# Locally Weighted Mean Phase Angle (LWMPA) Based Tone Mapping Quality Index
(TMQI-3)

Inaam Ul Hassan  1† Abdul Haseeb  2† Sarwan Ali3∗
1Department of Computer Science,
Lahore University of Management Sciences, Lahore, Pakistan
2Department of Computer Science, CECOS University,
Peshawar, Pakistan
3Department of Computer Science, Georgia State University,
Atlanta, GA 30303, USA
† Joint first authors
∗Corresponding author
email<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS>

###### Abstract

High Dynamic Range (HDR) images are the ones that contain a greater range of
luminosity as compared to the standard images. HDR images have a higher detail
and clarity of structure, objects and colour, which the standard images lack.
HDR images are useful in capturing scenes that pose high brightness, darker
areas, and shadows, etc. An HDR image comprises of multiple narrow-range-
exposure images combined into one high-quality image. As these HDR images
cannot be displayed on standard display devices, the real challenge comes
while converting these HDR images to Low dynamic range (LDR) images. The
conversion of HDR image to LDR image is performed using Tone-mapped operators
(TMOs). This conversion results in the loss of much valuable information in
structure, colour, naturalness and exposures. The loss of information in the
LDR image may not directly be visible to the human eye. To calculate how good
an LDR image is after conversion, various metrics have been proposed
previously. Some are not noise resilient, some work on separate colour
channels (Red, Green, and Blue one by one) and some lack capacity to identify
the structure. To deal with this problem, we propose a metric in this paper
called the Tone Mapping Quality Index (TMQI-3), which evaluate the quality of
the LDR image based on its objective score. TMQI-3 is noise resilient, takes
account of structure and naturalness, and works on all three colour channels
combined into one luminosity component. This eliminates the need to use
multiple metrics at the same time. We compute results for several HDR and LDR
images from the literature and show that our quality index metric performs
better than the baseline models.

###### keywords:

Tone Mapping, HDR, LDR, Mean Phase, Objective Quality Assessment, Tone Mapping
Operator

##  1 Introduction

The pictures taken from a camera is a combination of two components, the
luminance component and the chrominance component. The visibility of the
chrominance component is highly dependant on the intensity of white light
presented by the luminance component. [29, 15, 27]. The normal picture
consists of shadows and brighter portions. Sometimes due to brighter parts,
the details for shadow portions becomes very low, and in worst cases, the
objects in that portion of the image become near to invisible. This is termed
a Low Dynamic Range (LDR) image [24]. One way of balancing the brightness and
shadow areas, such that both are visible clearly and no detail is missed, is
called High Dynamic Range (HDR) Imaging [14].

According to [19], when we capture the high contrast image from a camera,
either dark images are saturated, or the bright ones are saturated in the
output image. This occurs because the sensors in our camera are minimal to
capture the luminosity due to shadows or over-brightness. Our conventional
cameras cannot provide a proper image, the same as a human eye. The idea of
HDR photograph is presented to us. Traditionally for creating an HDR
photograph, various LDR images are combined. But the prime issue is that most
of the devices are unable or incapable to show HDR images. Therefore, it is
important to convert the HDR image to the LDR image without any loss of
information.

The process of creating HDR images is done by capturing multiple shots (3 or
more) of various exposures and combining them into one better image. High-
dynamic-range imaging (HDRI) [30, 26] is a technique in which images are
processed or captured with a higher amount (or level) of luminosity. This
higher dynamic range of luminosity cannot be achieved through standard digital
imaging. To capture a HDR image, several narrow range exposed images are taken
and combined into one. A limited exposure range mainly results in the loss of
important information such as highlights and shadows. The method of capturing
HDR images is quite common and mostly the same in all techniques. Still, the
significant issue here is the representation of those High contrast and High
dynamic images onto our typical day to day devices which have minimal display
powers and are capable enough to display the LDR images only. Another crucial
part in HDR image rendering and displaying is Tone Mapping [22, 32].

Tone mapping corresponds to rendering an HDR image to a standard monitor or
printing device. This process is carried out by Tone mapping operators (TMO).
The need to rendering HDR to LDR is crucial because an HDR image cannot be
directly displayed on a standard display or printing device due to its high
contrast and colour ratios. There are different TMO proposed previously for
the conversion of HDR to LDR [17, 21, 20, 32]. A natural question that arises
is regarding the quality of images resulting from applying TMOs on them. Two
types of evaluation can be applied to the images, namely subjective evaluation
and objective evaluation. Subjective evaluation is done using human eyes,
while objective evaluation is done by analyzing the actual structure of the
images.

HDR contains a full noticeable scope of luminance and shading of the images
[34]. However, while converting images from HDR to LDR (using TMOs), there is
a possibility of missing essential image structures in the resultant LDR image
[35]. It is challenging for human eyes to catch these structural problems in
the LDR image (called subjective evaluation). This problem creates room for an
objective image quality measure of the tone mapped images so that the overall
quality of an image can be measured in detail.

Tone mapping quality index (TMQI) is used to objectively measure the quality
of the images (called objective evaluation) converted from HDR to LDR by the
TMOs. Previously, different objective quality measure are proposed such as
TMQI-1 [35], TMQI-2 [21], and Feature Similarity Index for Tone-Mapped Images
(FSITM) [25]. Although FSITM shows better results as compared to TMQI-1 and
TMQI-2, the major problem with it is that it works with Red, Blue, and Green
channels separately, rather than combined. We propose a new approach, called
TMQI-3, that overcomes the limitations of FSITM. Our contributions in this
paper are the following:

  1. 1.

We study different tone mapping operator and through experiments we identify
the strength and weaknesses of each operator.

  2. 2.

We propose an algorithm for objective quality assessment of images called
tone-mapped image quality index (TMQI-3).

  3. 3.

Our proposed quality index algorithm shows good correlations between a
subjective ranking score of the images and an objective ranking score computed
using TMQI-3.

##  2 Literature Review

Objective assessment is a common approach to analyse the output of algorithms
in many domains such as graph analytics [11, 1], protein sequence study [9, 5,
10, 4], smart grid [6, 8, 7], information processing [2], network security
[3], and pattern recognition [33]. In the images quality assessment domain, in
spite of operating on the original HDR pixel value, almost every tone mapping
algorithm perform the task on the logarithm of luminance values of HDR pixel.
Olivier Lezoray in [19] proposes to use manifold based ordering in which pixel
value is more non-linear rather than a log-luminance curve. The proposed
method learns the manifold of HDR image pixels to build a new HDR image
representation. That image representation is in the form of an index image
associated with an ordering of the HDR pixels’ vectors. It was done in the
following manner:

  1. 1.

Ordering of HDR vectors

  2. 2.

According to the order of first step The Novice depiction of HDR images are
performed

Authors in [23] propose a generic tone mapping algorithm that can be used in
the black-box analysis of existing TMO, backward compatibility of HDR image
compression, and the synthesis of new algorithms that are a combination of
existing operators. Their paper says that one model does the estimation of
TMOs comprise of tone curve accompanied by spatial modulation function.
Moreover, nearly the same image processing technique is used by approximately
all TMOs but the selection of parameters varies.

Eilertsen et. al. in [13], authors have done a broad comparison between
different TMOs and point out the drawbacks in video TMOs. Their prime focus is
on the descriptive analysis of the new changes and evolution in tone mapping
pipelines. They also devised a new and generic tone mapping algorithm that can
best suit the needs of future HDR videos

Kede et. al. In [21] the authors introduce TMOs that intend to pack high
dynamic range (HDR) pictures to Low Dynamic Range (LDR) to envision HDR photos
on standard presentations.They have proposed a significantly unique way to
deal with outline TMO. Rather than utilizing any predefined efficient
computational structure for tone mapping, they specifically explore all
pictures in the space of all photographs, hunting down the image that advances
an enhanced TMQI. Specifically, the first enhance the two building hinders in
TMQI—primary devotion and genuine expectation parts—prompting a TMQI-2 metric.
They then propose an iterative calculation that, on the other hand, enhances
the auxiliary loyalty and measurable expectation of the following picture.
Numerical and subjective tests show that the proposed analysis reliably
delivers better quality tone-mapped images, notwithstanding when the most
focused TMOs make the underlying pictures of the emphasis. Then, these
outcomes likewise approve the predominance of TMQI-2 over TMQI-1.

In [25] the authors suggested a feature similarity index for tone-mapped
images (FSITM) system that works on the local phase information of images. For
assessing the tone mapping operator (TMO), the suggested index compares the
calculated associated tone-mapped image by utilising the TMO method’s output
with the locally weighted mean phase angle map of an original high dynamic
range (HDR). For experiments, they have taken two sets of images after
assessing the results. It shows that the FSITM system outperforms the other
tone mapped quality index (TMQI) algorithms. Furthermore, they combine the
proposed system FSITM with TMQI, and show better results as compared to
typical TMQI’s.

Authors in [17] Introduce a new hybrid method that has been introduced by
combining two-hybrid tone mapping operators (local and global operators).
Several images are amalgamating into a single HDR which results in enhanced
HDR image. An enhancement map is constructed either with the threshold value
or the luminance value of the pixel. Using the enhanced map, the original
luminance map is separated from the base layer and detail layer by running
bilateral filtering (noise-reducing filter for images). The detail layer is
used to enhance the result of global tone mapping. The performance of hybrid
tone mapping is then compared to individual local and international operators,
and the results show that the hybrid operator gives better performance.

##  3 Proposed Approach

This section proposes our algorithm, TMQI-3, for objectively evaluating LDR
images (performance of TMO’s). In the literature, there are two types of
popular models for measuring the quality of images.

