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\begin_body

\begin_layout Title

\series bold
SOMO Identification
\begin_inset Newline newline
\end_inset


\begin_inset CommandInset line
LatexCommand rule
offset "0.5ex"
width "100col%"
height "1pt"

\end_inset


\end_layout

\begin_layout Author

\size footnotesize
R.
 Poteau,
 romuald.poteau@univ-tlse3.fr,
 LPCNO
\end_layout

\begin_layout Date

\size footnotesize
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
today
\end_layout

\end_inset


\end_layout

\begin_layout Section
Introduction
\end_layout

\begin_layout Standard
To identify singly occupied molecular orbitals (
\series bold
SOMOs
\series default
) in open-shell systems,
 we employed two complementary schemes:
 (i) orbital projection analysis,
 and (ii) cosine similarity mapping.
 Both approaches compare the sets of 
\begin_inset Formula $\alpha$
\end_inset

 and 
\begin_inset Formula $\beta$
\end_inset

 molecular orbitals obtained from unrestricted calculations.
 The projection scheme evaluates the squared overlap of each 
\begin_inset Formula $\alpha$
\end_inset

 orbital with the 
\begin_inset Formula $\beta$
\end_inset

 orbital space using the atomic orbital overlap matrix,
 allowing for a quantitative decomposition of each 
\begin_inset Formula $\alpha$
\end_inset

 orbital across the 
\begin_inset Formula $\beta$
\end_inset

 manifold.
 In contrast,
 the cosine similarity approach measures the angular similarity between 
\begin_inset Formula $\alpha$
\end_inset

 and 
\begin_inset Formula $\beta$
\end_inset

 orbitals based on their MO coefficients,
 identifying pairs of orbitals with nearly identical spatial character.
 Together,
 these methods help identify SOMOs as occupied 
\begin_inset Formula $\alpha$
\end_inset

 orbitals that lack a clear counterpart among the occupied 
\begin_inset Formula $\beta$
\end_inset

 orbitals.
 Instead,
 they show strong projections onto the 
\begin_inset Formula $\beta$
\end_inset

 virtual space and minimal overlap with the 
\begin_inset Formula $\beta$
\end_inset

 occupied space —
 a characteristic signature of magnetic (unpaired) molecular orbitals in open-shell systems.
\end_layout

\begin_layout Section
Similarity schemes
\end_layout

\begin_layout Subsection
Projection of occupied 
\begin_inset Formula $\alpha$
\end_inset

 MOs onto the 
\begin_inset Formula $\beta$
\end_inset

 orbital space
\end_layout

\begin_layout Standard
Given a Gaussian log file from an unrestricted DFT calculation,
 we extract the molecular orbital (MO) coefficients for both 
\begin_inset Formula $\alpha$
\end_inset

 and 
\begin_inset Formula $\beta$
\end_inset

 orbitals (
\family typewriter
pop=full
\family default
 keyword),
 along with the AO overlap matrix 
\begin_inset Formula $S$
\end_inset

 (
\family typewriter
iop(3/33=1)
\family default
 keyword).
 The analysis focuses on evaluating how each occupied 
\begin_inset Formula $\alpha$
\end_inset

 orbital projects onto the full space spanned by all 
\begin_inset Formula $\beta$
\end_inset

 orbitals,
 which includes both occupied and virtual ones.
\end_layout

\begin_layout Standard
Let 
\begin_inset Formula $\phi_{i}^{\alpha}\in\mathbb{R}^{1\times n_{\text{basis}}}$
\end_inset

 be the coefficient vector of the 
\begin_inset Formula $i$
\end_inset

-th occupied 
\begin_inset Formula $\alpha$
\end_inset

 orbital,
 and let 
\begin_inset Formula $\Phi^{\beta}\in\mathbb{R}^{N\times n_{\text{basis}}}$
\end_inset

 be the matrix of all 
\begin_inset Formula $\beta$
\end_inset

 orbitals stored row-wise,
 where 
\begin_inset Formula $N=n_{\beta}$
\end_inset

 is the total number of 
\begin_inset Formula $\beta$
\end_inset

 orbitals.
 The projection vector is computed as:
 
\begin_inset Formula 
\[
\mathbf{v}_{i}=\phi_{i}^{\alpha}\cdot S\cdot(\Phi^{\beta})^{T}\in\mathbb{R}^{1\times N}
\]

\end_inset

The squared norm 
\begin_inset Formula $\|\mathbf{v}_{i}\|^{2}$
\end_inset

 gives the total overlap of the 
\begin_inset Formula $\alpha$
\end_inset

 orbital with the 
\begin_inset Formula $\beta$
\end_inset

 space.
\end_layout

\begin_layout Standard
To differentiate between the contributions from occupied and virtual 
\begin_inset Formula $\beta$
\end_inset

 orbitals,
 we split the projection:
 
