# Electronic Vector Potential from the Exact Factorization of a Complex Wavefunction

Sara Giarrusso, Paola Gori-giorgi, Federica Agostini

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Sara Giarrusso, Paola Gori-giorgi, Federica Agostini. Electronic Vector Potential from the Exact Factorization of a Complex Wavefunction. ChemPhysChem, 2024, 10.1002/cphc.202400127. hal-04776390

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# Electronic Vector Potential from the Exact Factorization of a Complex Wavefunction

Sara Giarrusso,*[a] Paola Gori-Giorgi,[b, c] and Federica Agostini*[a]

We generalize the definitions of local scalar potentials named $v_{\text{kin}}$ and $v_{N-1}$, which are relevant to properly describe phenomena such as molecular dissociation with density-functional theory, to the case in which the electronic wavefunction corresponds to a complex current-carrying state. In such a case, an extra term in the form of a vector potential appears which cannot be gauged away. Both scalar and vector potentials are introduced via the exact factorization formalism which allows us to express the given Schrödinger equation as two coupled equations, one for the *marginal* and one for the *conditional*

amplitude. The electronic vector potential is directly related to the paramagnetic current density carried by the total wavefunction and to the diamagnetic current density in the equation for the marginal amplitude. An explicit example of this vector potential in a triplet state of two non-interacting electrons is showcased together with its associated circulation, giving rise to a non-vanishing geometric phase. Some connections with the exact factorization for the full molecular wavefunction beyond the Born-Oppenheimer approximation are also discussed.

## 1. Introduction

The exact factorization formalism[1,2] is an overarching strategy in quantum mechanics that is closely related to probability theory. It allows to express the given Schrödinger equation as two coupled equations by writing the wavefunction as a product of a *marginal* and *conditional* amplitude. Initially proposed as a way to tackle the full electron-nuclear problem beyond the Born-Oppenheimer approximation in a static setting,[3–5] the exact factorization has later been extended to the time-dependent one,[6,7] proving considerably successful as a trajectory-based method for *ab initio* molecular dynamics.[8–14]

In a purely electronic framework, the exact factorization formalism has been applied in different ways. In recent years, it has been used as an embedding strategy to separate a part of a system, which is considered easier to model and requires a “lower-level” approximation, from another part of the system, usually smaller (fragment), that requires to deal with stronger correlation effects and is in general still a many-body

quantity.[15,16] In the present work, we will adopt instead an application of this formalism – sometimes referred to as exact *electron* factorization, to distinguish it from its *electro-nuclear* counterpart – that sits well within the scope of density-functional theory (DFT), both in its pure Hohenberg-Kohn version[17] and in its more popular Kohn-Sham (KS) one.[18] In fact, this way of using the exact factorization on the electronic wavefunction is – to the best of our knowledge – also the oldest one.[19] Here, the marginal amplitude is played by the (square root of the) electron probability density [compare Eq. (4) below]. The choice of this mode of factorization is still interpretable as an embedding strategy. However compared to other types of embedding such as the ones mentioned earlier, it benefits from using as its main variable a marginal amplitude that lives in the three-dimensional space and can be physically observed, in line with the fundamental advantage underlying density-functional theory itself. In this context, the exact factorization has been adopted to gain physical insight in the effective potentials that map the many-body (electronic) problem into a one-body problem,[19–35] where the wording “effective potentials” is used here to refer to both the KS potential and the effective potential of orbital-free DFT.[36,37] The same analysis has been used by Staroverov and coworkers to design an algorithm to construct sensible effective potentials from correlated wavefunctions in finite basis sets.[38–42] There are stark features of these potentials, in the form of peaks and steps, that are extremely hard to model[21,30,43,44] but have been shown to be paramount for the proper description of phenomena such as molecular dissociation[45] or Mott-Hubbard transitions.[46] Such features have been extensively studied in simple systems for which accurate or exact solutions can be calculated. They have been attributed to specific terms of the total potential, named the kinetic potential, $v_{\text{kin}}$, and the $N-1$ potential, $v_{N-1}$, that were identified using the exact factorization and are responsible for the peaks and the step(s) features, respectively.[20,22–26,28–31,34,35] Few approximations targeting these

