10.3 Orbital Analysis 10.3.3 Donor–Acceptor Orbital Overlaps 10.4 Density Analysis

10.3.4 Occupied–Virtual Orbitals for Chemical Valence Analysis

(July 4, 2026)

10.3.4.1 Overview

In mean-field theory such as HF and KS-DFT, the energy of a quantum state is completely determined by the 1-PDM P. Therefore, the energy difference between any two quantum states can be completely determined by the density difference $\Delta\textbf{P}$ . Insightful interpretations about the orbital interactions can be obtained from the eigen-decomposition of $\Delta\textbf{P}$ , which gives the so-called natural orbitals for chemical valence (NOCV) ⁹⁵⁴ Mitoraj M. P., Michalak A., Ziegler T.
J. Chem. Theory Comput.
(2008), 5, pp. 962. Link . The NOCVs appear in pairs $\{|\psi_{\pm i}\rangle\}$ with associated eigenvalues $\pm\lambda_{i}$ . However, the NOCVs are mixtures of occupied and virtual orbitals, which makes them hard to interpret; in addition, the associated charge of each NOCV pair is erroneously large ¹²⁵⁰ Shen H., Wang Z., Head-Gordon M.
J. Chem. Theory Comput.
(2022), 18, pp. 7428. Link . An improvement over NOCV is the recently proposed occupied-virtual orbitals for chemical valence (OVOCV) ¹²⁴⁸ Shen H., Head-Gordon M.
J. Phys. Chem. A
(2024), 128, pp. 5202. Link , whose construction is summarized below.

10.3.4.2 OVOCV analysis between any two single-determinant states

In the orthonormal basis of occupied and virtual orbitals of initial state $I$ , the matrix representation of $\hat{P}_{I}$ is

{\color[rgb]{0,0,0}P_{I}}=\begin{pmatrix}1&0\\ 0&0\end{pmatrix}

(10.52)

where $1$ is $o\times o$ and the diagonal 0 matrix is $v\times v$ in dimension. Under the same basis, the representation of the final density operator $P_{F}$ can be constructed from $P_{I}$ as $P_{F}=\exp\{X\}P_{I}\exp\{-X\}$ , where

X=\begin{pmatrix}0&X_{ov}\\ -X_{ov}^{T}&0\end{pmatrix}

(10.53)

With the SVD of $X_{ov}$ ,

X_{ov}=U\Sigma V^{T}

(10.54)

where $\dim(U)=o\times o,\dim(\Sigma)=o\times v,\dim(V)=v\times v$ , we can write $\Delta P$ as

\Delta P=\begin{pmatrix}U&0\\ 0&V\end{pmatrix}\begin{pmatrix}-\sin^{2}{\Sigma^{\prime}}&-\cos{\Sigma^{\prime% }}\sin{\Sigma^{\prime}}&0\\ -\sin{\Sigma^{\prime}}\cos{\Sigma^{\prime}}&\sin^{2}{\Sigma^{\prime}}&0\\ 0&0&0\end{pmatrix}\begin{pmatrix}U^{T}&0\\ 0&V^{T}\end{pmatrix}

(10.55)

where $\Sigma^{\prime}=\mathrm{diag}(\sigma_{1},\sigma_{2},...,\sigma_{o})$ , and the diagonal 0 matrix is of dimension $v-o$ (assuming $o<v$ , which is the case for most practical calculations).

Since $\Delta P$ is rendered diagonal in each of its blocks in the $U, V$ representation, it is natural to define this choice as the Occupied-Virtual Orbitals for Chemical Valence (OVOCVs).

	$\displaystyle\|\phi^{i}_{\text{occ}}\rangle$	$\displaystyle=\sum_{k}^{o}U_{ki}\|\psi_{k}\rangle$		(10.56)
	$\displaystyle\|\phi^{i}_{\text{vir}}\rangle$	$\displaystyle=\sum_{a}^{v}V_{ai}\|\psi_{a}\rangle,$		(10.57)

Specifically, $|\phi^{i}_{\text{occ}}\rangle$ and $|\phi^{i}_{\text{vir}}\rangle$ are the occupied and virtual orbitals of the $i$ th OVOCV pair. Since the OVOCVs are obtained by unitary transformations of an orthonormal basis, they still form an orthonormal basis (with a much smaller dimension, as the virtual orbitals irrelevant to the density deformation are deleted).