\begin_inset Formula 
\begin{align*}
\mathbf{v}_{i}^{\text{occ}} & =\phi_{i}^{\alpha}\cdot S\cdot(\Phi_{\text{occ}}^{\beta})^{T}\\
\mathbf{v}_{i}^{\text{virt}} & =\phi_{i}^{\alpha}\cdot S\cdot(\Phi_{\text{virt}}^{\beta})^{T}
\end{align*}

\end_inset

We then compute:
\end_layout

\begin_layout Standard
\begin_inset Formula $\|\mathbf{v}_{i}^{\text{occ}}\|^{2}$
\end_inset

 = projection of 
\begin_inset Formula $\phi_{i}^{\alpha}$
\end_inset

 onto occupied 
\begin_inset Formula $\beta$
\end_inset

 orbitals
\end_layout

\begin_layout Standard
\begin_inset Formula $\|\mathbf{v}_{i}^{\text{virt}}\|^{2}$
\end_inset

 = projection of 
\begin_inset Formula $\phi_{i}^{\alpha}$
\end_inset

 onto virtual 
\begin_inset Formula $\beta$
\end_inset

 orbitals
\end_layout

\begin_layout Standard
The total projection norm is decomposed to analyze how concentrated or spread the projection is across 
\begin_inset Formula $\beta$
\end_inset

 orbitals:
 
\end_layout

\begin_layout Itemize
The three largest values among the squared projections 
\begin_inset Formula $v_{ij}^{2}$
\end_inset

 are summed to compute 
\series bold

\begin_inset Quotes eld
\end_inset

Top 1 contrib (%)
\series default

\begin_inset Quotes erd
\end_inset

,
 
\series bold

\begin_inset Quotes eld
\end_inset

Top 2 contrib (%)
\series default

\begin_inset Quotes erd
\end_inset

 and 
\series bold

\begin_inset Quotes eld
\end_inset

Top 3 contrib (%)
\series default

\begin_inset Quotes erd
\end_inset

.
 
\end_layout

\begin_layout Itemize
The 
\series bold
dominance ratio
\series default
 is defined as the largest single squared projection divided by the total projection norm:
 
\begin_inset Formula $\max_{j}v_{ij}^{2}/\|\mathbf{v}_{i}\|^{2}$
\end_inset

.
 
\end_layout

\begin_layout Itemize
The 
\begin_inset Quotes eld
\end_inset


\series bold

\begin_inset Formula $\beta$
\end_inset

 MOs >20%
\series default

\begin_inset Quotes erd
\end_inset

 column lists all 
\begin_inset Formula $\beta$
\end_inset

 orbitals contributing more than the specified percentage to the squared projection norm,
 along with their contribution in the format 
\begin_inset Formula $[j,p_{j}]$
\end_inset

,
 where 
\begin_inset Formula $j$
\end_inset

 is the index (1-based) and 
\begin_inset Formula $p_{j}$
\end_inset

 the percentage contribution.
 For the most important contribution,
 it is nothing else than the 
\series bold
dominance ratio
\series default
.
 It provides a direct quantitative decomposition of each 
\begin_inset Formula $\alpha$
\end_inset

 orbital onto the 
\begin_inset Formula $\beta$
\end_inset

 orbital basis.
 Each entry explicitly identifies the
\begin_inset Formula $\beta$
\end_inset

 orbital(s) that significantly compose the corresponding 
\begin_inset Formula $\alpha$
\end_inset

 orbital,
 along with their respective percentage contributions
\end_layout

\begin_layout Standard
An orbital is flagged as a 
\series bold
SOMO candidate
\series default
 if its projection onto the virtual 
\begin_inset Formula $\beta$
\end_inset

 space exceeds 0.5 and its projection onto the occupied 
\begin_inset Formula $\beta$
\end_inset

 space is below 0.5:
 
\begin_inset Formula 
\[
\|\mathbf{v}_{i}^{\text{virt}}\|^{2}>0.5\quad\text{and}\quad\|\mathbf{v}_{i}^{\text{occ}}\|^{2}<0.5
\]

\end_inset


\end_layout

\begin_layout Subsection
Cosine similarity of MOs
\end_layout

\begin_layout Standard
The identification of singly occupied molecular orbitals (SOMOs) can also be achieved through the computation of the cosine similarity between pairs of molecular orbitals (MOs) derived from unrestricted spin density functional theory (DFT) calculations.
 Specifically,
 we computed similarities between 
\begin_inset Formula $\alpha$
\end_inset

 and 
\begin_inset Formula $\beta$
\end_inset

 spin orbitals,
 taking into account the non-orthogonality of the basis set used in quantum chemical calculations.
\end_layout