  1. 1.

Peak signal-to-noise ratio

  2. 2.

Structural Similarity Index Metric (SSIM)

However, the above two quality measures assume that the reference and compared
images have the same dynamic range. Since that assumption is not valid in the
LDR images, we cannot directly apply these models in our research.

######  Definition 3.1.

TMQI-1 assesses the quality of individual LDR images based on combining an
SSIM-motivated structural fidelity measure with the statistical naturalness.
The expression for TMQI-1 is given in Equation (1).

$\text{TMQI-1}(x,y)=a[S(x,y)a+(1-a)N(y)]B$ (1)

$S$ represents structural fidelity, $N$ represents statistical naturalness,
$x$ represents HDR image, and $y$ represents the LDR image. The parameters $a$
and $B$ determine the sensitivity. The range of the parameter $a$ is the
following:

$0\leq a\leq 1$ (2)

######  Remark 3.2.

Note that the parameters structural fidelity and statistical naturalness are
upper bounded by $1$. Therefore, TMQI-1 is also upper bounded by $1$.

TMQI-1 was the first approach for measuring the LDR image’s quality across the
dynamic range to the best of our knowledge. It provides a better assessment
for the LDR images (compared to the traditional methods discussed above) as a
result of applying TMO on the HDR. However, TMQI-1 have some limitations such
as

  1. 1.

It can only be applied to greyscale images. However, most of the HDR images
nowadays are in colour.

  2. 2.

The statistical naturalness measure used in the TMQI-1 is based on the
intensity statistics only. However, many sophisticated statistical models in
the literature can also capture other properties of the image, such as
structural regularities in space, scale, and orientation.

######  Remark 3.3.

The problem of TMQI-1 only working with the greyscale images can be solved by
applying the TMQI-1 on each colour channel separately. Although this may allow
TMQI-1 to work on colour images, its performance will not be outstanding [35].

We have proposed a new approach in which we have combined $3$ existing
techniques differently to achieve better results in terms of subjective
evaluation of both TMQI-2 [21], and FSITM [25]. The approaches which we have
used as a reference for our research are the following.

Our proposed method (TMQI-3) take into account the following properties of the
images.

  1. 1.

Structural fidelity

  2. 2.

Statistical naturalness

  3. 3.

It also makes use of the mean phase angle of local weights

By combining the properties mentioned above, we could get a better objective
score for the input images.

We will now discuss all the above properties one by one.

###  3.1 Improved Structural Fidelity

The structural fidelity [21] of TMQI-1 can be computed using a sliding window
across the whole image. This process results in a quality map and hence
preserving the local structural detail of the image. Local structural fidelity
measure given by TMQI-1 is given in Equation. (3).

$S_{local}(x,y)=\frac{2\hat{\sigma}_{x}\hat{\sigma}_{y}+C_{1}}{\hat{\sigma}_{x}^{2}\hat{\sigma}_{y}^{2}+C1}.\frac{\sigma_{xy}+C_{2}}{\sigma_{x}\sigma_{y}+C2}$
(3)

where $\sigma_{x}$ and $\sigma_{y}$ denote the local standard deviations
(std), respectively. The $\sigma_{xy}$ denotes the covariance between two
corresponding patches. The two (positive) constant terms C1 and C2 are used to
avoid any possible instability. Overall structural fidelity is calculated
using Equation. Equation (4).

$S(X,Y)=\frac{1}{M}\sum_{i=1}^{M}S_{local}(x_{i},y_{i})$ (4)

The updates in structural fidelity are done through the gradient descent
method stated in Equation. (5).

$\hat{Y}_{k}=Y_{k}+\lambda\nabla_{Y}S(X,Y)|_{Y=Y_{k}}$ (5)

Where $Y_{k}$ is the image resulting from k iteration and $\lambda$ is the
step size. This works as the contrast visibility model for the local luminance
model.

###  3.2 Improved Statistical Naturalness

Model for statistical naturalness [21] proposed in TMQI-1 is given in Equation
(6).

$N(Y)=\frac{1}{K}P_{m}P_{d}$ (6)

where $P_{m}$ represents gaussian density function while $P_{d}$ represents
beta density function. However, Equations (6) has the following limitations.

  1. 1.

Gaussian density function and beta density function are considered independent
of image content, which may not entirely be true.

  2. 2.

Model for statistical naturalness is derived from high-quality images while
having no information regarding how an unnatural image may look.

The updates above can be abstractly defined by the equations below:

$Pm=\frac{1}{\sqrt{2\pi\theta_{1}}}\int_{-\infty}^{\mu}exp\Big{[}-\frac{(t-T_{1})^{2}}{2\theta^{2}_{1}}\Big{]}dt\mu\leq\mu_{e}$
(7)

$Pm=\frac{1}{\sqrt{2\pi\theta_{2}}}\int_{-\infty}^{2\mu_{r}-\mu}exp\Big{[}-\frac{(t-T_{2})^{2}}{2\theta^{2}_{2}}\Big{]}dt\mu>\mu_{e}$
(8)

$Pd=\frac{1}{\sqrt{2\pi\theta_{3}}}\int_{-\infty}^{\sigma}exp\Big{[}-\frac{(t-T_{3})^{2}}{2\theta^{2}_{3}}\Big{]}dt\sigma\leq\sigma_{e}$
(9)

$Pd=\frac{1}{\sqrt{2\pi\theta_{4}}}\int_{-\infty}^{2\sigma_{r}-\sigma}exp\Big{[}-\frac{(t-T_{4})^{2}}{2\theta^{2}_{4}}\Big{]}dt\sigma>\sigma_{e}$
(10)

######  Remark 3.4.

Note that the acceptable luminance changes saturate at both small and large
luminance levels without significantly tampering with the image visual
naturalness [21].

###  3.3 Use of Mean Phase Angle of Local Weights

An image quality measure based on the local phase information of an image is
proposed in [25]. Their model is noise independent; hence no parameter is
required for noise estimation.

######  Remark 3.5.

Note that multiple methods have already been proposed in the literature
related to the quality assessment of images using the phase. information [25,
36, 28, 12, 16, 31].

One drawback of the methods proposed in the literature that consider phase
information is that their results for evaluating tone-mapped images are not
reliable compared to other famous quality assessment metrics like SSIM. The
Locally Weighted Mean Phase Angle (LWMPA) is robust to noise and is computed
using the following expression according to [25].

$ph(x)=\text{arctan}\ 2[\sum_{\rho,r}e_{\rho r}(x),\sum_{\rho,r}o_{\rho
r}(x)]$ (11)

where $\rho$ represents scale and $r$ represents orientation of the image
[25]. The values for the pixels ($v_{ph}$) of $ph(x)$ are following

$\frac{-\pi}{2}\leq v_{ph}\leq\frac{+\pi}{2}$ (12)

The term $\frac{-\pi}{2}$ in Equation (12) represents a dark line while
$\frac{+\pi}{2}$ represents a bright line. The pixels of $ph(x)$ takes value
$0$ for the steps. For further detail, readers are referred to [18, 25].

The ph(x) in Equation (11) provides a good representation of image features.
This representation includes the edges of the objects within the image and the
shapes of those objects. From Equation (12), we know that ph(x) represents
both dark and bright lines. Therefore, it can be used to identify colours
within the image. This colour detection is a useful property for a TMO.

######  Remark 3.6.

Note that the LWMPA has the property to ignore the noise from the image, which
is not the case with the TMQI-1 that uses phase derived features.

FSITM works on Locally Weighted Mean Phase Angle (LWMPA). It works on separate
channels of Red, Blue, and Green. Not all of them combined. This is a
significant issue that we noticed in FSITM. The image should be judged as a
whole, not on separate channels only and according to the human eye’s
sensitivity. Combining the locally weighted mean phase angle with structural
fidelity and statistical naturalness requires it to be represented in
luminance. The reason is that the structural fidelity and statistical
naturalness work upon the luminance of the image. So to combine the LWMPA with
others, its quality index score should be mapped for luminance components,
too, since the human eye is much more responsive to the luminance rather than
RGB separately. To solve the problem that we identified in FSITM, we used the
YUV model. The equation of ‘Y’ is used to connect Red, Green and Blue to
luminance. We used this equation for the following reasons.

  1. 1.

The conversion of RGB to Y requires just a linear transform, which is very
easy to do and cheap to compute numerically

  2. 2.

‘Y’ is perceived as brightness, which is more sensitive to the human eye.
Luminance gives the measure of the amount of energy an observer perceives from
the light source. That is why separate RGB sensitivity values are selected in
Y.

Its equation for Y is the following:

$Y=0.299*R+0.587*G+0.114*B$ (13)

TMQI-1, TMQI-2, and FSITM work on different methods to evaluate the image.
Structural fidelity and statistical naturalness are considered in TMQI-2,
while LWMPA is considered in FSITM. Separately, these methods are suitable to
access some types of image features (but not an image as a whole). So they
should be combined into one quality assessment model. For this purpose, we
combine all of these to design our TMQI-3 and give equal weight to each
component, as given below:

$Q=(1/3)*N+(1/3)*F+(1/3)*L$ (14)

Where N, F, L denote statistical naturalness, structural fidelity, and LWMPA,
respectively.

The improvements in structural fidelity and statistical naturalness are
somehow provided in TMQI-2, but it lacks noise resilience like in SSIM and
other matrices. However, they give better results due to gradient ascent
method infidelity and solving parameter optimization problems for point-wise
intensity transformation in statistical naturalness. Further noise reduction
can be made using the phase angle map of locally weighted means because it is
robust and has error resilience. By combining improvements in structural
fidelity and statistical naturalness with phase angle map, we would be
improving the quality and hence enhance our quality index.

