Table : Average 3D Panoptic Quality on open ScanNet validation set. Mean
Panoptic Quality (PQ), Segmentation Quality (SQ) and Recongnition Quality (RQ)
are reported over all classes, as well as over things and stuff separately.

!  all stuff things PQ SQ RQ PQ SQ RQ PQ SQ RQ framerate (Hz)2 Results
reported in PanopticFusion1 [panopticfusion] offline, with CRF 33.5 73.0 45.3
58.4 70.7 80.9 30.8 73.3 41.3 - online, without CRF 29.7 71.2 41.1 56.7 69.5
79.6 26.7 71.4 36.8 4.30 Results with our method Greedy, 2.4 cm voxels 29.6
66.0 41.4 51.1 69.5 71.4 27.2 65.6 38.0 0.32 Greedy, 5 cm voxels 20.0 54.1
28.9 46.3 64.6 68.9 17.1 53.0 24.5 2.66 Greedy, 10 cm voxels 28.9 57.0 42.4
47.3 64.7 70.3 26.9 56.1 39.3 9.83 Hungarian, 2.4 cm voxels 33.5 65.4 47.6
53.4 70.0 74.3 31.3 64.9 44.6 0.33 Hungarian, 5 cm voxels 34.0 68.0 47.8 52.4
69.0 74.3 31.9 67.9 44.5 3.53 Hungarian, 10 cm voxels 31.7 62.3 47.0 47.2 66.5
68.8 30.0 61.9 44.5 11.63 Open validation set of the PanopticFusion paper
contains different ScanNet scenes than the validation set of this work.
PanopticFusion and our method were evaluated on different hardware, thus
framerates should not be directly compared between them. They are nevertheless
reported to clarify the effect of reducing image and voxel-grid resolutions.

Table  compares our method to PanopticFusion on the Panoptic Quality (PQ)
metric [panoptic_segmentation], which is a combination of Segmentation Quality
(SQ) and Recognition Quality (RQ). The results have been reported as an
average over all classes, as well as over stuff and thing classes separately.
The results of the original PanopticFusion [panopticfusion] have been reported
both with and without Conditional Random Field (CRF) regularisation. Because
the test data and panoptic segmentation model are not the same as reported in
[panopticfusion], we have also re-implemented the Greedy data-association
algorithm of PanopticFusion: detection segments are processed in descending
order based on size, by choosing the target corresponding to highest IoU. IoU
threshold for the greedy algorithm is set to $0.25$ as in the original work.
The results on the test closest to the original PanopticFusion results –
Greedy algorithm with $2.4~{}cm$ voxels – seem to be quite close to the
results reported in the original paper.

[b]0.32 [width=]fig/288_seg.pdf [b]0.32 [width=]fig/616_seg.pdf [b]0.32
[width=]fig/655_seg.pdf [b]0.32 [width=]fig/288_gt.pdf [b]0.32
[width=]fig/616_gt.pdf [b]0.32 [width=]fig/655_gt.pdf

Figure : scene $0288\\_00$ Figure : scene $0616\\_00$ Figure : scene
$0655\\_00$ Figure : Qualitative examples of typical scenes in the ScanNet
dataset. Top: panoptic reconstructions with our method, bottom: ground truth.
Colors of ’thing’ instances can be different from ground truth because they
are randomised.