In the OVOCV basis, $\Delta P$ is:

\Delta P=\begin{pmatrix}-\sin^{2}{\Sigma^{\prime}}&-\cos{\Sigma^{\prime}}\sin{% \Sigma^{\prime}}\\ -\sin{\Sigma^{\prime}}\cos{\Sigma^{\prime}}&\sin^{2}{\Sigma^{\prime}}\end{pmatrix}

(10.58)

The (diagonal) oo block is negative semi-definite, corresponding to $\sin^{2}\sigma_{i}$ electron loss from the $i\text{th}$ of $o$ initially occupied OVOCV levels. By contrast the (diagonal) vv block is equal and opposite, corresponding to promotion of $\sin^{2}\sigma_{i}$ electrons into the $i\text{th}$ of $o$ initially virtual OVOCV levels.

The energy difference between any two single-determinant states can be obtained as

\Delta E=\text{Tr}\{\hat{F}^{\text{eff}}\Delta\hat{P}\},

(10.59)

where $\hat{F}^{\text{eff}}$ is an effective fock operator obtained from a line-integral over the density matrices between the two states. Using the decomposition of $\Delta\hat{P}$ in OVOCV basis, we have

$\displaystyle\Delta E$	$\displaystyle=\sum_{i}^{o}\sin^{2}\sigma_{i}\left(\langle\phi^{i}_{\text{vir}}% \|\hat{F}^{\text{eff}}\|\phi^{i}_{\text{vir}}\rangle-\langle\phi^{i}_{\text{occ}% }\|\hat{F}^{\text{eff}}\|\phi^{i}_{\text{occ}}\rangle\right)$	(10.60)
	$\displaystyle-\sum_{i}^{o}\sin\sigma_{i}\cos\sigma_{i}\left(\langle\phi^{i}_{% \text{vir}}\|\hat{F}^{\text{eff}}\|\phi^{i}_{\text{occ}}\rangle+\langle\phi^{i}_% {\text{occ}}\|\hat{F}^{\text{eff}}\|\phi^{i}_{\text{vir}}\rangle\right)$
	$\displaystyle=\sum_{i}^{o}\Delta E_{i}$

The OVOCV analysis is coded for the spin polarized calculations, so UNRESTRICTED = TRUE must be set.

OVOCV_ANALYSIS

OVOCV_ANALYSIS
       Perform OVOCV analysis
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       0 Do not do OVOCV analysis 1 Do OVOCV analysis
RECOMMENDATION:
       None

OVOCV_QUADRATURE

OVOCV_QUADRATURE
       Number of quadrature points used for the line-integral of $\hat{F}^{\text{eff}}$ .
TYPE:
       INTEGER
DEFAULT:
       5
OPTIONS:
       -1 Printings for debug purpose 5 5 quadratures 9 9 quadratures 13 13 quadratures 17 17 quadratures 21 21 quadratures
RECOMMENDATION:
       Use 5 for ground state analysis, 9 or 13 for excited state analysis. Increase quadrature numbers to get higher accuracy in $\Delta E$

SAVE_OVOCV_INIT_DATA

SAVE_OVOCV_INIT_DATA
       Saves initial state for OVOCV analysis
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       0 Do not save initial state data 1 Save initial state data
RECOMMENDATION:
       Set to 1 at the initial state calculation

LOAD_OVOCV_INIT_DATA

LOAD_OVOCV_INIT_DATA
       Load initial state for OVOCV analysis, then do OVOCV analysis using current state as the final state
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       0 Do not load initial state data 1 Load initial state data
RECOMMENDATION:
       Set to 1 at the final state calculation

OVOCV_THRESH

OVOCV_THRESH
       Specifies the significance above which the OVOCVs will be saved
TYPE:
       INTEGER
DEFAULT:
       500
OPTIONS:
       $N$ OVOCVs that contributes more than $0.001\times N$ kJ/mol in energy decrease will be saved
RECOMMENDATION:
       None

SAVE_OVOCV

SAVE_OVOCV
       Save significant OVOCVs or not
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       0 Do not save significant OVOCVs 1 Save significant OVOCVs
RECOMMENDATION:
       Set to 1 to save OVOCVs. Note REMs for plotting cube files need also be set

Example 10.10.16 OVOCV analysis between a polarized state and the fully converged SCF state.

