Natural orbitals - eChem book

The one-particle density and the associated spin-orbital density matrix, $D^\mathrm{SO}$ , are related as follows

n(\mathbf{r}) \equiv \sum_{p,q} \psi^\dagger_p(\mathbf{r}) D_{pq}^\mathrm{SO} \psi_q(\mathbf{r})

(1)

A nonzero contribution to the density requires the spin of both orbitals to be identical and the summation can therefore be separated by introducing the $\alpha$ - and $\beta$ -spin densities and instead run over molecular orbitals

\begin{align*} n(\mathbf{r}) & = \sum_{p,q} \Big( \big[ \phi^\alpha_p(\mathbf{r}) \big]^\ast D_{pq}^\alpha \phi_q^\alpha(\mathbf{r}) + \big[ \phi^\beta_p(\mathbf{r}) \big]^\ast D_{pq}^\beta \phi_q^\beta(\mathbf{r}) \Big) \end{align*}

(2)

Natural orbitals is a single set of spatial orbitals in which the one-particle density is expanded

n(\mathbf{r}) = \sum_{p} \big[ \phi_p^\mathrm{NO}(\mathbf{r}) \big]^\ast D_{pp}^\mathrm{NO} \phi_p^\mathrm{NO}(\mathbf{r})

(3)

and the elements of the diagonal density matrix, $D_{pp}^\mathrm{NO}$ , in this basis are referred to as occupation numbers.

Restricted orbitals¶

In restricted calculations, we have a common set of MOs for the $\alpha$ - and $\beta$ -spin orbitals and can write

\begin{align*} n(\mathbf{r}) & = \sum_{p,q} \phi_p^\ast(\mathbf{r}) D_{pq} \phi_q(\mathbf{r}) \\ D_{pq} & = D_{pq}^\alpha + D_{pq}^\beta \end{align*}

(4)

The density matrix is Hermitian and can be diagonalized by a unitary transformation. We get

\begin{align*} \mathbf{D}^\mathrm{NO} & = \mathbf{U}^\dagger \mathbf{D} \mathbf{U} \\ \phi_p^\mathrm{NO}(\mathbf{r}) & = \sum_q \phi_q(\mathbf{r}) U_{qp} \end{align*}

(5)

In restricted closed (RHF) and open-shell Hartree–Fock (ROHF), the natural orbitals equal the HF orbitals and the occupation numbers equal to 0, 1, and 2 for unoccuppied, singly occupied, and doubly occupied MOs, respectively.

For the case of a multi-configurational wave function, we study the oxygen molecule as an example.

import multipsi as mtp
import numpy as np
import veloxchem as vlx

mol_str = """2

O   -0.6   0.0  0.0
O    0.6   0.0  0.0
"""
molecule = vlx.Molecule.read_xyz_string(mol_str)
basis = vlx.MolecularBasis.read(molecule, "cc-pvdz", ostream=None)
molecule.set_multiplicity(3)

rohf_drv = vlx.ScfRestrictedOpenDriver()
rohf_drv.ostream.mute()
rohf_results = rohf_drv.compute(molecule, basis)

molecule.show()

print(f" RHF energy: {rohf_drv.get_scf_energy() : 14.8f}")

 RHF energy:  -149.60946112

The MOs are occupied by nine $\alpha$ - and seven $\beta$ -spin electrons.

rohf_drv.molecular_orbitals.print_orbitals(molecule, basis)

                                                                                                                          
                                                 Spin Restricted Orbitals                                                 
                                                 ------------------------                                                 
                                                                                                                          
               Molecular Orbital No.   3:                                                                                 
               --------------------------                                                                                 
               Occupation: 2.000 Energy:   -1.66016 a.u.                                                                  
               (   1 O   2s  :    -0.34) (   1 O   3s  :    -0.27) (   1 O   1p+1:    -0.16)                              
               (   2 O   2s  :    -0.34) (   2 O   3s  :    -0.27) (   2 O   1p+1:     0.16)                              
                                                                                                                          