[a] S. Giarrusso, F. Agostini
Université Paris-Saclay, CNRS,
Institut de Chimie Physique UMR8000,
91405 Orsay, France
E-mail: federica.agostini@universite-paris-saclay.fr
sara.giarrusso@universite-paris-saclay.fr

[b] P. Gori-Giorgi
Department of Chemistry & Pharmaceutical Sciences and
Amsterdam Institute of Molecular and Life Sciences (AIMMS),
Faculty of Science, Vrije Universiteit,
De Boelelaan 1083, 1081HV Amsterdam, The Netherlands

[c] P. Gori-Giorgi
Microsoft Research AI4Science,
Evert van de Beekstraat 354, 1118CZ Schiphol, The Netherlands

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© 2024 The Authors. ChemPhysChem published by Wiley-VCH GmbH. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
specific terms of the total potential have been developed. For example, the GLLB functional[47] models an important part of the KS potential, called the response potential, that is directly related to the N − 1 potential. Although the strategy of developing approximations that model specifically these terms of the total potential, v$_{kin}$ and v$_{N}$$_{−}$$_{1}$ , shows prospects for overcoming long-standing problems of KSDFT[27,30] and is the subject of ongoing efforts, it is not the focus of the present work. Rather, in this work, we focus on the consequences of applying the exact factorization formalism to complex wave- functions. Indeed, the mentioned kinetic and N − 1 potentials have been defined and studied only for real wavefunctions. This restriction can be imposed with no loss of generality for ground states but it is limiting in the study of excited states where complex wavefunctions may occur. For complex wavefunctions, extra terms, in addition to the well-studied local potentials, are expected to appear, notably a vector potential.[32–35]

Recently, there has been a lot of effervescence towards excited states and different flavours of DFT that can describe them.[48] While time-dependent DFT is still a very popular method,[49–52] most calculations are performed using an adia- batic approximation, in which the instantaneous density is fed into a ground-state functional. Adiabatic functionals are incapa- ble of reproducing dynamical peaks and steps of the exact time-dependent KS potential, which in turn are essential to describe atoms and molecules driven far from their ground state.[53,54] Density-functional theories that are time-independent have also been developed to deal with excited states. One prominent approach is ensemble density-functional theory (EDFT), based on an ensemble of states giving an averaged density rather than a pure-state density.[55–58] While EDFT has experienced substantial recent developments crucial for its progress,[59–84] it suffers from drawbacks. Notably, when address- ing a high-lying excited state, it typically necessitates the inclusion of all lower-lying states in the ensemble. Additionally, modeling the weight dependence of the exchange-correlation functional poses challenges. More recently, orbital-optimized DFT, a time-independent framework in which the state of interest is selected by converging the KS self-consistent procedure to saddle points, has demonstrated relative success in computing certain kinds of excitations including those that are typically challenging for TDDFT.[85–88] Theoretically, func- tional approximations specific for the state of interest should be used,[89–91] however the development of such functionals is still in its early stages[70,75] and therefore most applications of DFT to excited states are performed using ground state functionals. On these bases, the development of state-specific functionals appears to be an area of DFT where progress would be particularly useful.

In turn, the exact electron factorization formalism can be adopted on ground and excited states likewise, as shown in the next section. However, in the context of DFT, it has been discussed only within the ground state theory. There, it proved to be a powerful tool in that it allows to clearly distinguish components of the effective potential that depend on slowly- varying features of the quantum state, typically easier to model, from those that are instead responsible for peaks and steps,

which are pivotal to generate a correlated density, but are missing from almost all of the approximate functionals available today. Hence, we expect this formalism to represent a privileged standpoint to devise approximations that can model such stark features for ground as well as for excited states. Yet, for certain excited states, it is important to consider not only the scalar potentials generating peaks and steps but also the role of the electronic vector potential mentioned earlier. As we will show, this vector potential is associated with current- carrying states and has never been studied in the context of DFT and the electronic wavefuction. It also represents the most striking difference to be found between ground and (certain) excited states when adopting the exact factorization formalism in a DFT setting. Therefore, this work’s main focus lies in the study of this term, which is absent from the ground state theory. In particular, we provide an explicit example of the electronic vector potential associated with a complex current- carrying state, in a simple model system.