The visual quality of the reconstruction varies mostly based on the resolution
of the voxel grid: the smaller the voxels are, the more detailed the meshes
become. Voxel size seems to affect computational requirements the most as
well. While image resolution could be reduced up to $0.1$ times the original
after segmentation with voxels larger than $5~{}cm$ without affecting the
quality of the outcome much, with smaller voxels there is a significant loss
of quality. When image resolution is only reduced to half of the original, the
quality is much better, but computational requirements grow respectively. In
tests with $5~{}cm$ and $10~{}cm$ voxels point cloud resolution was reduced to
$0.1$ times the original, and in tests with $2.4~{}cm$ voxels the resolution
was reduced to $0.5$ times the original. While our method seems to be roughly
on par with the version of PanopticFusion applying CRF, the one without CRF
seems to perform significantly worse than ours on PQ. This is further
supported by the tests on our re-implementation of their method. While both
PanopticFusion configurations perform better on ’stuff’ classes and
Segmentation Quality (SQ) than ours, they both seem to fall behind on ’thing’
classes and Recognition Quality (RQ). This implies that while our method’s
segmentation seems to be slightly worse, our data association strategy seems
to be performing better. Differences in SQ most likely originate from
differences in the image segmentation approaches. Some examples of panoptic
reconstruction on typical scenes found in ScanNet are presented in Figure .
Object instances seem to have been found and separated from one another
correctly most of the time, and segmentations look pretty accurate, even
though larger objects are often only seen partially in the video sequences.
Volumetric integration slows down a lot when voxel size is decreased taking on
average $46$, $106$ or $1188$ milliseconds for $10$, $5$ and $2.4$ centimetre
voxels respectively. Data-association with the Hungarian algorithm took on
average $47$, $177$ or $1875$ milliseconds for the same sizes, while greedy
matching took on average $53$, $247$ or $1949$ milliseconds. Most of this time
is spent on computing the IoUs, which takes more time with larger amounts of
voxels. Compared to the relations of these timings reported in the
PanopticFusion paper, our association algorithm is somewhat slower than the
original. Our implementation of the greedy algorithm is a bit slower than the
Hungarian method, most likely because the segments need to be sorted every
iteration. We can reach reconstruction quality similar to the version of
PanopticFusion with CRF regularisation even when decimating point clouds,
reducing voxel grid resolution and processing less frames. While the CRF is
only run every $10$ seconds, it is reported to take around $4.5$ seconds and
to slow down along increasing reconstruction size. For a real-time application
to get the benefits of the regularisation, one would need to tolerate long
delays during operation, thus we see it more useful as a post-processing tool
for cleaning the reconstruction offline.


# Molecular Berry curvatures and the adiabatic response tensors

Raffaele Resta<EMAIL_ADDRESS>Istituto Officina dei Materiali IOM-CNR,
Strada Costiera 11, 34151 Trieste, Italy http://www-dft.ts.infn.it/~resta/.
Donostia International Physics Center, 20018 San Sebastián, Spain

(DRAFT: run through LaTeX on at 22:28)

###### Abstract

Adiabatic transport in a many-electron system is expressed in terms of the
appropriate Berry curvature, owing to the Niu-Thouless theory [J. Phys A 17,
2453 (1984)]; the main equation is very compact and very general. I address
here three paradigmatic adiabatic response tensors—-the atomic polar tensor,
the atomic axial tensor, and the rotational $g$ factor—and I show that, for
all of them, the known formulas do not need an independent proof. They are
just case studies of the general expression, for different choices the
curvature’s two arguments.

##  I Introduction

The quantity which today is commonly addressed as a Berry curvature was
introduced in a milestone 1984 paper by Michael Berry;Berry84  at about the
same time, Niu and Thouless (NT) established the theory of adiabatic charge
transport in a many-electron system in terms of the same quantity.Niu84  The
curvature is a quasi-static quantity, in the sense that its definition
requires solely the instantaneous ground eigenstate; yet it encodes the
lowest-order effect of the excited states on the adiabatic evolution.

While the notion of Berry curvature has permeated several areas of condensed
matter physics,rap_a20 ; Xiao10 ; Vanderbilt  its role in molecular physics is
not generally appreciated. I am going to show here that the known expressions
for some molecular adiabatic response tensors are indeed paradigmatic
manifestations of the Berry curvature. When addressed by means of the NT
theory of adiabatic evolution, such tensors become simple case studies. Actual
derivation of the known formulas requires no independent proof: it is enough
to plug the definition of each tensor into the general NT expression for the
adiabatic evolution, based on the appropriate Berry curvature. I am going to
show how this works in three cases: the atomic polar tensor, the atomic axial
tensor, and the gyromagnetic factor. All three of these tensors are a measure
of how much electronic charge is “dragged” by the nuclear adiabatic motion,
and this is precisely what the Berry curvature allows casting in a very
compact formalism. It is also worth remarking that, when the nuclear motion is
harmonic at frequency $\omega$, the NT time-evolution of the many-electron
system is exact to linear order in the adiabaticity parameter $\omega$.