Our metric takes following $3$ inputs.

  1. 1.

An HDR image is used as a reference

  2. 2.

The LDR image being compared (either colour or greyscale image with its
dynamic range equal to 255)

  3. 3.

A local window for statistics

The default window for our approach is gaussian. The quality metric first
computes the structural fidelity and statistical naturalness of the image
using the method proposed by [21]. Then it computes the quality index for RGB
separately by the method provided in [25]. The main issue comes at the point
of combining these three. For this purpose, RGB quality index values are
mapped to luminance quality index values, as stated in the previous
paragraphs. They are combined according to the sensitivity of RGB values to
human eyes. Then after that, these three are combined with equal weights given
to each.

ombining these three different approaches comes from the fact that each method
checks the image from a different perspective. The structural fidelity focuses
upon the visibility of image details which further depends upon the sampling
density of the image, the distance between the image and the observer, the
resolution of the display, and the perceptual capability of the observer’s
visual system. For the statistical naturalness, the studies in [35] show that
among all attributes such as brightness, contrast, colour reproduction,
visibility, and reproduction of details, brightness and contrast have more
correlation with perceived naturalness. The real challenge of combining these
three together was due to the LWMPA from FSITM. The image is not observed
based on chrominance, luminance (brightness) or contrast, but based on the
three primary colours only, that is, Red, Green and Blue, and the quantities
of these three in an image. LWMPA is not susceptible to noise or error, which
makes it more robust in terms of quality. Combining these three results in a
Quality matrix with the capability of luminance, contrast, Structure,
Naturalness, and the phases of the image.

We would effectively distinguish the visible and invisible local contrast in
images and provide a good representation of image features indicating the
changes in colours and dark or bright lines.

##  4 Results and Discussion

In this section, we first give information about the dataset and other
implementation detail. Then we show our results and discuss the behaviour of
our proposed algorithm is compared to the baselines.

###  4.1 Experimental Setup

To evaluate our metric, we used the dataset proposed in [35]. It contains 15
HDR images, with each image having its associated 8 LDR images as well. The
LDR images are subjectively scored from the range of 1 to 8. A subjective
score of 1 means the image is best converted from HDR to LDR, and a score of 8
means the worst conversion. The subjective scores were obtained based on an
assessment of $20$ individuals. Implementation of our algorithm is done in
Matlab, and experiments are performed on Core I$3$ $2.4$ GHz with $8$ GB of
RAM.

The evaluation metric that we are using is Kendall’s rank-order correlation
coefficient (KRCC) [35]. KRCC is a non-parametric rank correlation metric
whose formula is the following:

$KRCC=\frac{N_{c}-N_{d}}{\frac{1}{2}N(N-1)}$ (15)

Where $N_{c}$ is the number of consistent rank order (concordant), and $N_{d}$
is the number of inconsistent rank order (discordant) pairs in the data set.

###  4.2 Kendall Correlation Based Results

Kendall’s correlation coefficient was run on the quality indices generated for
each method (TMQI-3, TMQI-2, TMQI-1, FSITM), and the results are shown in
Table 1. We can observe that the TMQI-3 process is comparable (also better for
some images) to the TMQI-2 approach. In reference to TMQI-1, the TMQI-3
performs better in many cases. In FSITM, TMQI-3 is better for image set 7 and
comparable for image set 1, 6, 11, and 15.

Image Set | TMQI-3 | TMQI-2 | TMQI-1 | FSITM
---|---|---|---|---
1 | 0.7857 | 0.7857 | 0.3571 | 0.7857
2 | 0.3571 | 0.2857 | 0.6429 | 0.7143
3 | 0.5714 | 0.5714 | 0.6429 | 0.7857
4 | 0.5000 | 0.5000 | 0.7143 | 0.7857
5 | 0.5000 | 0.5000 | 0.6429 | 0.6429
6 | 0.7857 | 0.7143 | 0.7143 | 0.7857
7 | 0.7857 | 0.7143 | 0.5714 | 0.7143
8 | 0.5714 | 0.5000 | 0.5714 | 0.6429
9 | 0.7143 | 0.7143 | 0.5714 | 0.8571
10 | 0.7857 | 0.7857 | 0.8571 | 0.8571
11 | 0.7143 | 0.7143 | 0.7143 | 0.7143
12 | 0.4286 | 0.4286 | 0.5714 | 0.5714
13 | 0.6071 | 0.6071 | 0.5357 | 0.6786
14 | 0.5714 | 0.5714 | 0.6429 | 0.6429
15 | 0.7857 | 0.7857 | 0.7857 | 0.7857
Average | 0.6309 | 0.6119 | 0.6357 | 0.7309
Min | 0.3571 | 0.2857 | 0.3571 | 0.5714
Max | 0.7857 | 0.7857 | 0.8571 | 0.8571
Std | 0.1402 | 0.1445 | 0.1138 | 0.0812
Table 1:  KRCC values between subjective score and different TMQI’s score
(higher score is better). For each of the 15 HDR images, we have 8 LDR images.
We computed each LDR image’s objective score separately and then reported
their average in this table (for all techniques). The Average, Minimum,
Maximum, and Standard Deviation values are also reported.

We can argue that although there is not any clear winner from Table 1. FSITM
looks to be better than the other methods. However, authors of the FSITM in
[25] argues that different TMO algorithms perform differently on additional
HDR images. Their behaviour depends on the (type of) HDR image to be
converted. From this uncertainty of the TMO’s, we can conclude that the best.
TMO approach must be found for each case (no TMO can be generalized on all HDR
images).

###  4.3 Visual Results on LDR Images

In this section, we explore the visual effect of different LDR images
generated from different TMO’s [25].

####  4.3.1 Indoor House Images:

These first set of LDR images in Figure 1 are of indoor houses generated from
their corresponding HDR images. Referring to the Table 2 we can identify that
the worst subjective score is given to the Figure 1(b) (which is 5.95).

######  Remark 4.1.

A higher subjective score (max 8) refers to a bad conversion from HDR to LDR,
while a lower subjective score (minimum 1) refers to better conversion.

We can observe that the best subjective score is of the Figure 1(c). For each
resultant Figure TMQI-3, TMQI-2, TMQI-1, and FSITM are run to produce
objective scores. The results can be seen in Table 2. We can observe the
correspondence between the objective and subjective scores in the associated
table.

######  Remark 4.2.

The objective score of a metric should be low if the subjective score is high
and high if the subjective score is low.

The aim of FSITM scores for Figure 1 (b) should be less than that of Figure 1
(a) because Figure 1 (b) have a higher subjective score. However, in reality,
the FSITM is giving the opposite of it. The same behavior is observed with
TMQI-1 regarding Figure 1 (b) and Figure 1 (a). The TMQI-2 and TMQI-3 perform
as expected, giving a lower objective score to the Figure with a higher
subjective score and vice versa.

|  | Objective scores
---|---|---
Figure | Subjective Score | TMQI-3 | TMQI-2 | TMQI-1 | FSITM
Fig. 1 (a) | 4.4 | 0.5484 | 0.4106 | 0.8016 | 0.8077
Fig. 1 (b) | 5.95 | 0.5194 | 0.3814 | 0.8799 | 0.8311
Fig. 1 (c) | 2 | 0.6645 | 0.5770 | 0.9191 | 0.8774
Table 2: Subjective and Objective score for different methods on indoor house
LDR image.

(a)

(b)

(c)

Figure 1: Indoor house LDR images created using different TMOs [35]. The
subjective and objective scores for each image are given in Table 2.

####  4.3.2 Open Area Wide Shots Images:

Figure 2 shows the LDR images of open area-wide computed by different TMOs
from the corresponding HDR image. The Figure 2 (a) has the highest subjective
score, Figure 2 (b) has the lowest and Figure 2 (c) is in the middle. The
Figure 2 (a) has the highest score because of its over brightness and unclear
image quality, and Figure 2 (b) has the lowest score because of its clear
foreground and background. (As stated earlier, a lower subjective score means
a better image, and a higher subjective score means the worst image.) The
corresponding objective scores of the metrics should be inversely related to
the subjective scores of each image, respectively. In this case, all the
metrics provide the right objective scores. The objective score and the
subjective scores for the Figures are shown in Table 3

|  | Objective scores
---|---|---
Figure | Subjective Score | TMQI-3 | TMQI-2 | TMQI-1 | FSITM
Fig. 2 (a) | 7.1 | 0.4547 | 0.3207 | 0.7004 | 0.6987
Fig. 2 (b) | 3.65 | 0.5774 | 0.4732 | 0.8328 | 0.8155
Fig. 2 (c) | 4.75 | 0.8637 | 0.9008 | 0.8589 | 0.8272
Table 3: Subjective and Objective score for different methods on open area
wide shots LDR image.

(a)

(b)

(c)

Figure 2: Open area wide shots LDR images created using different TMOs [35].
The subjective and objective scores for each image are given in Table 3.