In the time-dependent electro-nuclear formulation of the exact factorization, the development of approximations to compute the “effective potentials” of the theory, usually referred to as time-dependent potential energy surface and time-dependent vector potential, for molecules[92,93] has been achieved after assessing their behavior on prototypical case studies where the exact numerical solution of the problem was accessible.[94–99] Following the successful strategy of the electro- nuclear formulation, in this paper, we first reconstruct the time- independent exact electron factorization formalism in a general case manifesting a non-vanishing electronic vector potential and then we analyze such vector potential on a prototypical case study. Adopting an exactly-solvable two-electron non- interacting problem, we aim to characterize the most relevant features of the electronic vector potential, in particular demon- strating its relation to the property of the electronic wave- function of being complex and showing that it cannot be gauged away (thus it must be accounted for in an approximate treatment of the problem).

The paper is organized as follows. Section 2 introduces the definitions of the local scalar and vector potentials stemming from the exact factorization in the case of a complex wave- function and discusses how the vector potential is related to the current density. Section 3 illustrates the conditional ampli- tude and the electronic vector potential in the adopted model, while Section 4 discusses the related geometric phase. Finally, Section 5 gives some conclusions.

2. Exact Electron Factorization of a Complex Wavefunction

Let us consider the time-independent non-relativistic Schrö- dinger equation for the N -electron wavefunction:

$$
\hat{H}^N(r_1, \dots, r_N) \Psi_m(1, \dots, N) = E_m^N \Psi_m(1, \dots, N) \quad (1)
$$
with $i = \sigma_i r_i$ the spatial and spin coordinates of the i-th electron and where the Hamiltonian is the usual sum of the kinetic energy, the electron-electron repulsion and a (local) external scalar potential operator:

$$ \hat{H}^N(\mathbf{r}_1, \dots, \mathbf{r}_N) = \sum_{i=1}^{N} \left( -\frac{\nabla_{\mathbf{r}_i}^2}{2} + \sum_{j>i}^{N} \frac{1}{|\mathbf{r}_i - \mathbf{r}_j|} + v(\mathbf{r}_i) \right) \quad (2) $$

We consider a general eigenstate of the above equation, $\Psi_m(1, \dots, N)$, with associated eigenvalue $E_m^N$.

Following Refs. [19,20], we split $\hat{H}^N$ in three portions: the Hamiltonian for $N-1$ electrons, the one-body terms acting on electron 1 (taken as the reference), $\hat{h}$, and the interaction between electron 1 and all the others:

$$ \hat{H}^N(\mathbf{r}_1, \dots, \mathbf{r}_N) = \hat{H}^{N-1}(\mathbf{r}_2, \dots, \mathbf{r}_N) + \hat{h}(\mathbf{r}) + \sum_{i=2}^{N} \frac{1}{r_{1i}} \quad (3) $$

with $\hat{h}(\mathbf{r}) = -\frac{\nabla_{\mathbf{r}}^2}{2} + v(\mathbf{r})$ and $r_{1i} = |\mathbf{r} - \mathbf{r}_i|$. The wavefunction, solution of Eq. (1), is written as:

$$ \Psi_m(1, \dots, N) = \chi_m(\mathbf{r})\Phi_m(\sigma, 2, \dots, N; \mathbf{r}) \quad (4) $$

In the right hand side (RHS) of Eqs. (3) and (4), the subscript 1 from the coordinates of the reference electron is dropped, i.e. $\sigma_1 = \sigma$ and $\mathbf{r}_1 = \mathbf{r}$, a notation that will be kept for the rest of the paper. The functions $\chi_m$ and $\Phi_m$ in Eq. (4) are called marginal and conditional amplitude, respectively, and are constrained by the so-called partial normalization condition (PNC):

$$ \langle \Phi_m(\sigma, 2, \dots, N; \mathbf{r}) | \Phi_m(\sigma, 2, \dots, N; \mathbf{r}) \rangle_{\mathbf{r}} = 1 \quad \forall \mathbf{r} \quad (5) $$

where the Dirac brakets $\langle \dots | \dots \rangle_{\mathbf{r}}$ stand for $\int d\sigma d2 \dots dN$, i.e. integration over all the coordinates of the system but $\mathbf{r}$. The PNC ensures that the probability of finding the $N-1$ electrons anywhere in space with spin either up or down (including the spin of the reference one) is 1, regardless of where the reference electron is placed.