In Sec. II I give the general definition of a Berry curvature and I detail its
most basic features. In Sec. III I consider a generic time-independent
operator and the evolution of its expectation value over a given state when
such state is adiabatically varied in time: the NT theory—rooted in a Berry
curvature—provides a universal formula for such evolution. In Sec. IV I work
out in detail the three case studies mentioned above. A Conclusions Section is
not contemplated; in the Appendix I deal with the case where the ground
eigenstate is a Slater determinant of doubly occupied one-electron orbitals.

##  II The Berry curvature: fundamentals

Let $\hat{H}$ be a time-independent Hamiltonian and let $|\Psi_{n}\rangle$ its
eigenstates with eigenvalues $E_{n}$; $\hat{H}$ depends on a couple of generic
real parameters $(\lambda_{1},\lambda_{2})$: the eigenstates and eigenvalues
are parameter-dependent as well. We assume the ground state to be
nondegenerate for all $(\lambda_{1},\lambda_{2})$. The Berry curvature is by
definitionrap_a20 ; Xiao10 ; Vanderbilt

$\displaystyle\Omega(\lambda_{1},\lambda_{2})$ $\displaystyle=$ $\displaystyle
i(\langle\partial_{\lambda_{1}}\Psi_{0}|\partial_{\lambda_{2}}\Psi_{0}\rangle-\langle\partial_{\lambda_{2}}\Psi_{0}|\partial_{\lambda_{1}}\Psi_{0}\rangle)$
(1) $\displaystyle=$ $\displaystyle-2\,\mbox{Im
}\langle\partial_{\lambda_{1}}\Psi_{0}|\partial_{\lambda_{2}}\Psi_{0}\rangle.$

The parameters may have various physical interpretations and different
dimensions; the curvature has the inverse dimensions of the product
$\lambda_{1}\lambda_{2}$. For a macroscopic homogeneous system
$\Omega(\lambda_{1},\lambda_{2})$ is an extensive quantity.

The curvature admits a sum-over-states formula, first displayed in the
original Berry’s paper;Berry84

$\Omega(\lambda_{1},\lambda_{2})=-2\mbox{Im}\sum_{n\neq
0}\frac{\langle\Psi_{0}|\partial_{\lambda_{1}}\hat{H}|\Psi_{n}\rangle\langle\Psi_{n}|\partial_{\lambda_{2}}\hat{H}|\Psi_{0}\rangle}{(E_{0}-E_{n})^{2}}.$
(2)

The formula perspicuously shows that the Berry curvature encodes—to lowest
order—the effect of the excited states on the ground state when the quantum
system is transported in the parameter space. Eq. (2) has also the virtue of
showing that the curvature becomes ill defined whenever the ground state
becomes degenerate with the first excited state.

In the special case where $|\Psi_{0}\rangle$ is the Slater determinant of
doubly occupied single-particle orbitals—either Hartree-Fock or Kohn-
Sham—$\Omega(\lambda_{1},\lambda_{2})$ becomes a sum of single-particle
curvatures: the expression is presented in Appendix A.

The curvature is a geometrical quantity defined by the evolution of the
ground-state projector

${\hat{\rho}}=|\Psi_{0}\rangle\langle\Psi_{0}|$ (3)

in the parameter space, and can equivalently be expressed directly in terms of
${\hat{\rho}}$:

$\Omega(\lambda_{1},\lambda_{2})=-i\,\mbox{Tr
}\\{{\hat{\rho}}\,[\,\partial_{\lambda_{1}}{\hat{\rho}},\partial_{\lambda_{2}}{\hat{\rho}}\,]\\},$
(4)

where “Tr” indicates the trace over the Hilbert space. The form of Eq. (4)
explicitly displays the gauge-invariance of $\Omega$. A gauge-transformation
modifies $|\Psi_{0}\rangle$ by a phase factor, ergo leaves ${\hat{\rho}}$ and
the curvature unchanged.

