Theor Chem Acc (2016) 135:3
1 3
single-confi guration methods cannot be applied. However,
in cases where a multireference approach is necessary, it is
clear that the orbitals of a single confi guration, together with
a basis set that has been specifi cally optimized for a given
excited state, will prove more appropriate for the development of many-body correlation methods than orbitals
expanded in a basis set constructed for the ground state. Furthermore, it would be very useful to have an analogue of the
“HF + MP2” formalism for the description of excited states
which can be adequately described by a single Slater determinant. In doing so, we should take into account that the
basis functions that are used to construct molecular orbitals are typically optimized to describe the ground states of
atoms. Remembering that the ground and excited states are
often of quite different character, it is desirable to use different basis sets for different states. “The desirability of using
different basis sets for different states” was already pointed
out by Shull and Löwdin [ 9 ] in 1958. It is especially important for highly excited states. We shall show that our methodology allows a basis set to be optimized for the excited
state under consideration with essentially the same computational efforts as for the ground state. Such an approach
provides a compact and accurate representation of excited
state wave functions. On the other hand, fi nding a method
that offers a well-balanced treatment of both states is often
problematic. Accounting for electron correlation in excited
states is not as straightforward as in the ground state.
In this contribution, we further develop the “HF + MP2”
formalism for excited states focusing our attention on calculations of the ground state and excited state energies in a
balanced manner, i.e.,
1. Reference confi gurations are constructed by employing the same computational scheme. For example,
the ground and excited SCF functions are constructed
using the Hartree–Fock equations, whose solutions are
approximated in one particle basis sets optimized specifi cally to the state under consideration.
2. Correlation effects are taken into account using comparable schemes for the ground and excited states using,
for example, many-body Møller–Plesset-like perturbation theory.
Some preliminary results in this direction were reported
in papers [ 10 , 11 ]. Here, we extend the theory and practical calculations to highly doubly excited states and doubly
ionized core hole states. A simple and easily implemented
asymptotic projection (AP) method for taking orthogonality constraints into account, which has been proposed
earlier [ 12 – 14 ], allows one to perform the “HF + MP2”
scheme for the ground and excited states with essentially
the same computational costs. The AP method is based on
the properties of self-conjugate operators. It is general and
applicable to any problem that can be cast in the form of
an eigenvalue equation with some orthogonality constraints
imposed on the eigenvectors.
The present work is arranged as follows: in Sect. 2 ,
orthogonality constraints for single determinantal wave
functions and some existing methods to prevent “variational collapse” are briefl y discussed. Our orthogonalityconstrained HF method for excited states is presented in
Sect. 3 . Unlike existing self-consistent fi eld (SCF) techniques based on the Roothaan open-shell theory [ 15 ],
it does not involve off-diagonal Lagrange multipliers.
Additionally, equations for basis set optimization are also
derived. The well-defi ned Møller–Plesset-like perturbation
theory based on optimal excited orbitals generated by the
proposed HF method is the subject of Sect. 4 . In addition
single excitations do not contribute because the excited
state orbitals, like the ground state orbitals, satisfy the generalized Brillouin theorem. In Sect. 5 , we apply the formalism to highly doubly excited states of atoms as well as to
doubly ionized core hole states of diatomic molecules.
2 Specifi c features of SCF excited states
calculations
Quantum mechanics requires exact wave functions to be
orthogonal, but it makes no such demand on SCF functions. Indeed, consider the orthogonality condition for the
exact many-electron wave functions describing the ground
state, Ψ 0 , and the fi rst excited state Ψ 1 , i.e., (see also [ 11 ])
The exact ground state wave function, Ψ 0 , can be written
where Φ 0 is the many-electron ground state SCF wave
function and χ 0 is the correlation correction. Without loss
of generality, we can require
Similarly, the exact excited state wave function, Ψ 1 , can
be written
where Φ 1 is the many-electron excited state SCF wave
function and χ 1 is the corresponding correlation correction.