The PNC together with Eq. (4) determines only the modulus squared of the marginal amplitude, i.e. $|\chi_m|^2 = \rho_m/N$, but leaves an $\mathbf{r}$-dependent phase factor freedom in $\chi_m(\mathbf{r}) = \sqrt{\rho_m(\mathbf{r})/N}e^{i\langle\mathbf{r}\rangle}$ and $\Phi_m = \frac{\psi_e^{-i\langle\mathbf{r}\rangle}}{\sqrt{\rho_m/N}}$. For this latter equation to hold, we need to assume that the excited-state density $\rho_m$ has no nodes, which is always true if we exclude core excitations.

In this work, we choose $S(\mathbf{r}) = 0$ or:

$$ \chi_m(\mathbf{r}) = \sqrt{\rho_m(\mathbf{r})/N} \quad (6) $$

Making a choice for $S(\mathbf{r})$ is equivalent to setting the gauge of the electronic vector potential that will be introduced later. The gauge choice of Eq. (6) is not only in line with past literature[19,20,24–26,28–31] but, most importantly, an extremely convenient one, as will be evident in the following sections.

## 2.1. Equation for the Marginal Amplitude

To obtain an effective equation for the marginal amplitude, $\chi$, let us apply Eqs. (3) and (4) to Eq. (1), then left-multiply by $\Phi_m^*$ and integrate over all variables but $\mathbf{r}$. The nabla operator acting on the factorized wavefunction of Eq. (4) makes extra terms appear compared to the usual (Schrödinger-equation-like) expression. Note that, from the PNC, it follows that:

$$ \langle \nabla_r \Phi_m | \Phi_m \rangle_{\mathbf{r}} = - \langle \Phi_m | \nabla_r \Phi_m \rangle_{\mathbf{r}} \quad (7) $$

This in turn means that $\langle \Phi_m | \nabla_r \Phi_m \rangle_{\mathbf{r}}$ is purely imaginary. We then introduce the real-valued vector function $A_m(\mathbf{r})$, such that:

$$ \langle \Phi_m | \nabla_r \Phi_m \rangle_{\mathbf{r}} = i A_m(\mathbf{r}) \quad (8) $$

Moreover, choice Eq. (6) implies that any imaginary component contained in the wavefunction is transferred to the conditional amplitude alone, which can then be written as:

$$ \Phi_m(\sigma, 2, \dots, N; \mathbf{r}) = e^{i\theta_m(1\dots N)} R_m(\sigma, 2, \dots, N; \mathbf{r}) \quad (9) $$

with $\theta_m$ and $R_m$ both real. Therefore, the vector function $A_m$ becomes:

$$ A_m(\mathbf{r}) = \langle R_m | \nabla_r \theta_m | R_m \rangle_{\mathbf{r}} \quad (10) $$

From this point on, we will often refrain from specifying the state index $m$ to simplify the notation, but it remains understood that all the quantities derived from the full electronic wavefunction, including the local potentials that will be introduced, do depend on the particular state under consideration.

We shall specify the degree of excitation only in some cases in which it is particularly relevant, such as in Eqs. (18) and (19) below.

The final result of left-multiplying Eq. (1) by $\Phi_m^*$ and integrating over all variables but $\mathbf{r}$ is the following equation for the marginal amplitude [Eq. (6)]:

$$ \left( v_{\text{cond}} + v_{\text{kin}} + v_{N-1} + v - \frac{\nabla_r^2}{2} \right) \chi + i \left( \mathbf{A} \cdot \nabla_r + \frac{1}{2} \nabla_r \cdot \mathbf{A} \right) \chi = -I_p \chi \quad (11) $$

with $I_p = -(E^N - E^{N-1})$ the ionization potential,

$$ v_{\text{cond}} = \langle \Phi | \sum_{i=2}^{N} \frac{1}{r_{1i}} | \Phi \rangle_{\mathbf{r}} = \langle R | \sum_{i=2}^{N} \frac{1}{r_{1i}} | R \rangle_{\mathbf{r}} \quad (12) $$

the conditional potential,
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$$
\begin{align}
v_{\text{kin}} &= \frac{1}{2} \langle \nabla_r \Phi | \nabla_r \Phi \rangle_{\setminus r} \nonumber \\
&= \frac{1}{2} \langle \nabla_r R | \nabla_r R \rangle_{\setminus r} + \frac{1}{2} \langle R \nabla_r \theta | R \nabla_r \theta \rangle_{\setminus r} \tag{13/14}
\end{align}
$$