$comment
Calculate a polarized state
$end

$molecule
-1 1
--
0 1
C      -2.1241906261     0.4405879048    -0.6028769327
H      -2.2759080470     0.1247332923    -1.6303256895
H      -2.2970177347     1.4883974948    -0.3784045735
C      -2.1383367135    -0.5060266611     0.4271962891
H      -2.3004855320    -1.5544440597     0.1975704475
H      -2.3216355550    -0.1916949240     1.4498166705
--
-1 1
Pt     -0.1374350284    -0.0292366293    -0.0576671311
Cl     -0.1541352882    -1.7330117528    -1.6304294395
Cl      2.1808813192    -0.0260821423    -0.0234510188
Cl     -0.2063937942     1.6741234771     1.5139773781
$end

$rem
unrestricted        true
method               wb97x-v
scf_print_frgm       1
sym_ignore           1
symmetry             false
basis                def2-svp
ecp                  def2-ecp
scf_algorithm        diis
thresh               14
incfock              0
mem_total            16000
scf_final_print      2
scf_convergence      8
frgm_method          stoll
child_mp             true
child_mp_orders      232
ovocv_analysis       1
save_ovocv_init_data 1
$end

@@@

$molecule
read
$end

$rem
unrestricted        true
thresh              14
scf_convergence     8
scf_algorithm       diis
method              wb97x-v
basis               def2-svp
ecp                 def2-ecp
scf_guess           read
sym_ignore          true
symmetry            false
make_cube_files     true
plots               1
ovocv_analysis      1
load_ovocv_init_data 1
ovocv_quadrature    5
ovocv_thresh        10000
save_ovocv            0
$end

$plots
grid_points 50 50 50
$end

Example 10.10.17 OVOCV analysis between a DFT state with HF density and the DFT state with DFT density.

$comment
SCF for fragmo to get initial density for HF
$end

$molecule
1 2
--
0 1
He 0 0 0
--
1 2
He 0 0 3.0
$end

$rem
method              hf
scf_guess           fragmo
frgm_method         stoll
scf_convergence     7
basis               6-31G**
scf_algorithm       gdm
sym_ignore          true   #make sure symmetry is off
symmetry            false
internal_stability  true
unrestricted        true
$end

@@@

$comment
HF calculation with frag density
$end

$molecule
read
$end

$rem
method              hf
scf_guess           read
scf_convergence     7
basis               6-31G**
scf_algorithm       gdm
sym_ignore          true   #make sure symmetry is off
symmetry            false
internal_stability  true
unrestricted        true
ovocv_analysis      1
save_ovocv_init_data 1
$end

@@@

$comment
DFT calculation with HF density
$end

$molecule
read
$end

$rem
internal_stability  true
unrestricted        true
thresh              14
scf_convergence     7
scf_algorithm       gdm
method              PBE
basis               6-31G**
scf_guess           read
scf_max_cycles      500
sym_ignore          true
symmetry            false
IQMOL_FCHK          False
make_cube_files     true
plots               1
ovocv_analysis      1
load_ovocv_init_data 1
ovocv_quadrature    17
ovocv_thresh        500
save_ovocv            0
$end

$plots
grid_points 50 50 50
$end

10.3.4.3 OVOCV analysis for OO-DFT excitation analysis

With the development of SCF algorithms for non-aufbau determinants, it is often possible to accurately represent an excited state, such as a charge-transfer state or a doubly excited state, with a single Slater determinant using Orbital-Optimized Density Functional Theory (OO-DFT). It is therefore natural to analyze the electronic excitation using the OVOCV method ¹²⁴⁷ Shen H. et al.
J. Chem. Theory Comput.
(2025), 21, pp. 9525. Link , which gives a primary excitation due to a single (or double for double excitation) OVOCV pair $|\psi_{\text{D}}\rangle$ (donor) and $|\psi_{\text{A}}\rangle$ (acceptor), together with many polarized pairs to account for the orbital relaxation effect. To reveal the importance of orbital relaxations, we introduce an intermediate frozen state (FRZ), whose density is constructed from the complete depletion of donor orbital $|\psi_{\text{D}}\rangle$ and the complete occupation of acceptor orbital $|\psi_{\text{A}}\rangle$ , starting from the ground state density $\hat{P}_{\text{G}}$ :

\hat{P}_{\text{FRZ}}=\hat{P}_{\text{G}}-|\psi_{\text{D}}\rangle\langle\psi_{% \text{D}}|+|\psi_{\text{A}}\rangle\langle\psi_{\text{A}}|.

(10.61)

As a result, the connection between $\hat{P}_{\text{G}}$ and $\hat{P}_{\text{FRZ}}$ reveals the primary excitation, and the connection between $\hat{P}_{\text{FRZ}}$ and $\hat{P}_{\text{E}}$ reveals the orbital relaxation effect.