An important caveat is in order: the curvature is not gauge-invariant when the
parameter $\lambda_{1}$ is identified with a macroscopic magnetic field ${\bf
B}$, which is not a “normal” parameter on which the Hamiltonian explicitly
depends. When in Eq. (4) $\lambda_{1}$ is identified with ${\bf B}$, the
relevant entry is $\partial_{{\bf B}}{\hat{\rho}}$, which actually is the
functional derivative of ${\hat{\rho}}$ with respect to the vector potential
${\bf A}({\bf r})$, times the derivative of ${\bf A}({\bf r})$ with respect to
${\bf B}$: the latter factor is manifestly gauge-dependent.

Practical calculations are performed by projecting the Hamiltonian on a finite
basis set, and are notoriously plagued by gauge-dependence problems. I am not
discussing the issue in this work, ideally assuming a complete Hilbert space.

##  III Adiabatic transport

Let us suppose here that $\hat{H_{t}}$ is instead a time-dependent
Hamiltonian, and $|\Psi\rangle$ one of its solutions:

$\hat{H}_{t}|\Psi\rangle=i\hbar|\dot{\Psi}\rangle;$ (5)

the time-dependent energy is

$E(t)=\langle\Psi|\,\hat{H}_{t}\,|\Psi\rangle.$ (6)

Let us also assume that a generic observable $\hat{O}$ can be written as the
derivative of the Hamiltonian with respect to some parameter $\lambda_{1}$:

$\hat{O}=\partial_{\lambda_{1}}\hat{H}_{t},\qquad\mbox{time independent}.$ (7)

The $\lambda_{1}$-derivative of Eq. (6) yields

$\partial_{\lambda_{1}}E(t)=\langle\Psi|\,\hat{O}\,|\Psi\rangle+i\hbar\,(\,\langle\partial_{\lambda_{1}}\Psi|\dot{\Psi}\rangle-\langle\dot{\Psi}|\partial_{\lambda_{1}}\Psi\rangle\,).$
(8)

Next suppose that the time-dependence of $\hat{H_{t}}$ owes to the time
dependence of a parameter $\lambda_{2}$ entering it: a nuclear coordinate in
the cases of interest here. Upon replacing the exact time-dependent energy and
state vector in Eq. (8) with the instantaneous ground-state eigenvalue and
eigenvector one gets the adiabatic evolution of the ground-state observable in
terms of the Berry curvature as

$\langle
O(t)\rangle=\langle\Psi_{0}|\,\hat{O}\,|\Psi_{0}\rangle+\hbar\,\Omega(\lambda_{1},\lambda_{2})\,\dot{\lambda}_{2}(t),$
(9)

where $|\Psi_{0}\rangle$ and the curvature depend implicitly on time. To the
best of the author’s knowledge, a similar formulation first appeared in the
milestone NT paper.Niu84 ; Xiao10  In the following, for the sake of
simplicity, I am going to call Eq. (9) the “NT formula”.

When the Hamiltonian is time-reversal (T) invariant the ground state can be
taken as real in the Schrödinger representation; its derivative with respect
to a nuclear coordinate $\lambda_{2}$ is real as well. The observables of
interest in the following are purely imaginary. Therefore the first term in
Eq. (9) vanishes and the curvature, Eq. (1), admits a simplified expression;
the NT formula becomes in this special case

$\langle
O(t)\rangle=2i\hbar\,\langle\partial_{\lambda_{1}}\Psi_{0}|\partial_{\lambda_{2}}\Psi_{0}\rangle\,\dot{\lambda}_{2}(t),\qquad\mbox{T-invariant}.$
(10)

In view of the rest of this paper, it is important to notice that Eq. (10)
becomes exact in the adiabatic limit, i.e.. when $\dot{\lambda}_{2}$ is
infinitesimal. Ergo the expression

$\frac{\partial\langle
O(t)\rangle}{\partial\dot{\lambda}_{2}}=2i\hbar\,\langle\partial_{\lambda_{1}}\Psi_{0}|\partial_{\lambda_{2}}\Psi_{0}\rangle=\hbar\,\Omega(\lambda_{1},\lambda_{2})$
(11)

is also exact.

