7.2.3 Extended CIS (XCIS)

The motivation for the extended CIS procedure⁶²⁴ (XCIS) stems from the fact that ROCIS and UCIS are less effective for radicals that CIS is for closed shell molecules. Using the attachment/detachment density analysis procedure,³⁵³ the failing of ROCIS and UCIS methodologies for the nitromethyl radical was traced to the neglect of a particular class of double substitution which involves the simultaneous promotion of an $\alpha$ spin electron from the singly occupied orbital and the promotion of a $\beta$ spin electron into the singly occupied orbital. The spin-adapted configurations

$$|{\tilde{\mathrm{\Psi }}}_i^a(1)⟩=\frac{1}{\sqrt{6}}\left(|{\mathrm{\Psi }}_{\overline{i}}^{\overline{a}}⟩-|{\mathrm{\Psi }}_i^a⟩\right)+\frac{2}{\sqrt{6}}|{\mathrm{\Psi }}_{p\overline{i}}^{a\overline{p}}⟩$$

(7.12)

are of crucial importance. (Here, $a,b,c,\mathrm{\dots }$ are virtual orbitals; $i,j,k,\mathrm{\dots }$ are occupied orbitals; and $p,q,r,\mathrm{\dots }$ are singly-occupied orbitals.) It is quite likely that similar excitations are also very significant in other radicals of interest.

The XCIS proposal, a more satisfactory generalization of CIS to open shell molecules, is to simultaneously include a restricted class of double substitutions similar to those in Eq. (7.12). To illustrate this, consider the resulting orbital spaces of an ROHF calculation: doubly occupied ($d$), singly occupied ($s$) and virtual ($v$). From this starting point we can distinguish three types of single excitations of the same multiplicity as the ground state: $d\to s$, $s\to v$ and $d\to v$. Thus, the spin-adapted ROCIS wave function is

$$|{\mathrm{\Psi }}_{\mathrm{ROCIS}}⟩=\frac{1}{\sqrt{2}}\sum _{ia}^{dv}{a}_i^a\left(|{\mathrm{\Psi }}_i^a⟩+|{\mathrm{\Psi }}_{\overline{i}}^{\overline{a}}⟩\right)+\sum _{pa}^{sv}{a}_p^a|{\mathrm{\Psi }}_p^a⟩+\sum _{ip}^{ds}{a}_{\overline{i}}^{\overline{p}}|{\mathrm{\Psi }}_{\overline{i}}^{\overline{p}}⟩$$

(7.13)

The extension of CIS theory to incorporate higher excitations maintains the ROHF as the ground state reference and adds terms to the ROCIS wave function similar to that of Eq. (7.13), as well as those where the double excitation occurs through different orbitals in the $\alpha$ and $\beta$ space:

$$\begin{array}{c}\hfill |{\mathrm{\Psi }}_{\mathrm{XCIS}}⟩=\frac{1}{\sqrt{2}}\sum _{ia}^{dv}{a}_i^a\left(|{\mathrm{\Psi }}_i^a⟩+|{\mathrm{\Psi }}_{\overline{i}}^{\overline{a}}⟩\right)+\sum _{pa}^{sv}{a}_p^a|{\mathrm{\Psi }}_p^a⟩+\sum _{ip}^{ds}{a}_{\overline{i}}^{\overline{p}}|{\mathrm{\Psi }}_{\overline{i}}^{\overline{p}}⟩\\ \hfill +\sum _{iap}^{dvs}{\tilde{a}}_i^a(p)|{\tilde{\mathrm{\Psi }}}_i^a(p)⟩+\sum _{ia,p\ne q}^{dv,ss}{a}_{p\overline{i}}^{a\overline{q}}|{\mathrm{\Psi }}_{p\overline{i}}^{a\overline{q}}⟩\end{array}$$

(7.14)

XCIS is defined only from a restricted open shell Hartree-Fock ground state reference, as it would be difficult to uniquely define singly occupied orbitals in a UHF wave function. In addition, $\beta$ unoccupied orbitals, through which the spin-flip double excitation proceeds, may not match the half-occupied $\alpha$ orbitals in either character or even symmetry.

For molecules with closed shell ground states, both the HF ground and CIS excited states emerge from diagonalization of the Hamiltonian in the space of the HF reference and singly excited substituted configuration state functions. The XCIS case is different because the restricted class of double excitations included could mix with the ground state and lower its energy. This mixing is avoided to maintain the size consistency of the ground state energy.

With the inclusion of the restricted set of doubles excitations in the excited states, but not in the ground state, it could be expected that some fraction of the correlation energy be recovered, resulting in anomalously low excited state energies. However, the fraction of the total number of doubles excitations included in the XCIS wave function is very small and those introduced cannot account for the pair correlation of any pair of electrons. Thus, the XCIS procedure can be considered one that neglects electron correlation.

The computational cost of XCIS is approximately four times greater than CIS and ROCIS, and its accuracy for open shell molecules is generally comparable to that of the CIS method for closed shell molecules. In general, it achieves qualitative agreement with experiment. XCIS is available for doublet and quartet excited states beginning from a doublet ROHF treatment of the ground state, for excitation energies only.

$comment
   C6H5 phenyl radical C2v symmetry MP2(full)/6-31G* = -230.7777459
$end

$molecule
   0  2
   c1
   x1  c1  1.0
   c2  c1  rc2  x1  90.0
   x2  c2  1.0  c1  90.0  x1    0.0
   c3  c1  rc3  x1  90.0  c2    tc3
   c4  c1  rc3  x1  90.0  c2   -tc3
   c5  c3  rc5  c1   ac5  x1  -90.0
   c6  c4  rc5  c1   ac5  x1   90.0
   h1  c2  rh1  x2  90.0  c1  180.0
   h2  c3  rh2  c1   ah2  x1   90.0
   h3  c4  rh2  c1   ah2  x1  -90.0
   h4  c5  rh4  c3   ah4  c1  180.0
   h5  c6  rh4  c4   ah4  c1  180.0

   rh1  =    1.08574
   rh2  =    1.08534
   rc2  =    2.67299
   rc3  =    1.35450
   rh4  =    1.08722
   rc5  =    1.37290
   tc3  =   62.85
   ah2  =  122.16
   ah4  =  119.52
   ac5  =  116.45
$end

$rem
   BASIS            =  6-31+G*
   EXCHANGE         =  hf
   MEM_STATIC       =  80
   INTSBUFFERSIZE   =  15000000
   SCF_CONVERGENCE  =  8
   CIS_N_ROOTS      =  5
   XCIS             =  true
$end