               Molecular Orbital No.   4:                                                                                 
               --------------------------                                                                                 
               Occupation: 2.000 Energy:   -1.09511 a.u.                                                                  
               (   1 O   2s  :    -0.36) (   1 O   3s  :    -0.45) (   2 O   2s  :     0.36)                              
               (   2 O   3s  :     0.45)                                                                                  
                                                                                                                          
               Molecular Orbital No.   5:                                                                                 
               --------------------------                                                                                 
               Occupation: 2.000 Energy:   -0.73155 a.u.                                                                  
               (   1 O   3s  :     0.27) (   1 O   1p+1:    -0.46) (   1 O   2p+1:    -0.24)                              
               (   2 O   3s  :     0.27) (   2 O   1p+1:     0.46) (   2 O   2p+1:     0.24)                              
                                                                                                                          
               Molecular Orbital No.   6:                                                                                 
               --------------------------                                                                                 
               Occupation: 2.000 Energy:   -0.70486 a.u.                                                                  
               (   1 O   1p-1:     0.31) (   1 O   1p0 :     0.32) (   1 O   2p-1:     0.20)                              
               (   1 O   2p0 :     0.20) (   2 O   1p-1:     0.31) (   2 O   1p0 :     0.32)                              
               (   2 O   2p-1:     0.20) (   2 O   2p0 :     0.20)                                                        
                                                                                                                          
               Molecular Orbital No.   7:                                                                                 
               --------------------------                                                                                 
               Occupation: 2.000 Energy:   -0.70486 a.u.                                                                  
               (   1 O   1p-1:     0.32) (   1 O   1p0 :    -0.31) (   1 O   2p-1:     0.20)                              
               (   1 O   2p0 :    -0.20) (   2 O   1p-1:     0.32) (   2 O   1p0 :    -0.31)                              
               (   2 O   2p-1:     0.20) (   2 O   2p0 :    -0.20)                                                        
                                                                                                                          
               Molecular Orbital No.   8:                                                                                 
               --------------------------                                                                                 
               Occupation: 1.000 Energy:   -0.20397 a.u.                                                                  
               (   1 O   1p-1:     0.44) (   1 O   1p0 :    -0.31) (   1 O   2p-1:     0.33)                              
               (   1 O   2p0 :    -0.23) (   2 O   1p-1:    -0.44) (   2 O   1p0 :     0.31)                              
               (   2 O   2p-1:    -0.33) (   2 O   2p0 :     0.23)                                                        
                                                                                                                          
               Molecular Orbital No.   9:                                                                                 
               --------------------------                                                                                 
               Occupation: 1.000 Energy:   -0.20397 a.u.                                                                  
               (   1 O   1p-1:     0.31) (   1 O   1p0 :     0.44) (   1 O   2p-1:     0.23)                              
               (   1 O   2p0 :     0.33) (   2 O   1p-1:    -0.31) (   2 O   1p0 :    -0.44)                              
               (   2 O   2p-1:    -0.23) (   2 O   2p0 :    -0.33)                                                        
                                                                                                                          
               Molecular Orbital No.  10:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    0.47356 a.u.                                                                  
               (   1 O   2s  :     0.19) (   1 O   3s  :     1.28) (   1 O   1p+1:     0.40)                              
               (   1 O   2p+1:     1.23) (   2 O   2s  :    -0.19) (   2 O   3s  :    -1.28)                              
               (   2 O   1p+1:     0.40) (   2 O   2p+1:     1.23)                                                        
                                                                                                                          
               Molecular Orbital No.  11:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    1.08237 a.u.                                                                  
               (   1 O   3s  :    -0.99) (   1 O   1p+1:     0.71) (   1 O   2p+1:    -1.33)                              
               (   2 O   3s  :     0.99) (   2 O   1p+1:     0.71) (   2 O   2p+1:    -1.33)                              
                                                                                                                          