##  IV The adiabatic response tensors

###  IV.1 The electronic Hamiltonian

Within the Born-Oppenheimer (also called Born-Huang) approximation one
considers a bounded system of $N$ interacting electrons. The many-body
Hamiltonian is

$\hat{H}=\frac{1}{2m}\sum_{i=1}^{N}\left({\bf p}_{i}+\frac{e}{c}{\bf A}({\bf
r}_{i})\right)^{2}+\hat{V},$ (12)

where the potential $\hat{V}$ comprises one-body and two-body terms; it is a
multiplicative operator in the Schrödinger representation, and is a function
of the instantaneous nuclear positions $\\{{\bf R}_{s}\\}$. We assume a
singlet ground state and we neglect irrelevant spin variables. The Hamiltonian
is T-invariant at ${\bf A}=0$.

It is also expedient to define the two operators:

$\hat{\bf v}=\frac{1}{m}\sum_{i=1}^{N}{\bf p}_{i},\qquad\hat{{\bf
L}}=\frac{1}{m}\sum_{i=1}^{N}{\bf r}_{i}\times{\bf p}_{i}:$ (13)

velocity and angular momentum of the many-body system, respectively (at ${\bf
A}=0$). In the following of this work Greek subscripts are Cartesian indices,
and the sum over repeated indices is implicitly understood.

###  IV.2 Atomic polar tensor

When a given ion is transported with velocity ${\bf v}_{s}=\dot{\bf R}_{s}$,
the induced electrical current is

${\bf J}_{s}=eZ_{s}{\bf v}_{s}-e\langle{\bf v}(t)\rangle;$ (14)

the dimensionless atomic polar tensor (Born charge in solids and liquids) is
then defined as

$Z^{*}_{s,\alpha\beta}=\frac{1}{e}\partial_{v_{s\beta}}J_{s\alpha}=Z_{s}\delta_{\alpha\beta}-\partial_{v_{s\beta}}\langle{v}_{\alpha}(t)\rangle.$
(15)

Once this definition adopted, its microscopic expression immediately follows.
In the NT formula it is enough to identify $\lambda_{1}$ with $A_{\alpha}$ and
$\lambda_{2}$ with $R_{s\beta}$; the velocity is

$\hat{{\bf v}}=\frac{c}{e}\partial_{{\bf A}}\hat{H}.$ (16)

The atomic polar tensor is then

$Z^{*}_{s,\alpha\beta}=Z_{s}\delta_{\alpha\beta}+\frac{\hbar
c}{e}\Omega(A_{\alpha},R_{s\beta})..$ (17)

This equation is well known in molecular physics since the 1980s.Stephens85 ;
Stephens87 ; Buckingham87 ; Barron

What I deem new here is the very concise derivation, as well as the
realization that Eq. (17) is arguably the most straightforward case study of
the very general NT formula, Eq. (11). Eq. (17) is gauge-invariant and
invariant by translation of the coordinate origin.

###  IV.3 Atomic axial tensor

Let us suppose again that the nuclei are transported with velocities ${\bf
v}_{s}$. The magnetic moment of the molecule is, in Gaussian units,

${\bf m}=\frac{e}{2c}\left(\sum_{s}Z_{s}{\bf R}_{s}\times{\bf
v}_{s}-\langle{{\bf L}}(t)\rangle\right).$ (18)

A common coordinate origin must be set for the two terms in parenthesis,
and—for an arbitrary velocity distribution—the value of ${\bf m}$ depends on
the coordinate origin.

The atomic axial tensor of nucleus $s$ is defined as

$M_{s,\alpha\beta}=\frac{\partial m_{\beta}}{\partial
v_{s\alpha}}=\frac{e}{2c}[\,\varepsilon_{\alpha\beta\gamma}Z_{s}R_{s\gamma}-\partial_{v_{s\alpha}}\langle{L}_{\beta}(t)\rangle\,].$
(19)

Next, in order to exploit the NT formula, Eq. (11), all what is needed is
recognizing the operator $\hat{L}_{\beta}$ as the Hamiltonian derivative with
respect to some parameter. By adopting the central gauge centered on the
momentum origin the parameter is clearly the ${\bf B}$ field:

$\partial_{B_{\beta}}\hat{H}=\frac{e}{2c}\hat{L}_{\beta}.$ (20)

Upon identifying $\lambda_{1}$ with $B_{\beta}$ and $\lambda_{2}$ with
$R_{s\alpha}$, the NT formula of Eq. (11) immediately yields the well known
expression:Stephens85 ; Stephens87 ; Buckingham87 ; Barron