\begin_layout Standard
Let us denote two molecular orbital coefficient vectors as 
\begin_inset Formula $\Phi^{\alpha}$
\end_inset

 (for alpha-spin orbitals) and
\begin_inset Formula $\Phi^{\beta}$
\end_inset

 (for beta-spin orbitals).
 Each vector has dimensions corresponding to the number of basis functions used in the calculation,
 denoted by 
\begin_inset Formula $n_{\text{basis}}$
\end_inset

.
 Given the overlap matrix 
\begin_inset Formula $\boldsymbol{S}$
\end_inset

 (dimension 
\begin_inset Formula $n_{\text{basis}}\times n_{\text{basis}}$
\end_inset

),
 obtained from the quantum chemistry calculation,
 the scalar product between two coefficient vectors accounting for basis overlap is defined as:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\langle\Phi^{\alpha}|\Phi^{\beta}\rangle_{S}=\Phi^{\alpha T}\boldsymbol{S}\Phi^{\beta}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
Thus,
 the cosine similarity between two molecular orbitals 
\begin_inset Formula $\Phi^{\alpha}$
\end_inset

 and 
\begin_inset Formula $\Phi^{\beta}$
\end_inset

 accounting for the basis overlap matrix,
 
\begin_inset Formula $\boldsymbol{S}$
\end_inset

,
 is given by:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\text{cosine similarity}(\Phi^{\alpha},\Phi^{\beta})=\frac{\Phi^{\alpha T}\boldsymbol{S}\Phi^{\beta}}{\sqrt{\Phi^{\alpha T}\boldsymbol{S}\Phi^{\alpha}}\sqrt{\Phi^{\beta T}\boldsymbol{S}\Phi^{\beta}}}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
The similarity matrix constructed from these cosine similarities was then used to optimally match 
\begin_inset Formula $\alpha$
\end_inset

 and 
\begin_inset Formula $\beta$
\end_inset

 orbitals employing the Hungarian algorithm,
 ensuring maximal global similarity.
 Also known as the Kuhn–Munkres algorithm,
 it is a classic method used to solve the assignment problem:
 given a cost matrix,
 it finds the optimal one-to-one assignment (or matching) that minimizes (or maximizes) the total cost (or similarity).
 It is provided by the 
\family typewriter
scipy.optimize.linear_sum_assignment
\family default
 function.
 
\end_layout

\begin_layout Standard
Orbital pairs with high cosine similarity,
 particularly those involving occupied alpha-spin orbitals matched to virtual beta-spin orbitals (or vice versa),
 can also be identified as potential candidates for SOMOs.
 This method provides a robust and quantitatively precise approach to identifying SOMOs in unrestricted DFT calculations,
 facilitating detailed analyses of electronic structures in open-shell systems.
\end_layout

\begin_layout Subsection
Brief discussion
\end_layout

\begin_layout Standard
The projection technique quantifies how much each 
\begin_inset Formula $\alpha$
\end_inset

 orbital overlaps with the entire 
\begin_inset Formula $\beta$
\end_inset

 orbital space by computing the squared norm of the projection vector using the AO overlap matrix.
 This provides an absolute,
 physically meaningful measure of orbital mixing,
 especially relevant when analyzing partial spin contamination or magnetic character.
 In contrast,
 cosine similarity evaluates the angle between two orbital vectors,
 yielding a dimensionless similarity score between -1 and 1.
 It's more suited for comparing the shape of orbitals than their actual physical contribution to each other.
 While cosine similarity is useful for clustering and pattern recognition,
 the projection approach is generally more precise when it comes to quantifying actual contributions and mixing between spin orbitals,
 especially in systems with open-shell or near-degenerate character.
\end_layout

\begin_layout Section
Examples
\end_layout

\begin_layout Subsection
Formaldehyde (H
\begin_inset Formula $_{2}$
\end_inset

CO)
\end_layout

\begin_layout Standard
Table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Projection-of-alpha2beta-H2CO"
plural "false"
caps "false"
noprefix "false"
nolink "false"