####  4.3.3 Main Object Upfront Images:

Figure 3 shows the LDR images produced from HDR image from different TMOs.
Subjectively the best Figure is 3 (b) and the worst Figure is 3 (c) (lowest
and highest subjective scores respectively). The objective score and the
subjective scores for the images are shown in Table 4. We can notice that the
good image has a higher objective score, and the bad image have a lesser
objective score in all TMQI-3, TMQI-2, TMQI-1 and FSITM metrics. Thus they all
perform well on the images where the main object upfront is obvious than the
background. The only difference that can be noticed is that FSITM and TMQI-1
were not able to provide a larger objective-score gap for Figure 3 (b) and 3
(c). The difference is very minute where it should have been greater as the
subjective score of image 3 (c) is much greater than that of Figure 3 (b).
This is not the case with TMQI-2 and TMQI-3. They both provide better
differences in objective scores in correspondence to their subjective scores.

|  | Objective scores
---|---|---
Figure | Subjective Score | TMQI-3 | TMQI-2 | TMQI-1 | FSITM
Fig. 3 (a) | 1.65 | 0.7672 | 0.7337 | 0.9363 | 0.8862
Fig. 3 (b) | 1.45 | 0.8801 | 0.9051 | 0.9475 | 0.8926
Fig. 3 (c) | 5.6 | 0.6199 | 0.5544 | 0.9119 | 0.8332
Table 4: Subjective and Objective score for different methods on Main object
upfront LDR image.

(a)

(b)

(c)

Figure 3: Main object upfront LDR images created using different TMOs [35].
The subjective and objective scores for each image are given in Table 4.

####  4.3.4 Indoor with background scenery:

Figure 4 shows the LDR images produced from HDR image by different TMOs. The
results are given in Table 5. The best image subjectively so far is image 4
(b) and the worst is image 4 (c) (scores 1.55 and 5.65 respectively). The
objective score from TMQI-3, TMQI-2, TMQI-1, and FSITM are calculated on each
image. The objective score of a metric should be low if the subjective score
is high and high if the subjective score is low. The FSITM objective scores
for image (b) should be less than that of image (a) because the image (b) have
a higher subjective score. But in actual FSITM is giving the opposite of it.
The same is with TMQI-1 regarding figures (b) and (a) in this case. TMQI-2 and
TMQI-3 perform as expected, giving a lower objective score to the image with a
higher subjective score and giving a higher objective score to the image with
a lower subjective score.

|  | Objective scores
---|---|---
Figure | Subjective Score | TMQI-3 | TMQI-2 | TMQI-1 | FSITM-TMQI
Fig. 4 (a) | 1.55 | 0.8592 | 0.8704 | 0.9476 | 0.8942
Fig. 4 (b) | 3.9 | 0.8316 | 0.8166 | 0.9548 | 0.9050
Fig. 4 (c) | 5.65 | 0.3785 | 0.2142 | 0.8766 | 0.7484
Table 5: Subjective and Objective score for different methods on Indoor with
background scenery LDR image.

(a)

(b)

(c)

Figure 4: Indoor with background scenery LDR images created using different
TMOs [35]. The subjective and objective scores for each image are given in
Table 5.

####  4.3.5 Outdoor Natural Scenery Images:

This set of LDR images in Figure 5 are of outdoor natural scenery generated
produced from their respective HDR image through various TMOs. The results are
given in Table 6. It can be observed that the best image (with lowest
subjective score) is Figure 5 (a) and worst is Figure 5 (c). Their respective
subjective scores are 3.65 and 7.1. It can also be observed that TMQI-3,
TMQI-2, TMQI-1, and FSITM all produced the right results. The only difference
that can be noticed is that FSITM and TMQI-1 could not provide a larger
objective-score gap for each image. Whereas TMQI-2 and TMQI-3 were able to
provide better objective score differences with respect to the corresponding
subjective scores.

|  | Objective scores
---|---|---
Figure | Subjective Score | TMQI-3 | TMQI-2 | TMQI-1 | FSITM
Fig. 5 (a) | 3.65 | 0.8260 | 0.7768 | 0.9487 | 0.9332
Fig. 5 (b) | 4.75 | 0.7153 | 0.6142 | 0.8952 | 0.9035
Fig. 5 (c) | 7.1 | 0.4975 | 0.3156 | 0.8008 | 0.8167
Table 6: Subjective and Objective score for different methods on Outdoor
natural scenery LDR image.

(a)

(b)

(c)

Figure 5: Outdoor natural scenery created using different TMO’s [35]. The
subjective and objective scores for each image are given in Table 6.

##  5 Conclusion

We have proposed an objective quality index, called the Locally weighted mean
phase angle based tone mapping quality index (TMQI-3), which is based upon the
combination of three basic properties namely statistical naturalness,
structural fidelity, and locally weighted mean phase angle. These all three
properties were used separately in the literature. In this paper, we integrate
all of them in one quality index because they can provide better results (when
combined) while objectively evaluating an LDR image. The results are seemingly
accurate and better than those offered by previous quality matrices. Our
metric is noise resilient, includes structure and naturalness of image, and
and provide better objective scores. In future, we will focus on developing a
better weights assessment strategy for each individual component based on the
sensitivity and structure of each image separately.

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University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1,
A-1090 Vienna, Austria

E-mail<EMAIL_ADDRESS>


UChicago] Department of Chemistry, University of Chicago. UChicago] Department
of Chemistry, University of Chicago.

# Automatic State Interaction with Large Localized Active Spaces for
Multimetallic Systems

Valay Agarawal  Daniel King  Matthew R. Hermes<EMAIL_ADDRESS>[  Laura
Gagliardi<EMAIL_ADDRESS>[

###### Abstract

The localized active space self consistent field (LASSCF) method factorizes a
complete active space (CAS) wave function into an antisymmetrized product of
localized active space wave function fragments. Correlation between fragments
is then reintroduced through LAS state interaction (LASSI), in which the
Hamiltonian is diagonalized in a model space of LAS states. However, the
optimal procedure for defining the LAS fragments and LASSI model space is
unknown. We here present an automated framework to explore systematically
convergent sets of model spaces, which we call LASSI[$r$,$q$]. This method
requires the user to select only $r$, the number of electron hops from one
fragment to another and $q$, the number of fragment basis functions per
Hilbert space, which converges to CASCI in the limit of $r,q\to\infty$.
Numerical tests of this method on the tri-metal complexes
[Fe(III)Al(III)Fe(II)($\mu_{3}$-O)]6+ and [Fe(III)2Fe(II)($\mu_{3}$-O)]6+ show
efficient convergence to the CASCI limit with 4-10 orders of magnitude fewer
states.

##  1 Introduction

Transition metal systems pose a unique challenge for quantum chemistry due to
the near-degeneracy of the $d$-shell electrons. These near-degeneracies mean
that transition metal systems are often ill-described by any single electronic
configuration, and thus often require multiconfigurational treatment to obtain
accurate results. This is especially true in systems containing multiple
transition metals with some degree of interaction. While multiconfigurational
methods such as the complete active space self consistent field (CASSCF)
method1 are generally capable of describing a single metal quite well, they
are often computationally prohibitive when two or more metal centers are
present.2, 3

However, a large number of configuration state functions (CSFs) (or Slater
determinants (SDs) used in some codes) generated in the CAS method usually do
not contribute in a substantial way to the final CAS wave function; these
determinants are referred to as “deadwood”4. Efforts to remove the deadwood
from the wave function can be sorted broadly into two categories: “automated”
strategies such as selected configuration interaction (SCI) 5, 6, 7, 8, 9, 10
and density matrix renormalization group (DMRG) 11 which incorporate the
elimination of deadwood into the ansatz; and “manual” strategies such as
generalized active space SCF (GASSCF) 12, 13, 14, 15, restricted active space
SCF (RASSCF) 16, 17, the cluster mean field (cMF) 18 and localized active
space SCF (LASSCF) 19, 20 methods, which utilize chemical knowledge to wisely
eliminate deadwood prior to calculation. This latter category is clearly less
black box, but comes with the opportunity to be exponentially more efficient
prior to even beginning a calculation.

The LASSCF method (originally introduced as cMF 18 but independently re-
derived by us as a generalization of density matrix embedding theory 19, 20)
is an approximation to CASSCF in which the wave function is factorized as a
single antisymmetrized product of subspaces which are localized in real space.
This approximation itself is not qualitatively accurate for multimetallic
compounds involving interactions among the metal centers, even if they are
weak, for the simple reason that the spin operator ($\hat{S}^{2}$) is
nonlocal21. However, LASSCF states can serve as an efficient model space for
the localized active space state interaction (LASSI) method,22 which recovers
correlation between the subspaces. LASSI achieves this by projecting the
molecular Hamiltonian into a model space of LAS wave functions which share the
same set of orbitals in each subspace but possess different quantum numbers
(e.g., number of electrons, local spin, and excitation number).

The LASSI method can be regarded as the application of the active space
decomposition (ASD) method 23, 24, 25, 26, 27 to the LASSCF reference wave
function; or equivalently, a method intermediate between vector product state
(VPS)-CASCI and VPS-CASSCF28 (see also Ref. 29) in that orbitals are
optimized, but only for the reference state. However, all investigations into
these methods thus far have studied only organic chromophores for which the
appropriate model space is known in advance to consist of singlet excitations,
coupled triplet excited states, and single-electron charge transfer states.
For systems such as multimetallic molecules – key systems we would like to
treat with these product-form methods – the questions remain of what model
spaces are appropriate, how best to explore different model spaces, and how
quickly they can be expected to converge.

In previous applications of LASSI to transition metal systems, construction of
the model space was carried out entirely “manually”, i.e. the number of
electrons and spin for each fragment in each state was decided by the user.
This manual control facilitated a deeper understanding of the physics of the
test systems; for instance the different contributions to the exchange
coupling constant, $J$, for a simplified model22 of Kremer’s Dimer30.
Nevertheless, for larger systems, this manual state selection process is time
consuming, error prone, and non-scalable.