$\displaystyle M_{s,\alpha\beta}$ $\displaystyle=$
$\displaystyle\frac{e}{2c}\varepsilon_{\alpha\beta\gamma}Z_{s}R_{s\gamma}+\hbar\,\Omega(B_{\beta},R_{s\alpha})$
(21) $\displaystyle=$
$\displaystyle\frac{e}{2c}\varepsilon_{\alpha\beta\gamma}Z_{s}R_{s\gamma}+2i\hbar\langle\partial_{B_{\beta}}\Psi_{0}|\partial_{R_{s\alpha}}\Psi_{0}\rangle.$

In the present straightforward derivation it appears as one of the numerous
manifestations of the Berry curvature in adiabatic phenomena.

In order to arrive at the usual compact notation for the Berry
curvaturerap_a20 ; Xiao10 ; Vanderbilt  one casts the antisymmetric tensor
$M_{s,\alpha\beta}$ into its equivalent vector form

$\tilde{M}_{s\gamma}=\frac{1}{2}\varepsilon_{\gamma\alpha\beta}M_{s,\alpha\beta}:$
(22) $\tilde{{\bf M}}_{s}=\frac{e}{2c}Z_{s}{\bf
R}_{s}+i\hbar\,\langle\partial_{{\bf B}}\Psi_{0}|\times|\partial_{{\bf
R}_{s}}\Psi_{0}\rangle.$ (23)

The magnetic moment linearly induced by the displacement of the $s$-th nucleus
is then

${\bf m}_{s}=\tilde{{\bf M}}_{s}\times{\bf v}_{s}.$ (24)

Eqs. (21) and (23) are not invariant by translation of the coordinate origin;
it was early recognized that such ambiguity does not affect the measurable
properties. They are also gauge-dependent; it is nonetheless known how to
express the physical observables in gauge-invariant form.Stephens87

###  IV.4 Rotational $\bf g$ factor

In order to simplify the algebra, we fix here the rotation axis, taken as the
$z$-axis. Suppose a molecule is rigidly rotating around its center of mass;
the rotational $g$ factor is defined as the ratio between the total magnetic
moment of the molecule and its mechanical angular moment, expressed in
dimensionless units.

If $\dot{\vartheta}$ is the angular velocity, the mechanical moment is

${\cal M}_{z}=\sum_{s}M_{s}R_{s}^{2}\dot{\vartheta},$ (25)

where $R_{s}$ is the distance of the $s$-th nucleus from the axis and $M_{s}$
is its mass. The magnetic moment, analogously to Eq. (18), is

$m_{z}=\frac{e}{2c}\left(\sum_{s}Z_{s}R^{2}_{s}\dot{\vartheta}-\langle
L_{z}(t)\rangle\right).$ (26)

The $g$ factor can be cast as

$g=\frac{1}{\mu_{\rm B}}\frac{m_{z}}{{\cal M}_{z}}=\frac{1+\sigma}{\mu_{\rm
B}}\;\frac{\sum_{s}Z_{s}R^{2}_{s}}{\sum_{s}m_{s}R^{2}_{s}},$ (27)

where $\mu_{\rm B}$ is the Bohr magneton

$\mu_{\rm B}=\frac{e\hbar}{2mc}.$ (28)

The magnetic screening $\sigma$ measures how much the electrons are “left
behind” by the rotation of the nuclei. For $\sigma=0$ the electrons do not
contribute and the nuclei rotate as naked, while for $\sigma=-1$ the
electronic screening is complete and the molecule rotates as an uncharged
rigid body.

From Eq. (26) the screening factor is

$\sigma=-\frac{\partial_{\dot{\theta}}\langle
L_{z}(t)\rangle}{\sum_{s}Z_{s}R^{2}_{s}}.$ (29)

By adopting a gauge centered on the rotation axis, and exploiting Eq. (20),
the NT formula yields

$\sigma=-\frac{2\hbar
c}{e}\frac{\Omega(B_{z},\vartheta)}{\sum_{s}Z_{s}R^{2}_{s}},$ (30)

where the Berry curvature $\Omega(B_{z},\vartheta)$ is indeed a constant
(evaluated at ${\bf B}=0$, and $\vartheta$-independent).