               Molecular Orbital No.  12:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    1.10988 a.u.                                                                  
               (   1 O   1p-1:    -0.48) (   1 O   1p0 :     0.48) (   1 O   2p-1:     0.44)                              
               (   1 O   2p0 :    -0.44) (   2 O   1p-1:    -0.48) (   2 O   1p0 :     0.48)                              
               (   2 O   2p-1:     0.44) (   2 O   2p0 :    -0.44)                                                        
                                                                                                                          
               Molecular Orbital No.  13:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    1.10988 a.u.                                                                  
               (   1 O   1p-1:     0.48) (   1 O   1p0 :     0.48) (   1 O   2p-1:    -0.44)                              
               (   1 O   2p0 :    -0.44) (   2 O   1p-1:     0.48) (   2 O   1p0 :     0.48)                              
               (   2 O   2p-1:    -0.44) (   2 O   2p0 :    -0.44)                                                        
                                                                                                                          
               Molecular Orbital No.  14:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    1.15024 a.u.                                                                  
               (   1 O   3s  :     0.29) (   1 O   1p+1:     0.59) (   1 O   2p+1:    -0.82)                              
               (   2 O   3s  :     0.29) (   2 O   1p+1:    -0.59) (   2 O   2p+1:     0.82)

Main AO character of MOs

Active orbitals (8 electrons)

$3\sigma_u = p_z^1 + p_z^2$
$1\pi_g = p_{x,y}^1 - p_{x,y}^2$
$1\pi_u = p_{x,y}^1 + p_{x,y}^2$
$3\sigma_g = p_z^1 - p_z^2$

Inactive orbitals (8 electrons)

$2\sigma_u = 2s^1 - 2s^2$
$2\sigma_g = 2s^1 + 2s^2$
$1\sigma_u = 1s^1 - 1s^2$
$1\sigma_g = 1s^1 + 1s^2$

casscf_drv = mtp.McscfDriver()
orbital_space = mtp.OrbSpace(molecule, rohf_drv.mol_orbs)
orbital_space.cas(8, 6)

casscf_results = casscf_drv.compute(molecule, basis, orbital_space)

                                                                                                                          
                          Multi-Configurational Self-Consistent Field Driver
                         ====================================================
                                                                                                                          

               Active space definition:
               ------------------------
               Number of inactive (occupied) orbitals: 4
               Number of active orbitals:              6
               Number of virtual orbitals:             18

               This is a CASSCF wavefunction: CAS(8,6)

               CI expansion:
               -------------
               Number of determinants:      120


                                                                                                                          
               ╭────────────────────────────────────╮
               │          Driver settings           │
               ╰────────────────────────────────────╯
                                                                                                                          
                                                                                                                          
               Number of states              :   1
               State-specific calculation
                - State of interest          :   1
               Max. iterations               :   50
               BFGS window                   :   5
               Convergence thresholds:
                - Energy change              :   1e-08
                - Gradient sq. norm          :   1e-08
                                                                                                                          
                                                                                                                          
                          MCSCF Iterations
                         -------------------
                                                                                                                          
     Iter. | Average Energy | E. Change | Grad. Norm | CI Iter. | Trust rad. |  Time
     ---------------------------------------------------------------------------------
        1     -149.670911028     0.0e+00      4.1e-03          0        0.40    0:00:00
        2     -149.698155753    -2.7e-02      2.1e-03          0        0.40    0:00:00
        3     -149.702771606    -4.6e-03      1.7e-03          0        0.56    0:00:00
        4     -149.708302734    -5.5e-03      1.9e-04          0        0.39    0:00:00
        5     -149.708359855    -5.7e-05      1.3e-05          0        0.39    0:00:00
        6     -149.708366686    -6.8e-06      1.0e-06          0        0.47    0:00:00
        7     -149.708367125    -4.4e-07      1.2e-07          0        0.47    0:00:00
        8     -149.708367177    -5.2e-08      8.2e-09          0        0.56    0:00:00
        9     -149.708367181    -4.1e-09      9.7e-10          0        0.68    0:00:00
** Convergence reached in 9 iterations
       10     -149.708367181    -3.9e-10      4.4e-11          0        0.80    0:00:00
                                                                                                                          