In this paper, we propose a new method in which the LASSI model space
selection scheme is “automatic”. We call this method LASSI[$r$,$q$]. In this
approach, instead of selecting the individual configurations by hand, the user
simply must select the number of “electron hops” from one fragment to another,
$r$, and the number of locally excited states to construct for each fragment,
$q$. In the large-$r$, large-$q$ limit, LASSI[$r$,$q$] reproduces CASCI
(including the exponential cost). We perform test calculations on a triiron
complex [Fe(III)Fe(III)Fe(II)($\mu_{3}$-O)]6+, previously explored with
multireference methods in Ref. 31 and its aluminum substituted variant -
[Fe(III)Al(III)Fe(II)($\mu_{3}$-O)]6+ (henceforth “Fe3” and “AlFe2”), and show
that qualitative agreement with CASCI low-energy spectra is achieved by $r=1$
and $q=15$ at the most, corresponding to multiple orders of magnitude fewer
degrees of freedom than the CASCI limit. These calculations are described in
Sec. 4 below.

Through analysis of the LASSI eigenvectors of the Fe3 system using a minimal
(3$d$ only) active space, we furthermore identify more categories of LASSI
model states which are deadwood, and eliminate them systematically through a
further approximation which we call LASSI[$r$,$q_{\text{CT}}$] (CT stands for
charge transfer). This latter method, for the first time, permits us to carry
out LASSI calculations for which the corresponding conventional CASCI is
impossible: Fe3 with a 3$d$ and 4$d$ active space. We contrast these results
to those of a comparable DMRG calculation.

##  2 Theory

Considering only the active-space part of the wave function (i.e., treating
the doubly-occupied inactive electrons as part of the frozen core), the LAS
wave function,

$\ket{\text{LAS}}=\bigwedge_{K}^{\text{fragments}}\ket{0_{K}}_{\mathcal{P}},$
(1)

is a single antisymmetrized product of general many-electron fragment wave
functions $\ket{0_{K}}_{\mathcal{P}}$, defined in a particular Hilbert space
labeled $\mathcal{P}$, about which see below. The fragment wave functions are
the eigenstates of the projected Hamiltonian,

$\displaystyle\hat{H}_{\mathcal{P}K}$ $\displaystyle\equiv$
$\displaystyle\hat{\mathcal{P}}_{K}^{\dagger}\hat{H}\hat{\mathcal{P}}_{K},$
(2) $\displaystyle\hat{\mathcal{P}}_{K}$ $\displaystyle=$
$\displaystyle\bigotimes_{L\neq
K}^{\text{fragments}}\ket{0_{L}}_{\mathcal{P}},$ (3) $\displaystyle 0$
$\displaystyle=$
$\displaystyle\left(\hat{H}_{\mathcal{P}K}-E_{n\mathcal{P}K}\right)\ket{n_{K}}_{\mathcal{P}},$
(4)

where $n\geq 0$ is an integer indexing the excitation number of the particular
fragment. Each fragment Hamiltonian, $\hat{H}_{\mathcal{P}K}$, describes the
$K$th fragment in the mean field of all other fragments in their respective
ground states. The LAS states in total are then indexed as

$\ket{\vec{n}}_{\mathcal{P}}\equiv\bigwedge_{K}^{\text{fragments}}\ket{n_{K}}_{\mathcal{P}}$
(5)

where $\vec{n}=\\{n_{1},n_{2},n_{3},\ldots\\}$ is the “excitation number
vector” (ENV) of nonnegative integers that index the various eigenstates of
the $\hat{H}_{\mathcal{P}K}$ operators.

The Hilbert space indexed by $\mathcal{P}$ in which the roots of the projected
Hamiltonians are sought (henceforth the ‘rootspace’) is defined by the number
of electrons in each fragment, the spin quantum numbers of each fragment, and
the point group irreducible representation of each fragment wave function. In
other words, $\mathcal{P}$ is a compound index representing the space table
below, in which different rows represent different fragments and different
columns represent different types of quantum number,

$\mathcal{P}=\begin{bmatrix}N_{\uparrow}^{(1)}&N_{\downarrow}^{(1)}&S^{(1)}&\Gamma^{(1)}\\\
N_{\uparrow}^{(2)}&N_{\downarrow}^{(2)}&S^{(2)}&\Gamma^{(2)}\\\
N_{\uparrow}^{(3)}&N_{\downarrow}^{(3)}&S^{(3)}&\Gamma^{(3)}\\\
\vdots&\vdots&\vdots&\vdots\end{bmatrix},$ (6)

with $N_{\uparrow}$ for spin-up electrons, $N_{\downarrow}$ for spin-down
electrons, $S$ for spin magnitude, and $\Gamma$ for point-group irrep. These
fictitious, fragment-indexed quantum numbers are not conserved by the exact,
correlated many-electron wave function of the whole system, but they are (in
practice) conserved by the approximate $\ket{\text{LAS}}$ wave function. Much
like a single determinant, a single LAS state is generally not an eigenstate
of $\hat{S}^{2}$ unless at most one fragment has a nonzero $S^{(K)}$.

In addition to breaking spin symmetry, a single LAS state omits all
entanglement between fragments; fragments interact exclusively _via_ a mean-
field potential. This approximation is obviously quantitatively inaccurate,
and can also be qualitatively inaccurate in certain cases 19. However,
interaction and entanglement between fragments, as well as global
$\hat{S}^{2}$ symmetry, can be restored by constructing a linear combination
of multiple states of the LAS type, which we refer to as LASSI 22. The LASSI
wave functions are linear combinations of the LAS states
$\ket{\vec{n}}_{\mathcal{P}}$:

$\ket{\text{LASSI}_{i}}\equiv\sum_{\mathcal{P}}^{\text{rootspaces}}\sum_{\vec{n}}^{\text{ENVs}}\ket{\vec{n}}_{\mathcal{P}}C^{(i)}_{\mathcal{P}\vec{n}}$
(7)

where $C^{(i)}_{\mathcal{P}\vec{n}}$ is an eigenvector coefficient of the
Hamiltonian projected into the model space of LAS states:

$\braket{\text{LASSI}_{i}}{\hat{H}}{\vec{n}}_{\mathcal{P}}=E_{\text{LASSI}}^{(i)}C^{(i)}_{\mathcal{P}\vec{n}},$
(8)

###  2.1 Analyzing LASSI results

In Sec. 4 below, we draw a distinction between significant model states and
“deadwood” model states by analyzing the LASSI eigenvector,
$C^{(i)}_{\mathcal{P}\vec{n}}$, which can be decomposed as

$C^{(i)}_{\mathcal{P}\vec{n}}=\sqrt{w_{\mathcal{P}}^{(i)}}u^{(i),\mathcal{P}}_{\vec{n}},$
(9)

with the normalized factors

$\displaystyle\sum_{\mathcal{P}}^{\text{rootspaces}}w_{\mathcal{P}}^{(i)}$
$\displaystyle=$ $\displaystyle 1\ \forall\ i$ (10)
$\displaystyle\sum_{\vec{n}}^{\text{ENVs}}(u^{(i),\mathcal{P}}_{\vec{n}})*u^{(i),\mathcal{P}}_{\vec{n}}$
$\displaystyle=$ $\displaystyle 1\ \forall\ i,\mathcal{P}.$ (11)

Weights ($w_{\mathcal{P}}^{(i)}$) close to zero indicate that an entire
rootspace is deadwood in the $i$th LASSI eigenstate. For rootspaces with
nonzero weights, the ENV coefficients ($u^{(i),\mathcal{P}}_{\vec{n}}$) can be
used to compute the single-fragment density matrix in the excitation number
basis,

$\displaystyle d_{m,m^{\prime}}^{(i),K\mathcal{P}}$ $\displaystyle\equiv$
$\displaystyle\sum_{\vec{n}\in\\{n_{K}=m\\}}^{\text{ENVs}}\sum_{\vec{n}^{\prime}\in\\{n_{K}=m^{\prime}\\}}^{\text{ENVs}}\prod_{L\neq
K}^{\text{fragments}}\delta_{n_{L}n^{\prime}_{L}}$ (12)
$\displaystyle\times(u^{(i),\mathcal{P}}_{\vec{n}})^{*}u^{(i),\mathcal{P}}_{\vec{n}^{\prime}}.$

In the application considered below, we examine spin-state energy gaps as
obtained from energy differences of different LASSI eigenstates, and for this
purpose, the weights and density matrices are averaged over the lowest-energy
$i_{\textrm{max}}+1$ LASSI eigenstates,

$\displaystyle w_{\mathcal{P}}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{i_{\textrm{max}}+1}\sum_{i=0}^{i_{\textrm{max}}}w_{\mathcal{P}}^{(i)},$
(13) $\displaystyle d_{m,m^{\prime}}^{K\mathcal{P}}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{i_{\textrm{max}}+1}\sum_{i=0}^{i_{\textrm{max}}}d_{m,m^{\prime}}^{(i),K\mathcal{P}},$
(14)

where in the spin-ladder applications below, $i_{\textrm{max}}+1$ is the
number of states in the spin ladder.

From the (state-averaged) single-fragment density matrix, we extract the
average excitation number of the $K$th fragment in the $\mathcal{P}$th
rootspace,

$\bar{n}_{K\mathcal{P}}=\sum_{m=0}m\times d^{K\mathcal{P}}_{m,m},$ (15)

as well as the von Neumann entropy,

$\displaystyle
s_{K\mathcal{P}}=-\sum_{m}\left(\rho_{m}^{K\mathcal{P}}\log\rho_{m}^{K\mathcal{P}}\right),$
(16)

where $\\{\rho_{m}^{K\mathcal{P}}\\}$ are the eigenvalues of the matrix
$d^{K\mathcal{P}}$. If $\bar{n}_{K\mathcal{P}}$ and $s_{K\mathcal{P}}$ are
close to zero, then excited states of $K$th fragment in the $\mathcal{P}$th
rootspace are deadwood.

