SCF Excited States Using
The Maximum Overlap Method
Andrew T. B.
aThe
a
Gilbert ,
Nicholas A.
b
Besley
and Peter M. W.
a
Gill
Australian National University, Canberra 0200 ACT, Australia
bThe
University of Nottingham, Nottingham NG7 2RD, U.K.
Introduction
Properties
The Maximum Overlap Method (MOM) provides a simple approach for finding
excited-state solutions to self-consistent field (SCF) equations. Instead of using
the aufbau principle, the MOM maximizes the overlap between the occupied
orbitals on successive SCF iterations. This prevents variational collapse to the
ground state and guides the SCF process towards the nearest, rather than the
lowest-energy, solution. These higher energy solutions of the SCF equations can
be thought of as single-determinant approximations to the excited states of the
system. The MOM therefore provides a simple low-cost route to excited states
using either Hartree-Fock or Density Functional theory.
The resulting excited-state solutions can be treated in the same way as the
ground-state solution and, in particular, derivatives of excited-state energies can
be computed using existing ground state derivative code. This allows geometries
and frequencies to be easily computed.
π∗
Algorithm
On each iteration of the SCF procedure, the current MO coefficient matrix is used
to build a Fock (or Kohn-Sham) matrix and a generalized eigenvalue equation
must be solved:
old
F C
new
= SC
new
π
n
!
The new occupied MOs must be selected, and this is usually achieved by following
the aufbau principle. An alternative protocol, which we call the Maximum Overlap
Method (MOM), states that the new occupied orbitals should be those that
overlap maximally with the span of the old occupied orbitals.
The projection of the jth orbital onto the space spanned by the previous occupied
orbitals is given by
a2
Here we have computed the IR spectra for three excited states of the fluorenone
molecule using B3LYP/6-31G* and have compared these to the experimental
excited-state spectrum. The three computed spectra are all distinct and allow us
to determine that it is the a2π* state that is being measured experimentally.
) N " n old % , new
Pj = ! + ! $ ! Cµi ' Sµ( .C( j
& ( * µ # i
N
Charge Transfer Using DFT
and we therefore occupy the new orbitals with the largest projection values.
Each solution has a basin of attraction associated with it and in order to obtain a
particular solution the initial guess orbitals must lie within its basin. Initial
guesses can be obtained by converging the ground state (cation) and exciting
electrons from occupied to virtual orbitals.
The failure of TD-DFT with standard functionals to provide a satisfactory
treatment of charge-transfer states is a well-known problem. For such states, TDDFT underestimates the excitation energy substantially and fails to yield the
correct 1/R dependence of the charge-transfer states. MOM-based calculations of
excited charge-transfer states do not suffer from the electron-transfer selfinteraction problem, and thus offer a promising route to the study of chargetransfer states within DFT.
Orthogonality?
e-
The SCF states are genuine solutions of the SCF equations, but they are not
orthogonal because they are eigenfunctions of different Hamiltonians (the Fock
operator depends on the solution of the equations).
R
The overlap between two single-determinant wavefunctions is given by
(
†
! iHF ! HF
=
det
C
j
i SC j
)
We have used this to test the quasi-orthogonality of the SCF wavefunctions and
find overlaps are typically between 0.00-0.07 except if the states are of different
symmetry, in which case they are strictly orthogonal.
We contend that
orthogonality is not a necessity for excited-state approximations.
Here we consider the π→π* transition of a C2H4˙˙˙˙C2F4 complex. The excitation
energy obtained using time-dependent B3LYP is almost independent of the intermolecular distance R. Using the MOM and either HF, B3LYP or MP2 yields the
qualitatively correct 1/R dependence as the charged species are separated.
Excitation Energies
Vertical excitation energies and deviations (in eV) for several states of
acetaldehyde were computed using the 6-311(2+,2+)G(d,p) basis set and
various excited-state methods. Experimental values were taken from Ref [2].
A summary of the results for several other small systems is also shown.
The MOM results are competitive with the conventional
methods, but perform better for the Rydberg states.
This is particularly true when we compare the timedependent B3LYP and MOM-based B3LYP results.
Conclusions
The Maximum Overlap Method has many advantages over existing singledeterminant excited-state methods such as CIS, CIS(D) and TD-DFT:
•
•
•
•
•
It
It
It
It
It
is easy to implement
yields simple molecular orbital approximations for excited states
can be coupled with Hartree-Fock, DFT and MP2 levels of theory
inherits existing ground-state derivative theory
avoids problems with Rydberg and charge-transfer states using TD-DFT
Conventional
CIS CIS(D) TD-DFT
Singlets 1.11 0.45
0.59
Triplets 0.89
All
1.01
0.37
0.45
0.41
0.52
MOM-based
HF
MP2
B3LYP
Singlets 1.15 0.38
0.33
Triplets 1.13 0.39
0.22
All
1.14
0.38
0.28
References
[1]
[2]
A. T. B. Gilbert, N. A. Besley and P. M. W. Gill, Journal of Physical
Chemistry A, DOI:10.1021/jp801738f, (2008)
M. B. Robin, Higher Excited States of Polyatomic Molecules, Academic
Press: New York, (1985)