               Final results
               -------------
                                                                                                                          
* State 1
  - S^2    : 2.00  (multiplicity = 3.0 )
  - Energy : -149.70836718142888
  - Natural orbitals
1.96300 1.96300 1.96098 1.03638 1.03638 0.04027
                                                                                                                          
                                                 Spin Restricted Orbitals                                                 
                                                 ------------------------                                                 
                                                                                                                          
               Molecular Orbital No.   3:                                                                                 
               --------------------------                                                                                 
               Occupation: 2.000 Energy:   -1.47278 a.u.                                                                  
               (   1 O   2s  :     0.34) (   1 O   3s  :     0.36) (   2 O   2s  :     0.34)                              
               (   2 O   3s  :     0.36)                                                                                  
                                                                                                                          
               Molecular Orbital No.   4:                                                                                 
               --------------------------                                                                                 
               Occupation: 2.000 Energy:   -1.08415 a.u.                                                                  
               (   1 O   2s  :    -0.36) (   1 O   3s  :    -0.46) (   2 O   2s  :     0.36)                              
               (   2 O   3s  :     0.46)                                                                                  
                                                                                                                          
               Molecular Orbital No.   5:                                                                                 
               --------------------------                                                                                 
               Occupation: 1.963 Energy:   -0.68821 a.u.                                                                  
               (   1 O   1p-1:     0.35) (   1 O   1p0 :    -0.28) (   1 O   2p-1:     0.22)                              
               (   1 O   2p0 :    -0.17) (   2 O   1p-1:     0.35) (   2 O   1p0 :    -0.28)                              
               (   2 O   2p-1:     0.22) (   2 O   2p0 :    -0.17)                                                        
                                                                                                                          
               Molecular Orbital No.   6:                                                                                 
               --------------------------                                                                                 
               Occupation: 1.963 Energy:   -0.68821 a.u.                                                                  
               (   1 O   1p-1:    -0.28) (   1 O   1p0 :    -0.35) (   1 O   2p-1:    -0.17)                              
               (   1 O   2p0 :    -0.22) (   2 O   1p-1:    -0.28) (   2 O   1p0 :    -0.35)                              
               (   2 O   2p-1:    -0.17) (   2 O   2p0 :    -0.22)                                                        
                                                                                                                          
               Molecular Orbital No.   7:                                                                                 
               --------------------------                                                                                 
               Occupation: 1.961 Energy:   -0.88323 a.u.                                                                  
               (   1 O   1p+1:    -0.49) (   1 O   2p+1:    -0.23) (   2 O   1p+1:     0.49)                              
               (   2 O   2p+1:     0.23)                                                                                  
                                                                                                                          
               Molecular Orbital No.   8:                                                                                 
               --------------------------                                                                                 
               Occupation: 1.036 Energy:   -0.20043 a.u.                                                                  
               (   1 O   1p-1:     0.44) (   1 O   1p0 :    -0.31) (   1 O   2p-1:     0.32)                              
               (   1 O   2p0 :    -0.23) (   2 O   1p-1:    -0.44) (   2 O   1p0 :     0.31)                              
               (   2 O   2p-1:    -0.32) (   2 O   2p0 :     0.23)                                                        
                                                                                                                          