the kinetic potential, and

$$
\begin{align}
\tilde{v}_{N-1} &= \langle \Phi | \hat{H}^{N-1} | \Phi \rangle_{\setminus r} \nonumber \\
&= \langle R | \sum_{i=2}^{N} \left( \frac{1}{\mathbf{r}_{1i}} + v(\mathbf{r}_i) \right) | R \rangle_{\setminus r} \tag{15/16} \\
&\quad + \frac{1}{2} \sum_{i=2}^{N} \left( \langle \nabla_{\mathbf{r}_i} R | \nabla_{\mathbf{r}_i} R \rangle_{\setminus r} + \langle R \nabla_{\mathbf{r}_i} \theta | R \nabla_{\mathbf{r}_i} \theta \rangle_{\setminus r} \right) \nonumber
\end{align}
$$

with $v_{N-1} = \tilde{v}_{N-1} - E^{N-1}$ the $N-1$ potential. An appropriate constant shift to the potential $\tilde{v}_{N-1}$ is added such that, when $|r| \to \infty$, $v_{N-1}$ goes to zero. When the distance of the reference electron from the bulk of the system is large enough, one may expect that the conditional amplitude $\Phi_m$ associated with the $m$-th excited $N$-electron state $\Psi_m$ collapses to the corresponding $m$-th excited $N-1$-electron state $\Psi_m^{N-1}$. Hence, the appropriate constant would correspond to the energy of the state $\Psi_m^{N-1}$, that is $E^{N-1}$ above, where the subscript $m$ is implied. However, the specific ion state to which the conditional amplitude collapses may depend on the direction along which the reference electron is removed. This requires a discussion of the possible presence of nodal planes in the conditional amplitude which is a delicate subject also for ground states.[100,101] In turn, nodal planes are expected to occur more frequently for higher-energy states so it is important to keep this possibility in mind. The possible presence of nodal planes means that the constant shift that gauges $v_{N-1}$ to zero asymptotically along one direction may not gauge it to zero along other directions. (Note that here we are discussing a scalar potential gauge, which is simply a constant shift, while later on we will discuss the gauge of the vector potential which is related to the phase-factor freedom that defines our conditional and marginal amplitudes.) Concerning the $|r| \to \infty$ behavior of the other scalar potentials, it is easy to see that the conditional potential is vanishing asymptotically as $N^{-1}$ while the external potential as $Z/r$. Finally, whether the kinetic potential vanishes isotropically is a question that requires again to consider the possible presence of nodal planes. Excluding such pathological cases, one expects that the variation of the conditional amplitude with respect to the position of the reference electron, embodied by the kinetic potential, goes to zero, as the conditional amplitude collapses to the ionized state.

Definitions (12)–(16) of $v_{\text{kin}}$, $v_{\text{cond}}$, $v_{N-1}$ correspond to the usual definitions of these potentials[20,25,30,31] when the conditional amplitude is purely real, $\Phi(\sigma, 2, \dots, N; \mathbf{r}) = R(\sigma, 2, \dots, N; \mathbf{r})$, and are extended in this paper to the complex case for $\theta(1, \dots, N) \neq 0, 2\pi, \dots$.

Concerning the vector function $A(\mathbf{r})$, note that the term in parenthesis in the second line of Eq. (11) needs to vanish, i.e.:

$$
\mathbf{A} \cdot \nabla_r \chi = -\frac{\chi}{2} \nabla_r \cdot \mathbf{A} \qquad (17)
$$

as a result of the RHS of Eq. (11) being real [per Eq. (6)].

Following the above discussion on both scalar and vector potentials, when $|r| \to \infty$, Eq. (11) becomes simply:

$$
-\frac{\nabla_r^2}{2} \chi_m(\mathbf{r}) \sim -l_{p,m} \chi_m \qquad (18)
$$

which implies the following decay of the marginal amplitude:

$$
\chi_m(\mathbf{r}) \sim e^{-\sqrt{2|l_{p,m}|}r} \qquad (19)
$$

This same derivation was used in Ref. [19] to prove that the decay of the ground-state density is directly related to the first ionization potential. Here, we generalize this conclusion also to excited-state densities: their asymptotic decay is directly related to their corresponding first ionization potential, at least in the absence of nodal planes or along directions far from them. This result is in line with the conclusions of Ref. [90] where it is shown that an excited-state density (obtained from a Coulomb external potential as is the case here) retains the information of the degree $m$ of its excitation under the assumption of isotropy.