###  2.2 LASSI[$r,q$]

If the LASSI model space consists of all possible rootspaces described by all
possible space tables [Eq. (6)] along with all possible excitation number
vectors, then LASSI reproduces CASCI, with the same exponential-scaling
computational cost. However, the conjecture underlying LAS methods in general
(that electrons interact more strongly within fragments than between
fragments) suggests that a small approximate model space could reproduce CASCI
qualitatively with orders of magnitude fewer model states than the number of
determinants or CSFs in the CAS. In order for this to be practical, such an
approximate model space must be generable semi-automatically. As mentioned
above, when LASSI was introduced in Ref. 22, model states were constructed by
the user entering all relevant space tables by hand; additionally, only the
ground-state ENV ($\vec{0}=\\{0_{1},0_{2},0_{3},\ldots\\}$) was considered in
any rootspace. Here we explore model space construction more systematically.

First, we stipulate that all model spaces should be chosen to restore spin
symmetry: LASSI wave functions should be $\hat{S}^{2}$ eigenvalues. This is
accomplished by permuting

$M^{(K)}\equiv\frac{1}{2}\left(N^{(K)}_{\uparrow}-N^{(K)}_{\downarrow}\right)$
(17)

for each fragment in all possible ways consistent with the fixed whole-
molecule total number of spin up and spin down electrons, as well as the
constraint $M^{(K)}\leq|S^{(K)}|$, and including all such permutations in any
model space. Omitting point-group symmetry for the current work, the space
table [Eq. (6)] can now be simplified to

$\mathcal{P}=\begin{bmatrix}N^{(1)}&S^{(1)}\\\ N^{(2)}&S^{(2)}\\\
N^{(3)}&S^{(3)}\\\ \vdots&\vdots\end{bmatrix},$ (18)

with the understanding that implicitly, all possible orientations of nonzero
fragment spins are included in the model space.

Second, we construct simplified space tables [Eq. (18)] by systematically
moving one electron from any one fragment to any other fragment up to $r$
times in all possible ways, where $r$ is a non-negative integer chosen by the
user. For each increment or decrement of $N^{(K)}$, the corresponding local
spin $S^{(K)}$ can (in general) either increase or decrease by $\frac{1}{2}$.
A schematic for this is presented in Table 1.

Rootspace depiction |
---|---
$r=0$
(a) |  | $\underline{{\color[rgb]{1,0,0}\uparrow\downarrow}}\;\underline{{\color[rgb]{1,0,0}\uparrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{1,0,0}\uparrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{1,0,0}\uparrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{1,0,0}\uparrow}\phantom{\uparrow}}\;$ |  | $\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;$ | $\mathcal{P}=\begin{bmatrix}N^{(1)}&S^{(1)}\\\ N^{(2)}&S^{(2)}\end{bmatrix}$
(b) |  | $\underline{{\color[rgb]{1,0,0}\uparrow\downarrow}}\;\underline{{\color[rgb]{1,0,0}\uparrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{1,0,0}\uparrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{1,0,0}\uparrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{1,0,0}\downarrow}\phantom{\uparrow}}\;$ |  | $\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\uparrow}\phantom{\uparrow}}\;$
(c) |  | $\underline{{\color[rgb]{1,0,0}\uparrow\downarrow}}\;\underline{{\color[rgb]{1,0,0}\uparrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{1,0,0}\uparrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{1,0,0}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{1,0,0}\downarrow}\phantom{\uparrow}}\;$ |  | $\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\uparrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\uparrow}\phantom{\uparrow}}\;$
(d) |  | $\underline{{\color[rgb]{1,0,0}\uparrow\downarrow}}\;\underline{{\color[rgb]{1,0,0}\uparrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{1,0,0}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{1,0,0}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{1,0,0}\downarrow}\phantom{\uparrow}}\;$ |  | $\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\uparrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\uparrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\uparrow}\phantom{\uparrow}}\;$
(e) |  | $\underline{{\color[rgb]{1,0,0}\uparrow\downarrow}}\;\underline{{\color[rgb]{1,0,0}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{1,0,0}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{1,0,0}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{1,0,0}\downarrow}\phantom{\uparrow}}\;$ |  | $\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\uparrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\uparrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\uparrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\uparrow}\phantom{\uparrow}}\;$
$r=1$
(f) |  | $\underline{{\color[rgb]{1,0,0}\uparrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{1,0,0}\uparrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{1,0,0}\uparrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{1,0,0}\uparrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{1,0,0}\downarrow}\phantom{\uparrow}}\;$ |  | $\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\uparrow}{\color[rgb]{1,0,0}\downarrow}}\;$ | $\begin{bmatrix}N^{(1)}-1&S^{(1)}\pm 0.5\\\ N^{(2)}+1&S^{(2)}-0.5\end{bmatrix}$
(g) |  | $\underline{{\color[rgb]{1,0,0}\uparrow\downarrow}}\;\underline{{\color[rgb]{1,0,0}\uparrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{1,0,0}\uparrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{1,0,0}\uparrow}\phantom{\uparrow}}\;\underline{\phantom{\uparrow}\phantom{\uparrow}}\;$ |  | $\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\uparrow}{\color[rgb]{1,0,0}\downarrow}}\;$
(h) |  | $\underline{{\color[rgb]{1,0,0}\uparrow\downarrow}}\;\underline{{\color[rgb]{1,0,0}\uparrow}{\color[rgb]{0,0,1}\downarrow}}\;\underline{{\color[rgb]{1,0,0}\uparrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{1,0,0}\uparrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{1,0,0}\uparrow}\phantom{\uparrow}}\;$ |  | $\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{\phantom{\uparrow}\phantom{\uparrow}}\;$ | $\begin{bmatrix}N^{(1)}+1&S^{(1)}-0.5\\\ N^{(2)}-1&S^{(2)}-0.5\end{bmatrix}$
$r=2$
(i) |  | $\underline{{\color[rgb]{1,0,0}\uparrow}{\color[rgb]{0,0,1}\downarrow}}\;\underline{{\color[rgb]{1,0,0}\uparrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{1,0,0}\uparrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{1,0,0}\uparrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{1,0,0}\downarrow}\phantom{\uparrow}}\;$ |  | $\underline{\phantom{\uparrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\uparrow}{\color[rgb]{1,0,0}\downarrow}}\;$ | $\begin{bmatrix}N^{(1)}&S^{(1)}\\\ N^{(2)}&S^{(2)}-1\end{bmatrix}$
(j) |  | $\underline{{\color[rgb]{1,0,0}\uparrow\downarrow}}\;\underline{{\color[rgb]{1,0,0}\uparrow}{\color[rgb]{0,0,1}\downarrow}}\;\underline{{\color[rgb]{1,0,0}\uparrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{1,0,0}\uparrow}\phantom{\uparrow}}\;\underline{\phantom{\uparrow}\phantom{\uparrow}}\;$ |  | $\underline{\phantom{\uparrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\uparrow}{\color[rgb]{1,0,0}\downarrow}}\;$ | $\begin{bmatrix}N^{(1)}&S^{(1)}-1\\\ N^{(2)}&S^{(2)}-1\end{bmatrix}$
(k) |  | $\underline{{\color[rgb]{1,0,0}\uparrow\downarrow}}\;\underline{{\color[rgb]{1,0,0}\uparrow}{\color[rgb]{0,0,1}\downarrow}}\;\underline{{\color[rgb]{1,0,0}\uparrow}{\color[rgb]{0,0,1}\downarrow}}\;\underline{{\color[rgb]{1,0,0}\uparrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{1,0,0}\uparrow}\phantom{\uparrow}}\;$ |  | $\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{{\color[rgb]{0,0,1}\downarrow}\phantom{\uparrow}}\;\underline{\phantom{\uparrow}\phantom{\uparrow}}\;\underline{\phantom{\uparrow}\phantom{\uparrow}}\;$ | $\begin{bmatrix}N^{(1)}+2&S^{(1)}-1\\\ N^{(2)}-2&S^{(2)}-1\end{bmatrix}$
Table 1: Examples of the rootspaces included in a LASSI calculation for a
LASSCF reference with the red Fe2+ (6e,5o) atom and blue Fe3+ (5e,5o) atoms,
depicted in terms of representative single determinants, the corresponding
simplified space table [Eq. 18], and the $r$ value at which they appear for
AlFe2 systems. (a)–(e) are the various cases of the local spin polarizations,
$2M^{(K)}\equiv N^{(K)}_{\uparrow}-N^{(K)}_{\downarrow}$. (f)–(h) are examples
of $r=1$ rootspaces generated from (b) and are not exhaustive of all possible
spin polarizations. Rootspace (f) adds 0.5 to $S^{(1)}$, and (g) decreases
$S^{(1)}$ by 0.5 since $S$ corresponds to the number of singly occupied
molecular orbitals divided by 2. (i)–(k) are examples of $r=2$ rootspaces.
Specifically, (i), (j) and (k) rootspaces are generated by moving electron
from the blue atom to the red atom from (f), (g) and (h) rootspaces
respectively. $r=2$ rootspaces are not exhaustive.

Third, we stipulate a maximum number of fragment roots $q$, obtain up to the
$q$th eigenstate of each fragment Hamiltonian in each rootspace
$\hat{H}_{\mathcal{P}K}$ (or all eigenstates if there are fewer than $q$), and
build all possible ENVs from this product space of these sets.