               Molecular Orbital No.   9:                                                                                 
               --------------------------                                                                                 
               Occupation: 1.036 Energy:   -0.20043 a.u.                                                                  
               (   1 O   1p-1:    -0.31) (   1 O   1p0 :    -0.44) (   1 O   2p-1:    -0.23)                              
               (   1 O   2p0 :    -0.32) (   2 O   1p-1:     0.31) (   2 O   1p0 :     0.44)                              
               (   2 O   2p-1:     0.23) (   2 O   2p0 :     0.32)                                                        
                                                                                                                          
               Molecular Orbital No.  10:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.040 Energy:    0.67653 a.u.                                                                  
               (   1 O   2s  :    -0.24) (   1 O   3s  :    -0.39) (   1 O   1p+1:    -0.74)                              
               (   1 O   2p+1:    -0.26) (   2 O   2s  :     0.24) (   2 O   3s  :     0.39)                              
               (   2 O   1p+1:    -0.74) (   2 O   2p+1:    -0.26)                                                        
                                                                                                                          
               Molecular Orbital No.  11:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    0.91053 a.u.                                                                  
               (   1 O   3s  :    -1.60) (   1 O   1p+1:     0.38) (   1 O   2p+1:    -1.81)                              
               (   2 O   3s  :     1.60) (   2 O   1p+1:     0.38) (   2 O   2p+1:    -1.81)                              
                                                                                                                          
               Molecular Orbital No.  12:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    1.11712 a.u.                                                                  
               (   1 O   1p-1:     0.23) (   1 O   1p0 :    -0.64) (   1 O   2p-1:    -0.21)                              
               (   1 O   2p0 :     0.59) (   2 O   1p-1:     0.23) (   2 O   1p0 :    -0.64)                              
               (   2 O   2p-1:    -0.21) (   2 O   2p0 :     0.59)                                                        
                                                                                                                          
                                                                                                                          
                                                Dipole moment for state 1                                                 
                                               ---------------------------                                                
                                                                                                                          
                                   X   :         0.000000 a.u.         0.000000 Debye                                     
                                   Y   :        -0.000000 a.u.        -0.000000 Debye                                     
                                   Z   :         0.000000 a.u.         0.000000 Debye                                     
                                 Total :         0.000000 a.u.         0.000000 Debye                                     
                                                                                                                          
                                                                                                                          
Total MCSCF time: 00:00:02

print(f"CAS(8,6) energy: {casscf_results["energies"][0] : 12.8f}")

CAS(8,6) energy: -149.70836718

The wave function is predominantly a sum of two determinants with $\alpha$ -spin occupation of the $\pi_u$ and $\pi_g$ molecular orbitals, respectively.

casscf_results["ci_vectors"].print(0)

Determinant     coef.    weight 
222aa0          0.972    0.945 
aa2220         -0.155    0.024

The occupation numbers are (i) equal to 2 for the four inactive orbitals, (ii) close to 2 for $3\sigma_g$ and $1\pi_u$ , (iii) close to 1 for $1\pi_g$ , and (iv) close to 0 for $3\sigma_u$ .

print(casscf_results["natural_occupations"][0])

[1.96299851 1.96299851 1.96098176 1.03637599 1.03637599 0.04026925]

Unrestricted orbitals¶

In unrestricted calculations, we have separate sets of MOs for the $\alpha$ - and $\beta$ -spin orbitals. However, one set can be expressed in terms of the other according to

\phi^\beta_q(\mathbf{r}) = \sum_p \phi^\alpha_p(\mathbf{r}) S_{pq}^{\alpha\beta}

(6)

where $S^{\alpha\beta}$ is the overlap between the two sets of orbitals that, in turn, can be expressed in terms of the overlap matrix in the AO basis

S_{pq}^{\alpha\beta} = \sum_{\delta,\gamma} \big[c_{\delta p}^\alpha\big]^\ast c^\beta_{\gamma q} S_{\delta \gamma}

(7)

The one-particle density becomes

\begin{align*} n(\mathbf{r}) & = \sum_{p,q} \big[ \phi^\alpha_p(\mathbf{r}) \big]^\ast D_{pq} \phi_q^\alpha(\mathbf{r}) \\ D_{pq} & = D_{pq}^\alpha + \sum_{r,s} \big[ S^{\alpha\beta}_{pr} \big]^\ast D_{rs}^\beta S^{\alpha\beta}_{qs} \end{align*}

(8)

and the natural orbitals and occupation numbers are obtained from a diagonalization of this total density matrix.

