N −∆)
) −2Trη( CoccSCocc
−I N ) , (1.26)
2
where µ is the undetermined multiplier associated with the constraint Eq. (1.24), whereas the
symmetric matrix η contains the multipliers associated with the MO orthonormality constraints.
Differentiating this Lagrangian with respect to the MO coefficients and setting the result equal to
zero, we arrive at the level-shifted Roothaan–Hall equations:
( F − µ SD S) C ( µ ) = SC ( µ ) λ ( µ ). (1.27)
0 0 occ occ
Since the density matrix, Eq. (1.8), is invariant to unitary transformations among the occupied MOs
in C occ ( µ ), we may transform this eigenvalue problem to the canonical basis:
( F − µ SD S) C ( µ ) = SC ( µ ) ε ( µ ) , (1.28)
0 0 occ occ
where the diagonal matrix ε(µ) contains the orbital energies. Note that, since D 0 S projects onto the
part of C occ that is occupied in D 0 (see ref. 46 ), the level-shift parameter µ shifts only the energies of
the occupied MOs. Therefore, the role of µ is to modify the difference between the energies of the
occupied and virtual MOs - in particular, the HOMO–LUMO gap.
Clearly, the success of the trust region Roothaan–Hall (TRRH) method will depend on our ability to
make a judicious choice of the level-shift parameter µ in Eq. (1.28). In our standard TRRH
implementation, we determine µ by requiring that D(µ) does not differ much from D 0 in the sense of
2
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