7.2 Uncorrelated Wave Function Methods

7.2.1 Single Excitation Configuration Interaction (CIS)

The derivation of the CI-singles energy and wave function216, 270 begins by selecting the HF single-determinant wave function as reference for the ground state of the system:

ΨHF=1n!det{χ1χ2χiχjχn} (7.1)

where n is the number of electrons, and the spin orbitals

χi=μNcμiϕμ (7.2)

are expanded in a finite basis of N atomic orbital basis functions. Molecular orbital coefficients {cμi} are usually found by SCF procedures which solve the Hartree-Fock equations

𝐅𝐂=𝜺𝐒𝐂, (7.3)

where S is the overlap matrix, C is the matrix of molecular orbital coefficients, ε is a diagonal matrix of orbital eigenvalues and F is the Fock matrix with elements

Fμυ=Hμυ+λσicμicυi(μλ||υσ) (7.4)

involving the core Hamiltonian and the anti-symmetrized two-electron integrals

(μμ||λσ)=ϕμ(𝐫1)ϕν(𝐫2)(1r12)[ϕλ(𝐫1)ϕσ(𝐫2)-ϕσ(𝐫1)ϕλ(𝐫2)]d𝐫1d𝐫2 (7.5)

On solving Eq. (7.3), the total energy of the ground state single determinant can be expressed as

EHF=μυPμυHFHμυ+12μυλσPμυHFPλσHF(μλ||υσ)+Vnuc (7.6)

where PHF is the HF density matrix and Vnuc is the nuclear repulsion energy.

Equation (7.1) represents only one of many possible determinants made from orbitals of the system; there are in fact n(N-n) possible singly substituted determinants constructed by replacing an orbital occupied in the ground state (i, j, k,) with an orbital unoccupied in the ground state (a, b, c,). Such wave functions and energies can be written

Ψia=1n!det{χ1χ2χaχjχn} (7.7)
Eia=EHF+εa-εi-(ia||ia) (7.8)

where we have introduced the anti-symmetrized two-electron integrals in the molecular orbital basis

(pq||rs)=μυλσcμpcυqcλrcσs(μλ||υσ) (7.9)

These singly excited wave functions and energies could be considered crude approximations to the excited states of the system. However, determinants of the form Eq. (7.7) are deficient in that they:

  • do not yield pure spin states

  • resemble more closely ionization rather than excitation

  • are not appropriate for excitation into degenerate states

These deficiencies can be partially overcome by representing the excited state wave function as a linear combination of all possible singly excited determinants,

ΨCIS=iaaiaΨia (7.10)

where the coefficients {aia} can be obtained by diagonalizing the many-electron Hamiltonian, A, in the space of all single substitutions. The appropriate matrix elements are:

Aia,jb=Ψia|H|Ψjb=(εa-εj)δijδab-(ja||ib) (7.11)

According to Brillouin’s, theorem single substitutions do not interact directly with a reference HF determinant, so the resulting eigenvectors from the CIS excited state represent a treatment roughly comparable to that of the HF ground state. The excitation energy is simply the difference between HF ground state energy and CIS excited state energies, and the eigenvectors of A correspond to the amplitudes of the single-electron promotions.

CIS calculations can be performed in Q-Chem using restricted (RCIS),216, 270 unrestricted (UCIS), or restricted open-shell618 (ROCIS) spin orbitals.

Example 7.1  A basic CIS excitation energy calculation on formaldehyde at the HF/6-31G* optimized ground state geometry, which is obtained in the first part of the job. Above the first singlet excited state, the states have Rydberg character, and therefore a basis with two sets of diffuse functions is used.

$molecule
   0  1
   C
   O  1  CO
   H  1  CH  2  A
   H  1  CH  2  A  3  D

   CO =   1.2
   CH =   1.0
   A  = 120.0
   D  = 180.0
$end

$rem
   JOBTYPE   =  opt
   EXCHANGE  =  hf
   BASIS     =  6-31G*
$end

@@@

$molecule
   read
$end

$rem
   EXCHANGE      =  hf
   BASIS         =  6-311(2+)G*
   CIS_N_ROOTS   =  15            Do 15 states
   CIS_SINGLETS  =  true          Do do singlets
   CIS_TRIPLETS  =  false         Don’t do Triplets
$end