OVOCV_EXCITE_ANALYSIS

OVOCV_EXCITE_ANALYSIS
       Perform OVOCV analysis for OO-DFT excitation analysis
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       0 Do not do OVOCV OO-DFT analysis 1 Do OVOCV OO-DFT analysis
RECOMMENDATION:
       None

OVOCV_EXCITE_TYPE

OVOCV_EXCITE_TYPE
       Specify the excitation type to construct the intermediate FRZ state
TYPE:
       INTEGER
DEFAULT:
       1
OPTIONS:
       1 Construct FRZ state using the largest OVOCV pair in $\alpha$ space 2 Construct FRZ state using the largest OVOCV pair in $\beta$ space 3 Construct FRZ state using the largest OVOCV pair in both $\alpha$ and $\beta$ space (for double excitation)
RECOMMENDATION:
       Use 1 or 2 for single excitation, 3 for double excitation

Example 10.10.18 OVOCV OO-DFT analysis for long distance charge transfer between NH ${}_{3}$ and F ${}_{2}$ .

$molecule
0 1
N     0.000000000     0.000000000     -2.149985070
H     -0.469826010     -0.813762490     -2.534400940
H     -0.469826010     0.813762490     -2.534400940
H     0.939652030     0.000000000     -2.534400940
F     0.000000000     0.000000000     15.850014930
F     0.000000000     0.000000000     17.275484090
$end

$comment
IMOM we use for TRIPLET EXCITATIONS or MIXED-SPIN configurations
$end

$rem
thresh              14
scf_convergence     8
method              hf
basis               def2-TZVP
scf_algorithm       diis_gdm
sym_ignore          true   #make sure symmetry is off
symmetry            false
XC_GRID             50000194
unrestricted        true
ovocv_analysis      1
save_ovocv_init_data 1
$end

@@@

$rem
unrestricted        true
thresh              14
scf_convergence     6
method              hf
basis               def2-TZVP
mom_method          imom
mom_start           1
scf_guess           read
scf_max_cycles      500
sym_ignore          true
symmetry            false
XC_GRID             50000194
make_cube_files     true
plots               1
ovocv_analysis      1
load_ovocv_init_data 1
ovocv_quadrature    9
ovocv_thresh        15000
ovocv_excite_analysis 1
ovocv_excite_type     1
save_ovocv            0
$end

$molecule
read
$end

$occupied
1 2 3 4 5 6 7 8 9 10 11 12 13 15
1 2 3 4 5 6 7 8 9 10 11 12 13 14
$end

$plots
grid_points 50 50 50
$end

Example 10.10.19 OVOCV OO-DFT analysis for LMCT in Fe(H ${}_{2}$ O) ${}^{3+}_{6}$ .

$molecule
+3 6
--
+3 6
Fe     -0.5022247054     1.2927779200     0.0001938411
--
0 1
H       2.1134166395     2.0808631740    -0.0009010689
H      -0.5027205697    -1.3235141738    -0.7847973080
H      -3.1183219511     2.0796505928    -0.0004236834
H       0.2840444564     1.2933014045     2.6166394903
H      -0.5027751731     3.9101762183     0.7846214605
O       1.5358193871     1.2947592260    -0.0014374082
H       2.1156949943     0.5103151332    -0.0009786075
O      -0.5019595039     3.3312830798    -0.0005116891
H      -0.5019009551     3.9098531329    -0.7858415694
O      -0.5013746615     1.2930914301     2.0381118720
H      -1.2864874118     1.2932385232     2.6170717164
O      -2.5401948736     1.2938974534     0.0009627059
H      -3.1195714968     0.5090879007     0.0019042859
O      -0.5018840213    -0.7448372467     0.0005186723
H      -0.5008290080    -1.3235744599     0.7857827770
O      -0.5020354665     1.2915906981    -2.0378380237
H       0.2835925140     1.2915440728    -2.6160740707
H      -1.2869601935     1.2917959205    -2.6170203924
$end

$rem
jobtype             sp
method              wb97x-v
basis               def2-tzvp
scf_max_cycles      1000
scf_algorithm       gdm
sym_ignore          true
symmetry            false
thresh              14
scf_convergence     7
unrestricted        true
ovocv_analysis      1
save_ovocv_init_data 1
$end