Again, without loss of generality, we can require
Substituting ( 2 ) and ( 4 ) into ( 1 ), we have
(1)
Ψ 0 | Ψ 1 = 0
(2)
Ψ 0 = Φ 0 + χ 0
(3)
Φ 0 | χ 0 = 0
(4)
Ψ 1 = Φ 1 + χ 1
(5)
Φ 1 | χ 1 = 0
(6)
Ψ 0 | Ψ 1 = Φ 0 | Φ 1 + Φ 0 | χ 1
+ χ 0 | Φ 1 + χ 0 | χ 1 = 0
190
Reprinted from the journal
1 3
single-confi guration methods cannot be applied. However,
in cases where a multireference approach is necessary, it is
clear that the orbitals of a single confi guration, together with
a basis set that has been specifi cally optimized for a given
excited state, will prove more appropriate for the development of many-body correlation methods than orbitals
expanded in a basis set constructed for the ground state. Furthermore, it would be very useful to have an analogue of the
“HF + MP2” formalism for the description of excited states
which can be adequately described by a single Slater determinant. In doing so, we should take into account that the
basis functions that are used to construct molecular orbitals are typically optimized to describe the ground states of
atoms. Remembering that the ground and excited states are
often of quite different character, it is desirable to use different basis sets for different states. “The desirability of using
different basis sets for different states” was already pointed
out by Shull and Löwdin [ 9 ] in 1958. It is especially important for highly excited states. We shall show that our methodology allows a basis set to be optimized for the excited
state under consideration with essentially the same computational efforts as for the ground state. Such an approach
provides a compact and accurate representation of excited
state wave functions. On the other hand, fi nding a method
that offers a well-balanced treatment of both states is often
problematic. Accounting for electron correlation in excited
states is not as straightforward as in the ground state.
In this contribution, we further develop the “HF + MP2”
formalism for excited states focusing our attention on calculations of the ground state and excited state energies in a
balanced manner, i.e.,
1. Reference confi gurations are constructed by employing the same computational scheme. For example,
the ground and excited SCF functions are constructed
using the Hartree–Fock equations, whose solutions are
approximated in one particle basis sets optimized specifi cally to the state under consideration.
2. Correlation effects are taken into account using comparable schemes for the ground and excited states using,
for example, many-body Møller–Plesset-like perturbation theory.
Some preliminary results in this direction were reported
in papers [ 10 , 11 ]. Here, we extend the theory and practical calculations to highly doubly excited states and doubly
ionized core hole states. A simple and easily implemented
asymptotic projection (AP) method for taking orthogonality constraints into account, which has been proposed
earlier [ 12 – 14 ], allows one to perform the “HF + MP2”
scheme for the ground and excited states with essentially
the same computational costs. The AP method is based on
the properties of self-conjugate operators. It is general and
applicable to any problem that can be cast in the form of
an eigenvalue equation with some orthogonality constraints
imposed on the eigenvectors.
The present work is arranged as follows: in Sect. 2 ,
orthogonality constraints for single determinantal wave
functions and some existing methods to prevent “variational collapse” are briefl y discussed. Our orthogonalityconstrained HF method for excited states is presented in
Sect. 3 . Unlike existing self-consistent fi eld (SCF) techniques based on the Roothaan open-shell theory [ 15 ],
it does not involve off-diagonal Lagrange multipliers.
Additionally, equations for basis set optimization are also
derived. The well-defi ned Møller–Plesset-like perturbation
theory based on optimal excited orbitals generated by the
proposed HF method is the subject of Sect. 4 . In addition
single excitations do not contribute because the excited
state orbitals, like the ground state orbitals, satisfy the generalized Brillouin theorem. In Sect. 5 , we apply the formalism to highly doubly excited states of atoms as well as to
doubly ionized core hole states of diatomic molecules.
2 Specifi c features of SCF excited states
calculations
Quantum mechanics requires exact wave functions to be
orthogonal, but it makes no such demand on SCF functions. Indeed, consider the orthogonality condition for the
exact many-electron wave functions describing the ground
state, Ψ 0 , and the fi rst excited state Ψ 1 , i.e., (see also [ 11 ])
The exact ground state wave function, Ψ 0 , can be written
where Φ 0 is the many-electron ground state SCF wave
function and χ 0 is the correlation correction. Without loss
of generality, we can require
Similarly, the exact excited state wave function, Ψ 1 , can
be written
where Φ 1 is the many-electron excited state SCF wave
function and χ 1 is the corresponding correlation correction.
Again, without loss of generality, we can require
Substituting ( 2 ) and ( 4 ) into ( 1 ), we have
(1)
Ψ 0 | Ψ 1 = 0
(2)
Ψ 0 = Φ 0 + χ 0
(3)
Φ 0 | χ 0 = 0
(4)
Ψ 1 = Φ 1 + χ 1
(5)
Φ 1 | χ 1 = 0
(6)
Ψ 0 | Ψ 1 = Φ 0 | Φ 1 + Φ 0 | χ 1
+ χ 0 | Φ 1 + χ 0 | χ 1 = 0
190
Reprinted from the journal