Let us consider the unrestricted Hartree–Fock state of the oxygen molecule.

uhf_drv = vlx.ScfUnrestrictedDriver()
uhf_drv.ostream.mute()
uhf_results = uhf_drv.compute(molecule, basis)

print(f"UDFT energy: {uhf_drv.get_scf_energy() : 14.8f}")

UDFT energy:  -149.62899231

norb = basis.get_dimensions_of_basis()
n_alpha_electrons = 9
n_beta_electrons = 7

S = uhf_results["S"]
C_alpha = uhf_results["C_alpha"]
C_beta = uhf_results["C_beta"]

D_alpha = np.zeros((norb, norb))
for i in range(n_alpha_electrons):
    D_alpha[i, i] = 1

D_beta = np.zeros((norb, norb))
for i in range(n_beta_electrons):
    D_beta[i, i] = 1

S_ab = np.einsum("ap, bq, ab -> pq", C_alpha, C_beta, S)
D = D_alpha + np.einsum("pr, qs, rs -> pq", S_ab, S_ab, D_beta)

occ_numbers, U = np.linalg.eigh(D)

print("Orbital  Occ. number")
for orb, occ in enumerate(list(occ_numbers)[::-1]):
    if occ > 1e-3:
        print(f"{orb + 1 : >2} {occ : 12.3f}")

Orbital  Occ. number
 1        2.000
 2        2.000
 3        2.000
 4        1.999
 5        1.997
 6        1.994
 7        1.994
 8        1.000
 9        1.000
10        0.006
11        0.006
12        0.003

We can also get the natural orbitals and occupation numbers from the natural_orbitals method in VeloxChem.

natural_orbs = uhf_drv.natural_orbitals(uhf_results)
natural_orbs.print_orbitals(molecule, basis)

                                                                                                                          
                                                 Spin Restricted Orbitals                                                 
                                                 ------------------------                                                 
                                                                                                                          
               Molecular Orbital No.   3:                                                                                 
               --------------------------                                                                                 
               Occupation: 4.000 Energy:   -1.09255 a.u.                                                                  
               (   1 O   1p+1:    -0.47) (   1 O   2p+1:    -0.22) (   2 O   1p+1:     0.47)                              
               (   2 O   2p+1:     0.22)                                                                                  
                                                                                                                          
               Molecular Orbital No.   4:                                                                                 
               --------------------------                                                                                 
               Occupation: 3.998 Energy:   -1.34641 a.u.                                                                  
               (   1 O   2s  :     0.33) (   1 O   3s  :     0.37) (   1 O   1p+1:    -0.16)                              
               (   2 O   2s  :     0.33) (   2 O   3s  :     0.37) (   2 O   1p+1:     0.16)                              
                                                                                                                          
               Molecular Orbital No.   5:                                                                                 
               --------------------------                                                                                 
               Occupation: 3.994 Energy:   -1.09919 a.u.                                                                  
               (   1 O   2s  :     0.36) (   1 O   3s  :     0.45) (   2 O   2s  :    -0.36)                              
               (   2 O   3s  :    -0.45)                                                                                  
                                                                                                                          
               Molecular Orbital No.   6:                                                                                 
               --------------------------                                                                                 
               Occupation: 3.988 Energy:   -0.70203 a.u.                                                                  
               (   1 O   1p0 :     0.44) (   1 O   2p0 :     0.28) (   2 O   1p0 :     0.44)                              
               (   2 O   2p0 :     0.28)                                                                                  
                                                                                                                          