We use the formulation “LASSI[$r,q$]” to refer to LASSI calculations whose
model spaces are constructed in this way. Note that the CASCI limit is
achieved as both parameters approach infinity:

$\lim_{r\to\infty}\lim_{q\to\infty}\text{LASSI}[r,q]\equiv\text{CASCI}.$ (19)

###  2.3 Efficient Selection with LASSI[$r,q_{\text{CT}}$]

We will show evidence in Sec. 4 from the results of LASSI[$r$,$q$]
calculations that it is profitable to use different number of eigenstates for
different fragments in different rootspaces, because whole categories of model
states can be shown to be “deadwood” by examining the LASSI eigenvectors. In
particular, we will show that to obtain qualitative agreement with CASCI, it
is often not necessary to include “local excitations” (i.e., $n_{K}>0$),
_except_ where a particular fragment has a different number of electrons than
in the reference LAS wave function.

For instance, for a molecule with three fragments, if the reference ($r=0$)
space table is written as Eq. (18), so that a single $r=1$ space
($\mathcal{Q}$) table might be

$\mathcal{Q}=\begin{bmatrix}N^{(1)}+1&S^{(1)}+\frac{1}{2}\\\
N^{(2)}-1&S^{(2)}+\frac{1}{2}\\\ N^{(3)}&S^{(3)}\end{bmatrix},$ (20)

then only in the first and second fragments of this rootspace do local excited
states, $n_{K}>0$, contribute to low-energy LASSI eigenstates.

We will therefore define “LASSI[$r,q_{\text{CT}}$]” as an efficient
approximate alternative to LASSI[$r,q$], where “CT” stands for “charge
transfer” and indicates that the model space includes up to the $q$th local
eigenstate _only_ if the number of electrons in the given fragment in the
given rootspace differs from that in the reference rootspace; otherwise, only
the ground eigenstate of $\hat{H}_{\mathcal{P}K}$ is included. Such
calculations are indicated in the text by the Roman subscript CT on the
integer $q$ value; for instance “LASSI[$1,5_{\text{CT}}$].”

##  3 Computational Methods

In this work, we studied two compounds reported in Ref. 31:
[Fe(III)Al(III)Fe(II)($\mu_{3}$-O)]6+ and [Fe(III)2Fe(II)($\mu_{3}$-O)]6+.
They are interesting chemical species because they are the building blocks in
several metal-organic frameworks. The first system is a simplified model due
to replacement of one Fe3+ atom with a closed shell Al3+, reducing the number
of active orbitals that need to be considered. The geometries of the systems
were taken from Ref. 31 without further optimization. The structure and active
spaces of the systems under study are depicted in Fig. 1. To generate an
initial guess for all MR calculations, we perform a high-spin ($S=9/2$ for
AlFe2 or $S=7$ for Fe3) restricted open shell Hartree-Fock (ROHF) calculation,
followed by atomic valence active space (AVAS) localization32 for 3$d$ and
4$d$ orbitals of all Fe atoms. Note that AVAS generates larger-than-intended
active spaces, and required active spaces are hand-picked to generate the
initial guess for all calculations. At the LASSCF level, the orbitals are
optimized for the high-spin state ($S=9/2$ for AlFe2 system and $S=7$ for Fe3
system), and these optimized LASSCF orbitals are used in all subsequent LASSI
and CASCI calculations. Below, we refer to the CASCI results as LAS-CASCI.

We used the cc-pVDZ 33 basis set for all C and H atoms, and cc-pVTZ 33 for all
Al, Fe and O atoms. Since we use both 3$d$ and 4$d$ orbitals for Fe atoms, we
need at-least triple $\zeta$ basis set for all Fe atoms. Additionally, we used
cc-pVTZ for Al and O atoms since previous studies show contribution from O
atoms even when orbitals on O are not taken into active space 31. The H and C
atoms are modeled with relatively smaller basis set (cc-pVDZ) since they do
not have contributions to the active space and it helps with reducing cost.
LASSCF and LASSI are implemented in mrh 34. CASSCF and CASCI calculations were
performed using PySCF 35.

Figure 1: AlFe2 and Fe3 system under study. Yellow atoms are Fe, red atoms are
O, pink atoms are C, blue atoms are H. The green atom is Al for AlFe2 system
and Fe for Fe3 system. All active orbitals are $3d$ and $4d$ iron orbitals.
All Fe2+ fragments were assigned a spin magnitude of $S=2$ (quintet) and all
Fe3+ fragments were assigned a spin magnitude of $S=\frac{5}{2}$ (sextet) in
the reference LASSCF wave function. When optimizing orbitals using ROHF or
LASSCF, the AlFe2 system was fixed at
$M\equiv\frac{1}{2}(N_{\uparrow}-N_{\downarrow})=\frac{9}{2}$ and the Fe3
system was fixed at $M=7$. When carrying out LASSI calculations, AlFe2 was
initialized at $M=\frac{1}{2}$ and Fe3 was initialized at $M=0$.

##  4 Results and discussion

###  4.1 Aluminium Di Iron system

This system consists of one $d^{6}$ Fe2+center, one $d^{5}$ Fe3+ center and a
$d^{0}$ Al3+ center. Since Al3+ is closed shell, we do not consider any active
space on it. We explore two different active spaces for this system as shown
in Fig. 1, namely (11e,10o) with all 3$d$ orbitals of both Fe atoms and
(11e,20o) with all 3$d$ and 4$d$ orbitals of both Fe atoms. In these different
parameters, we investigate the dependence of the $J$ coupling according to the
Yamaguchi formula36

$J=-\frac{E_{HS}-E_{LS}}{\braket{S^{2}}_{HS}-\braket{S^{2}}_{LS}}$ (21)

for the sake of brevity. The full spin ladder is reported in the SI table II
and III.

In Fig. 2, we compare the performance of LASSI with its theoretical limit LAS-
CASCI, under two variations. We first vary $r$ and $q$ for the (11e,10o)
active space. LASSI[0,1] includes no charge transfer rootspaces and only
ground state eigenfunctions of the projected Hamiltonian in each fragment
(equivalent to (a)–(e) rootspaces in table 1). This generates only enough
model states to ensure that the LASSI eigenstates are eigenfunctions of the
$\hat{S}^{2}$ operator and wrongly predicts ferromagnetic interaction ($J>0$)
as compared to LAS-CASCI due to the complete omission of kinetic exchange. As
$r$ and $q$ increase, the agreement between LASSI and LAS-CASCI improves. The
correct negative sign of $J$ is observed for $r>0$; $q>4$. At LASSI[1,5] (Fig.
2: green curve with upside down triangles), we observe agreement within CASCI
to within 1 cm-1, from a wave function built from 250 model states. At
LASSI[2,10] (1780 states) (purple curve with triangles), the difference is
less than 0.1 cm-1, at the price of requiring about 7 times as many model
states compared to LASSI[1,5]. No performance improvement was observed for
LASSI[3,$q$] series. LASSI[2,10] wave function still has another 30 times
fewer model states than the 52920 single determinants in the CASCI limit. All
corresponding absolute energies are in SI table II and III. For comparison,
CASSCF predicts a $J$ coupling of -7.3 cm-1, as opposed to the LAS-CASCI limit
of -3.8 cm-1, which suggests that any inaccuracy in the LASSI[1,5] result is
more likely attributable to the LASSCF orbitals than to the LASSI
approximation.

Figure 2: LASSI[1,$q$] and CASCI predicted $J$ coupling for an AlFe2 system
with active spaces of (11e,10), and (11e,20o). For (11e,10o) active space,
$J_{\text{LASSI}[0,q]}=+116.69cm^{-1}\;\forall\;q=\\{1,10\\}$ and
$J_{\text{LASSI}[2,q]}\approx
J_{\text{LASSI}[3,q]}\;\forall\;q=\\{1,10\\}$.The LAS-CASCI result
for(11e,10o) is -3.82 cm-1 and for (11e,20o) is -6.245 cm-1.

The second variation explored is the active space dependence of $J$. For all
three active space choices, agreement of $J$ to within 3 cm-1 was achieved
with vastly fewer model states than the number of single determinants in the
corresponding CASCI: 7425 model states for the LASSI[1,15](11e,20o)
calculation, as compared to $\approx 6\times 10^{8}$ determinants in the
CASCI(11e,20o) wave function. Clearly, the convergence with respect to $q$ is
less rapid in the larger active space. However, we note that the von Neumann
entropies [Eq. (16)] for the LASSI[1,15](11e,20o)
($\text{max}(s_{K\mathcal{P}})=0.95$) results are similar to the
LASSI[1,5](11e,10o) ($\text{max}(s_{K\mathcal{P}})=0.95$) results. (See
supporting information for a complete list.) This means that the entanglement
between the two fragments does not increase in the larger active space,
suggesting that the higher required $q$ in the larger active space is due
rather to the less-accurate modeling of each fragment individually by the
LASSI model states constructed from the eigenstates of
$\hat{H}_{\mathcal{P}K}$.

###  4.2 Tri-Iron system

This system consists of one $d^{6}$ Fe2+ center and two $d^{5}$ Fe3+ centers.
We explore the (16e,15o) active space of 3$d$ orbitals as well as the
(16e,30o) active space including 3$d$ and 4$d$ orbitals. This system is more
challenging for MR methods due to the size of active space, and as a reminder,
DFT predicts that all Fe centers are identical31. We calculate $S=0,1,...,7$
spin states for each active space for all methods.