@@@

$molecule
+3 6
--
+2 5
Fe     -0.5022247054     1.2927779200     0.0001938411
--
+1 2
H       2.1134166395     2.0808631740    -0.0009010689
H      -0.5027205697    -1.3235141738    -0.7847973080
H      -3.1183219511     2.0796505928    -0.0004236834
H       0.2840444564     1.2933014045     2.6166394903
H      -0.5027751731     3.9101762183     0.7846214605
O       1.5358193871     1.2947592260    -0.0014374082
H       2.1156949943     0.5103151332    -0.0009786075
O      -0.5019595039     3.3312830798    -0.0005116891
H      -0.5019009551     3.9098531329    -0.7858415694
O      -0.5013746615     1.2930914301     2.0381118720
H      -1.2864874118     1.2932385232     2.6170717164
O      -2.5401948736     1.2938974534     0.0009627059
H      -3.1195714968     0.5090879007     0.0019042859
O      -0.5018840213    -0.7448372467     0.0005186723
H      -0.5008290080    -1.3235744599     0.7857827770
O      -0.5020354665     1.2915906981    -2.0378380237
H       0.2835925140     1.2915440728    -2.6160740707
H      -1.2869601935     1.2917959205    -2.6170203924
$end

$rem
jobtype             sp
scf_guess           fragmo
FRGM_METHOD         STOLL
SCF_PRINT_FRGM      TRUE
method              wb97x-v
basis               def2-tzvp
scf_max_cycles      1000
scf_algorithm       gdm
sym_ignore          true
symmetry            false
thresh              14
scf_convergence     7
unrestricted        true
$end

$rem_frgm
SCF_CONVERGENCE     5
SCF_ALGORITHM       DIIS_GDM
THRESH              14
$end

@@@

$molecule
read
$end

$rem
jobtype             sp
method              wb97x-v
basis               def2-tzvp
scf_guess           read
mom_method          imom
mom_start           1
sym_ignore          true
symmetry            false
thresh              14
scf_convergence     7
scf_max_cycles      2000
unrestricted        true
make_cube_files     true
plots               1
ovocv_analysis      1
load_ovocv_init_data 1
ovocv_quadrature    9
ovocv_thresh        20000
ovocv_excite_analysis 1
ovocv_excite_type     2
save_ovocv            0
$end

$plots
grid_points 50 50 50
$end

Example 10.10.20 OVOCV OO-DFT analysis for double excitation in trans-butadiene.

$molecule
0 1
C       1.7631203268    -0.4019108032    -0.3481693109
C       0.6037765052     0.2512052721    -0.3293638273
C      -0.6037682643    -0.2511908395     0.3294491374
C      -1.7631216007     0.4019093543     0.3482256846
H      -2.6370356062    -0.0041538614     0.8460657302
H      -1.8733953929     1.3678634226    -0.1376167636
H       2.6369974511     0.0040901125    -0.8461249062
H       1.8735775331    -1.3675549056     0.1382472290
H      -0.5187645613    -1.2176355085     0.8243565706
H       0.5187966092     1.2176897568    -0.8241975438
$end

$rem
method              wB97X-V
basis               def2-tzvp
scf_algorithm       diis_gdm
sym_ignore          true   #make sure symmetry is off
symmetry            false
max_scf_cycles      200
unrestricted        true
ovocv_analysis      1
save_ovocv_init_data 1
$end

@@@

$rem
unrestricted        true
method              wB97X-V
basis               def2-tzvp
scf_guess           read
scf_max_cycles      1000
mom_method          imom
mom_start           1
sym_ignore          true
symmetry            false
make_cube_files     true
plots               1
ovocv_analysis      1
load_ovocv_init_data 1
ovocv_quadrature    9
ovocv_thresh        1000
ovocv_excite_analysis 1
ovocv_excite_type     3
save_ovocv            0
$end

$molecule
read
$end

$occupied
1:14 16
1:14 16
$end

$plots
grid_points 50 50 50
$end

	$\displaystyle\|\phi^{i}_{\text{occ}}\rangle$	$\displaystyle=\sum_{k}^{o}U_{ki}\|\psi_{k}\rangle$		(10.56)
	$\displaystyle\|\phi^{i}_{\text{vir}}\rangle$	$\displaystyle=\sum_{a}^{v}V_{ai}\|\psi_{a}\rangle,$		(10.57)

Search Results

10.3.4 Occupied–Virtual Orbitals for Chemical Valence Analysis

10.3.4.1 Overview

10.3.4.2 OVOCV analysis between any two single-determinant states

10.3.4.3 OVOCV analysis for OO-DFT excitation analysis