\end_inset

 presents simplified projection data of occupied 
\begin_inset Formula $\alpha$
\end_inset

 orbitals onto 
\begin_inset Formula $\beta$
\end_inset

 orbitals for the lowest triplet state,
 T
\begin_inset script subscript

\begin_layout Plain Layout
1
\end_layout

\end_inset

,
 of formaldehyde.
 It is adapted from the dataframe created by the 
\family typewriter
project_occupied_alpha_onto_beta()
\family default
 function.
 Orbitals identified as SOMO indicate significant projection onto virtual 
\begin_inset Formula $\beta$
\end_inset

 orbitals and negligible projection onto occupied 
\begin_inset Formula $\beta$
\end_inset

 orbitals.
\end_layout

\begin_layout Standard
\begin_inset Float table
placement H
alignment document
wide false
sideways true
status open

\begin_layout Plain Layout
\align center

\size small
\begin_inset Tabular
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<row>
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\size small
\begin_inset Formula $\boldsymbol{\alpha}$
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\series bold
 MO 
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\series bold
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Occ 
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\begin_inset Formula $\boldsymbol{\alpha}$
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Energy (Ha) 
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\series bold
\size small
P² 
\begin_inset Formula $\boldsymbol{\beta_{\mathrm{virt}}}$
\end_inset

 
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\end_inset
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P² 
\begin_inset Formula $\boldsymbol{\beta_{\mathrm{occ}}}$
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\series bold
\size small
\begin_inset Formula $\boldsymbol{\beta}$
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 MO* 
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Occ 
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SOMO?
 
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 MOs >20%
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 96.2%
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</lyxtabular>

\end_inset


\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "tab:Projection-of-alpha2beta-H2CO"

\end_inset

Projection of 
\begin_inset Formula $\alpha$
\end_inset

 molecular orbitals onto 
\begin_inset Formula $\beta$
\end_inset

 space for formaldehyde (H
\begin_inset Formula $_{2}$
\end_inset

CO),
 highlighting SOMOs.
 
\series bold
\size small
P² 
\begin_inset Formula $\boldsymbol{\beta_{\mathrm{virt}}}$
\end_inset


\series default
\size default
 and 
\series bold
\size small
P² 
\begin_inset Formula $\boldsymbol{\beta_{\mathrm{occ}}}$
\end_inset


\series default
\size default
 are 
\begin_inset Formula $\|\mathbf{v}_{i}^{\text{occ}}\|^{2}$
\end_inset

 and 
\begin_inset Formula $\|\mathbf{v}_{i}^{\text{virt}}\|^{2}$
\end_inset

 ,
 respectively (see text)
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
A heatmap can generated to visualize the main projection contributions of 
\begin_inset Formula $\alpha$
\end_inset

 molecular orbitals onto 
\begin_inset Formula $\beta$
\end_inset

 orbitals.
 The color intensity reflects the percentage contribution of each 
\begin_inset Formula $\beta$
\end_inset

 orbital to the total projection norm of a given 
\begin_inset Formula $\alpha$
\end_inset

 orbital.
 Only contributions above 20% were retained for clarity.
 Red dashed lines indicate the HOMO–LUMO frontier for both spin channels (Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Similarity-of-alphaWithBeta-H2CO"
plural "false"
caps "false"
noprefix "false"
nolink "false"

\end_inset

a).
 This analysis is very close to the heatmap generated after the cosine similarity between 
\begin_inset Formula $\alpha$
\end_inset

 and 
\begin_inset Formula $\beta$
\end_inset

 MOs (Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Similarity-of-alphaWithBeta-H2CO"
plural "false"
caps "false"
noprefix "false"
nolink "false"

\end_inset

b)
\end_layout

\begin_layout Standard
\begin_inset Float figure
placement H
alignment document
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename H2CO_T1_g09_wOverlaps_projection_heatmap-C.png
	lyxscale 30
	width 8cm

\end_inset


\begin_inset Graphics
	filename H2CO_T1_g09_wOverlaps_heatmap-C.png
	lyxscale 30
	width 8cm

\end_inset


\end_layout

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\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:Similarity-of-alphaWithBeta-H2CO"

\end_inset

similarity of 
\begin_inset Formula $\alpha$
\end_inset

 and 
\begin_inset Formula $\beta$
\end_inset

 MOs of the first triplet state of H
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

CO around the HOMO-LUMO frontier.
 (a) Projection of 
\begin_inset Formula $\alpha$
\end_inset

 MOs onto the full space spanned by all 
\begin_inset Formula $\beta$
\end_inset

 orbitals (only contributions above 20% were retained for clarity);
 (b) Cosine similarity.
 Dashed lines mark the HOMO/LUMO boundaries for 
\begin_inset Formula $\alpha$
\end_inset

 (horizontal) and 
\begin_inset Formula $\beta$
\end_inset

 (vertical) spin orbitals.
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\end_body
\end_document