To the best of the author’s knowledge, $\sigma$ was first recognized as a
Berry curvature in Ref. Ceresoli02, ; the form of Eq. (30) owes to Ref.
Stengel18, . Eq. (30) can be cast into a more familiar sum over states by
means of Eq. (2).

##  Appendix A Independent-electron Berry curvature

In the mean-field case (either Hartree-Fock or Kohn-Sham) the singlet ground
state is a Slater determinant of doubly occupied single-particle orbitals
$|\psi_{j}\rangle$ with eigenvalues $\epsilon_{j}$. We assume that the
parameters do not affect the two body terms in the Hamiltonian, ergo the
$\lambda$-derivatives entering Eq. (2) are one-body operators:

$\partial_{\lambda_{1}}\hat{H}=\sum_{i=1}^{N}O_{1}({\bf r}_{i}),$ (31)

and analogously for $\partial_{\lambda_{2}}\hat{H}$. By the Slater-Condon
rules the matrix elements in Eq. (2) are converted into one-body elements of
$O_{1}$ and $O_{2}$ between occupied and unoccupied orbitals; the energy
differences are easily expressed as well, and Eq. (2) becomes

$\Omega(\lambda_{1},\lambda_{2})=2\,i\sum_{\stackrel{{\scriptstyle
j=\mbox{occupied}}}{{j^{\prime}=\mbox{unoccupied}}}}\frac{\langle\psi_{j}|O_{1}|\psi_{j^{\prime}}\rangle\langle\psi_{j^{\prime}}|O_{2}|\psi_{j}\rangle}{(\epsilon_{j}-\epsilon_{j^{\prime}})^{2}}+\mbox{cc},$
(32)

where the factor of two accounts for spin, and “cc’ stays for complex
conjugate. Straightforward manipulations yield the independent-electron
version of Eq. (1) as

$\Omega(\lambda_{1},\lambda_{2})=2\,\tilde{\Omega}(\lambda_{1},\lambda_{2}),$
(33) $\tilde{\Omega}(\lambda_{1},\lambda_{2})=-2\,\mbox{Im
}\sum_{j}\theta(\mu-\epsilon_{j})(\,\langle\partial_{\lambda_{1}}\psi_{j}|\partial_{\lambda_{2}}\psi_{j}\rangle\,),$
(34)

where $\mu$ is the Fermi level.

$\tilde{\Omega}$ can be expressed in terms of the one-body ground-state
projector

$P=\sum_{j}\theta(\mu-\epsilon_{j})|\psi_{j}\rangle\langle\psi_{j}|:$ (35)
$\tilde{\Omega}(\lambda_{1},\lambda_{2})=-i\,\mbox{Tr
}\\{P\,[\,\partial_{\lambda_{1}}P,\partial_{\lambda_{2}}P\,]\\}.$ (36)

This shows that $\tilde{\Omega}$ enjoys a generalized form of gauge
invariance: it is in fact invariant by any unitary mixing of the occupied
orbitals between themselves.

## Acknowledgments

I have thoroughly discussed some key points of the present paper with Stefano
Baroni, Ivo Souza, Massimiliano Stengel, and David Vanderbilt. Work supported
by the Office of Naval Research (USA) Grant No. N00014-20-1-2847.

## References

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  * (2) Q. Niu and D. J. Thouless, J. Phys A 17, 2453 (1984).
  * (3) R. Resta, J. Phys.: Condens. Matter 12, R107 (2000).
  * (4) D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010).
  * (5) D. Vanderbilt, Berry Phases in Electronic Structure Theory (Cambridge University Press, Cambridge, 2018).
  * (6) P. J. Stephens, J. Phys. Chem. 89, 748 (1985).
  * (7) P. J. Stephens, J. Phys. Chem. 91, 1712 (1987).
  * (8) A. D. Buckingham, P. W. Fowler, and P. A. Galwas, Chem. Phys. 112, 1 (1987).
  * (9) L. Barron, Molecular Light Scattering and Optical Activity, 2nd Edition (Cambridge University Press Cambridge 2004).
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  * (11) Sec. V in: M. Stengel and D. Vanderbilt, Phys. Rev. B 98, 125133 (2018).