               Molecular Orbital No.   7:                                                                                 
               --------------------------                                                                                 
               Occupation: 3.988 Energy:   -0.70203 a.u.                                                                  
               (   1 O   1p-1:     0.44) (   1 O   2p-1:     0.28) (   2 O   1p-1:     0.44)                              
               (   2 O   2p-1:     0.28)                                                                                  
                                                                                                                          
               Molecular Orbital No.   8:                                                                                 
               --------------------------                                                                                 
               Occupation: 2.000 Energy:   -0.20095 a.u.                                                                  
               (   1 O   1p-1:    -0.53) (   1 O   2p-1:    -0.38) (   2 O   1p-1:     0.53)                              
               (   2 O   2p-1:     0.38)                                                                                  
                                                                                                                          
               Molecular Orbital No.   9:                                                                                 
               --------------------------                                                                                 
               Occupation: 2.000 Energy:   -0.20095 a.u.                                                                  
               (   1 O   1p0 :    -0.53) (   1 O   2p0 :    -0.38) (   2 O   1p0 :     0.53)                              
               (   2 O   2p0 :     0.38)                                                                                  
                                                                                                                          
               Molecular Orbital No.  10:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.012 Energy:    1.21925 a.u.                                                                  
               (   1 O   1p-1:     0.57) (   1 O   1p0 :    -0.38) (   1 O   2p-1:    -0.49)                              
               (   1 O   2p0 :     0.32) (   2 O   1p-1:     0.57) (   2 O   1p0 :    -0.38)                              
               (   2 O   2p-1:    -0.49) (   2 O   2p0 :     0.32)                                                        
                                                                                                                          
               Molecular Orbital No.  11:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.012 Energy:    1.21925 a.u.                                                                  
               (   1 O   1p-1:     0.38) (   1 O   1p0 :     0.57) (   1 O   2p-1:    -0.32)                              
               (   1 O   2p0 :    -0.49) (   2 O   1p-1:     0.38) (   2 O   1p0 :     0.57)                              
               (   2 O   2p-1:    -0.32) (   2 O   2p0 :    -0.49)                                                        
                                                                                                                          
               Molecular Orbital No.  12:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.006 Energy:    0.65627 a.u.                                                                  
               (   1 O   2s  :     0.41) (   1 O   3s  :     0.41) (   1 O   1p+1:     0.32)                              
               (   1 O   2p+1:     0.74) (   1 O   1d+2:    -0.20) (   2 O   2s  :    -0.41)                              
               (   2 O   3s  :    -0.41) (   2 O   1p+1:     0.32) (   2 O   2p+1:     0.74)                              
               (   2 O   1d+2:     0.20)                                                                                  
                                                                                                                          
               Molecular Orbital No.  13:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.002 Energy:    2.07548 a.u.                                                                  
               (   1 O   1s  :     0.38) (   1 O   2s  :     0.94) (   1 O   3s  :    -0.80)                              
               (   1 O   2p+1:     0.18) (   1 O   1d+2:    -0.35) (   1 O   1d0 :     0.20)                              
               (   2 O   1s  :     0.38) (   2 O   2s  :     0.94) (   2 O   3s  :    -0.80)                              
               (   2 O   2p+1:    -0.18) (   2 O   1d+2:    -0.35) (   2 O   1d0 :     0.20)                              
                                                                                                                          
               Molecular Orbital No.  14:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    1.39353 a.u.                                                                  
               (   1 O   1s  :    -0.23) (   1 O   2s  :    -0.50) (   1 O   3s  :     0.17)                              
               (   1 O   1p+1:    -0.62) (   1 O   2p+1:     0.83) (   2 O   1s  :    -0.23)                              
               (   2 O   2s  :    -0.50) (   2 O   3s  :     0.17) (   2 O   1p+1:     0.62)                              
               (   2 O   2p+1:    -0.83)