Spin Ladder (in kJ/mol)
---
Spin | CASPT2† | CASSCF† | LAS-CASCI | LASSI[1,5] | LASSI[1,5${}_{\text{CT}}$]
$S=7$ | 22.8 | 6.6 | 5.0 | 4.9 | 4.9
$S=6$ | 21.0 | 4.8 | 3.5 | 3.4 | 3.4
$S=5$ | 13.1 | 3.1 | 2.2 | 2.2 | 2.2
$S=4$ | 6.2 | 1.6 | 1.2 | 1.2 | 1.2
$S=3$ | - | 0.5 | 0.5 | 0.4 | 0.4
$S=2$ | - | 0.9 | -0.1 | -0.1 | -0.1
$S=1$ | - | 0.0 | -0.1 | -0.1 | -0.1
$S=0$ | 0.0 | 0.0 | 0.0 | 0.0 | 0.0
States | n/a | $\approx 4.1\times 10^{7}$ | 6910 | 3014
Table 2: Spin ladder of Fe3 system with (16e,15o) active space. $S=0$ state is
set as a reference. Results of LAS-CASCI and LASSI[1,5] compared with previous
CASSCF and CASPT2 results from Ref. 31 with the same (16e,15o) active space.
Empty CASPT2 rows are indicative of results not available in Ref. 31. Absolute
and relative energies LASSI[1,5] and LASSI[1,5${}_{\text{CT}}$] differ by less
than 0.01kJ/mol. LASSI[1,5${}_{\text{CT}}$] has 3014 states. †: Results from
Ref. 31 use relativistic all-electron ANO-RCC triple-$\zeta$ basis sets for
all O and Fe atoms, and double-$\zeta$ for C and H atoms.

Table 2 shows the LASSI results for the (16e,15o) active space against the
LAS-CASCI limit. The absolute values are reported in SI table XI. LASSI[0,1]
incorrectly predicts a high-spin ground state, while the LAS-CASCI predicts a
triply-degenerate low-spin ground state (i.e., singlet, triplet, and quintet).
The LASSI[1,5] (6910 model states) results differ from the LAS-CASCI
($4.1\times 10^{7}$ determinants) results only by 0.1 kJ/mol while using four
orders of magnitude fewer states. LASSI[1,5] and LASSI[1,5${}_{\text{CT}}$]
(3014 model states) differ by less than 0.01kJ/mol. Additionally, CASPT237
predicts an energy difference between the $S=0$ and $S=7$ states of 23 kJ/mol,
indicating the importance of dynamical correlation.

Charge | $\Delta S$ | $\bar{n}_{\mathcal{P}K}$
---|---|---
| $\begin{bmatrix}{\color[rgb]{1,0,0}S^{(1)}}\\\ {\color[rgb]{0,0,1}S^{(2)}}\\\ {\color[rgb]{0.0,0.42,0.24}S^{(3)}}\end{bmatrix}=\begin{bmatrix}2\\\ 2.5\\\ 2.5\end{bmatrix}$ | $\begin{bmatrix}{\color[rgb]{1,0,0}0.00}\\\ {\color[rgb]{0,0,1}0.00}\\\ {\color[rgb]{0.0,0.42,0.24}0.00}\end{bmatrix}$
| $\begin{bmatrix}{\color[rgb]{1,0,0}-0.5}\\\ {\color[rgb]{0,0,1}-0.5}\\\ {\color[rgb]{0.0,0.42,0.24}0}\end{bmatrix}^{*}$ | $\begin{bmatrix}{\color[rgb]{1,0,0}1.04}\\\ {\color[rgb]{0,0,1}0.91}\\\ {\color[rgb]{0.0,0.42,0.24}0.00}\end{bmatrix}$
| $\begin{bmatrix}{\color[rgb]{1,0,0}-0.5}\\\ {\color[rgb]{0,0,1}-0.5}\\\ {\color[rgb]{0.0,0.42,0.24}0}\end{bmatrix}$ | $\begin{bmatrix}{\color[rgb]{1,0,0}1.18}\\\ {\color[rgb]{0,0,1}2.02}\\\ {\color[rgb]{0.0,0.42,0.24}0.00}\end{bmatrix}$
| $\begin{bmatrix}{\color[rgb]{1,0,0}0}\\\ {\color[rgb]{0,0,1}-0.5}\\\ {\color[rgb]{0.0,0.42,0.24}-0.5}\end{bmatrix}$ | $\begin{bmatrix}{\color[rgb]{1,0,0}0.00}\\\ {\color[rgb]{0,0,1}1.19}\\\ {\color[rgb]{0.0,0.42,0.24}1.40}\end{bmatrix}$
Figure 3: Summary table of LASSI[1,5](16e,15o) results for the Fe3 system in
different rootspaces. The first column denotes a pictorial representation of
the rootspace, in which the charge on each iron is that after the electron is
transferred in the direction of the arrow. The change in spin on each iron in
the rootspace is shown in the 2nd column. The third column shows the average
excitation number [$\bar{n}_{K\mathcal{P}}$, Eq. (15)] on each fragment in the
given rootspace and it’s all possible spin polarizations.

However, the LASSI[1,$q$] approach appears to create significant deadwood in
these low-energy wave functions. We find that 48 of the 182 rootspaces
generated at the $r=1$ level have zero average weight [$w_{\mathcal{P}}$, Eq.
(13)]; these correspond to rootspaces where the charge transfer Fe2+
$\rightarrow$ Fe3+ results in increased $S$ on Fe2+ center. Additionally, we
find that the average excitation number [$\bar{n}_{K\mathcal{P}}$, Eq. (15)]
is zero if the corresponding fragment does not participate in a charge
transfer compared to the reference state (Figure 3). In other words, for any
$r=1$ rootspace (cf. Table 1), a “spectator” fragment does not need local
excitations. This observation motivates the development of
LASSI[$r$,$q_{\text{CT}}$], as described in Sec. 2.3. The result for
LASSI[1,5${}_{\text{CT}}$] is quantitatively the same as LASSI[1,5], but with
fewer than half the model states of LASSI[1,5].

Figure 4: Spin ladder for Fe3 system with (16e,30o) active space.
[1,$q_{\text{CT}}$] are the corresponding LASSI calculations. The state $S=2$
is set as a reference. DMRGCI is placed on the abscissa at the number of
single determinants in the corresponding hypothetical LAS-CASCI calculation.
LASSI[1,8${}_{\text{CT}}$] recovers LAS-DMRGCI spin ladder qualitatively.

We now discuss results with (16e,30o) active space presented in Fig. 4. Since
one cannot perform CASSCF or CASCI in this active space, we have used DMRG 38
to calculate the reference value starting from optimized orbitals of
LASSCF(16e,30o) (SI table XII). DMRG was performed with a bond dimension
$M=500$. We show DMRGCI results in Fig. 4 on the abscissa at 34 trillion
states corresponding to the size of a (16e,30o) CAS calculations. We see that
the qualitative features of the DMRG(16e,30o) are generally reproduced at the
LASSI[1,8${}_{\text{CT}}$] level (22808 model states), except that LASSI
predicts a quintet ground state and DMRG predicts a degenerate singlet,
triplet, and quintet.

##  5 Conclusions and Outlook

We have introduced LASSI[$r$,$q$], a systematic hierarchy of post-LASSCF
methods in which the CASCI limit is recovered in the limit of large $r$ and
$q$. Test calculations on the spin-state energy ladder of a triiron compound
Fe3 and its aluminum-substituted variant AlFe2 show that LASSI[$r$,$q$]
qualitatively reproduces CASCI for $r=1$ and $q$ no greater than 15, and
usually as low as 7, leading to a compactification of the wave function by
several orders of magnitude.

Close examination of the LASSI[1,5](16e,15o) wave functions of Fe3 led to the
identification of another large category of deadwood model states, which are
systematically removed in the more approximate LASSI[$r$,$q_{\text{CT}}$]
hierarchy of methods. Using this latter model, we report for the first time
practical LASSI calculations on a trimetallic species with both 3$d$ and 4$d$
orbitals in the model space, a calculation which is impossible for
conventional CASCI but which gives qualitatively similar results to the
corresponding (16e,30o) DMRG-CI calculation. Additionally, LASSI allows us to
understand the wave function and draw meaningful conclusions, which is
difficult with DMRGCI.

The convergence with respect to $q$ or $q_{\text{CT}}$ appears to be sensitive
to the size of the active space, in both systems. This is especially
pronounced for the AlFe2 system. However, the fact that there is essentially
no difference in the values of the von Neumann entropies of the LASSI[1,$q$]
wave functions between the smaller and larger active spaces suggests that the
fragments are not more strongly entangled in the latter, in a physical sense,
despite the larger $q$ required to achieve agreement with CASCI. Generating
more compact LASSI wave functions is a primary target for future work.

##  6 Acknowledgment

We would like to thank Dr. Rishu Khurana, Dr. Cong Liu and Dr. Christopher
Knight for helpful discussions. This work is supported as part of the
Computational Chemical Sciences Program, under Award DE-SC0023382, funded by
the U.S. Department of Energy, Office of Basic Energy Sciences, Chemical
Sciences, Geosciences, and Biosciences Division.

##  7 Supplemental Information

The supplemental information contains geometries of compounds studied, all
converged active space orbitals for all active spaces for all compounds and
absolute energies of LASSI, LAS-CASCI and LAS-DMRGCI calculations performed in
this work.

##  8 Conflict of interest

The authors declare no competing conflict of interest.

##  9 Data availability

Input and output files, converged orbitals and geometries are available on
Ref. 39.

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