Before moving on to discuss the relation of the electronic vector potential to the current density in Sec. 2.2, we address the matter of the gauge of the vector potential in the remainder of this section.

Eq. (11) can also be rearranged as:

$$
\frac{[-i\nabla_r + \mathbf{A}]^2}{2} \chi + (v^G + v_{\text{cond}} + v_{N-1} + v)\chi = -l_p\chi \qquad (20)
$$

This form is perhaps more familiar[6,32] and leads one to identify **A** as a magnetic-like vector potential. Indeed, Eq. (20) for the marginal amplitude is formally equivalent to a Schrödinger equation for a one-body quantum system (such as the hydrogen atom) subject to an external magnetic field. However, **A**, along with the scalar potential in parenthesis in the RHS of Eq. (20), encodes the effect of the conditional amplitude and, thus, of the remaining $N-1$ electrons on the reference one.

Note that, among the scalar potentials, the geometric potential $v^G$ appears. This is defined as:[32,35]

$$
v^G = \frac{1}{2} \sum_{k=x,y,z} g_{kk} = \frac{1}{2} \sum_{k=x,y,z} \langle \partial_k \Phi | \hat{P}_\perp | \partial_k \Phi \rangle_{\setminus r} \qquad (21)
$$

with $g_{kk}$ a Fubini-Study metric tensor involving the projector operator $\hat{P}_\perp = 1 - |\Phi\rangle\langle\Phi|$ and the derivative with respect to the $k$-th component of $\mathbf{r}$, $\partial_k$. In our gauge, definition (21) gives simply:

$$
v^G = v_{\text{kin}} - \frac{|\mathbf{A}|^2}{2} \qquad (22)
$$
The reason why the geometric potential of Eq. (21) has been introduced is that it can be interpreted as a quantum geometric (diagonal) tensor independent of the gauge.

To dig deeper into the meaning of the gauge choice in the exact electron factorization, we take a few steps back and derive the equation for the marginal before imposing any gauge: the phase factor in the marginal amplitude and in the conditional amplitude can be anything, not just zero. We call it $S'(r)$. To distinguish these different marginal and conditional amplitudes from the $\chi$ and $\Phi$ that we have considered so far, we will label them $\chi'$ and $\Phi'$. Because the electronic vector potential depends on the conditional amplitude, we shall consider also a primed electronic vector potential $\mathbf{A}'$. In formulas, we are considering the following relations:

$$ \chi' \rightarrow \chi e^{iS'(r)} \quad \Phi' \rightarrow \Phi e^{-iS'(r)} \quad \mathbf{A}' \rightarrow \mathbf{A} + \nabla_r S'(r) \quad (23) $$

Repeating the same steps as before but with $\chi'$ and $\Phi'$, we get to an equation for the marginal which is almost identical to Eq. (20):

$$ \frac{(-i\nabla_r + \mathbf{A}')^2}{2} \chi' + (v^G + v_{\text{cond}} + v'_{N-1} + v)\chi' = -l_p\chi' \quad (24) $$

In the above equation, $v^G$ is gauge-invariant by construction, $v$ is the given external potential of the original Schrödinger equation, while $v_{\text{cond}}$ is gauge invariant because its definition involves a multiplicative potential and therefore this potential is blind to any phase factor, as visible already in Eq. (12). Finally, the potential $v'_{N-1}$ defined as:

$$ v'_{N-1} = \langle \Phi' | \hat{H}^{N-1} | \Phi' \rangle_r - E^{N-1} \quad (25) $$

contains the Laplacian operator inside $\hat{H}^{N-1}$ and therefore it is overall gauge-dependent. Although the single terms $\mathbf{A}', \chi'$ and $v'_{N-1}$ depend on the particular choice for $S'(r)$, the overall Eq. (24) is form-invariant under a gauge transformation of the type Eq. (23).

For some applications, it may be useful to work with a gauge-invariant potential $v^G$. However, this does not remove the gauge-dependence on the $N-1$ potential or on the electronic vector potential. Thus, if one's interest is to design approximations for these local scalar and vector potentials, making a sensible choice for their gauge and adhering to it across different approximations is of utmost importance. Since it is undeniably convenient to work with a marginal which is simply the square-root of the electron density, we believe that our definitions Eqs. (14)–(16) are bona fide definitions in which the choice of a particular gauge choice does not restrict their applicability but rather enhances it. Further advantages of the chosen gauge are to be discussed in the next section.

## 2.2. Relation with the Current Density

The appearance of a non-vanishing vector potential in the equation for the marginal wavefunction can be directly related

to the appearance of a non-vanishing current density carried by the total wavefunction, as is the case in the electro-nuclear formulation.[97,99,102]

In a DFT setting, the theoretical framework which considers the electronic current density as a basic variable was introduced in the 80s and is now known as current-density-functional theory.[103,104] In general, an external vector potential $\mathbf{A}_{\text{ext}}$ induces a current density and allows to define the mechanical momentum operator $\pi_r = -i\nabla_r + \mathbf{A}_{\text{ext}}(r)$. The induced current density, $j_{\psi}(r)$, carried by an eigenstate $\psi$ of the Hamiltonian under study is:

$$ j_{\psi}(r) = N \text{Re} \langle \psi | \pi_r \psi \rangle_r \quad (26) $$

Such current density $j_{\psi}$ is usually split into a paramagnetic contribution, $j_{\psi,p}$, which comes from the canonical momentum operator $\mathbf{p}_r = -i\nabla_r$.

$$ j_{\psi,p}(r) = N \text{Re} \langle \psi | \mathbf{p}_r \psi \rangle_r \quad (27) $$

and a diamagnetic contribution, $j_{\psi,d}(r)$, which is the remaining term, involving $\mathbf{A}_{\text{ext}}$.

Plugging $\Psi$ [Eq. (4)] in the definition of the paramagnetic density [Eq. (27)] and comparing with Eq. (8), one finds that the effective vector potential appearing in the exact factorization yields the physical paramagnetic current density (divided by the electron density):

$$ \mathbf{A}(r) = \frac{j_{\psi,p}}{\rho}(r) \quad (28) $$

This straightforward relation is a remarkable advantage of the chosen gauge as it allows to directly relate the effective vector potential of the “reduced” system, described by the marginal amplitude, to a physical property of the full (many-body) system.

Furthermore, by substituting Eq. (28) in Eq. (17), one obtains:

$$ \nabla_r \cdot j_{\psi,p} = 0 \quad (29) $$

This result indicates that, although the paramagnetic current associated with the $N$-electron wavefunction $\Psi$, $j_{\psi,p}$, may be non-zero, its divergence is exactly zero. Furthermore, since in the original Schrödinger equation for $\Psi$ [Eq. (1)], there is no external vector potential $\mathbf{A}_{\text{ext}}$, the diamagnetic contribution associated with $\Psi$ is zero, $j_{\psi,d} = 0$.

Eq. (29) together with $j_{\psi,d} = 0$ satisfies the continuity equation for the total current associated with $\Psi$, $j_{\psi}(r) = j_{\psi,p} + j_{\psi,d}$, in the static setting, i.e.:

$$ \partial_t \rho(r) = -\nabla_r \cdot j_{\psi}(r) = 0 \quad (30) $$

On the other hand, focusing now on the equation for the marginal wavefunction [Eqs. (11) or (20)], containing the effective vector potential $\mathbf{A}$, the paramagnetic term is zero
because of the choice of gauge which makes $\chi$ real, $\mathbf{j}_{\chi,p} = 0$, while the diamagnetic term is non-zero and equal to:

$$ \mathbf{j}_{\chi,d}(\mathbf{r}) = \mathbf{A}(\mathbf{r})\chi^2(\mathbf{r}) \qquad (31) $$

In other words, the (physical) paramagnetic current density in the original many-body problem [Eq. (1)] is transformed into a diamagnetic-type current density in the effective reformulation represented by the Schrödinger equation for the marginal wavefunction. Note that, in the static setting chosen in this work, the terms involving the vector potential $\mathbf{A}$ in Eq. (11) need to cancel out, as mentioned [Eq. (17)]. However, the concept of an electronic vector potential appearing in the reduced Schrödinger equation for the marginal amplitude may be useful in a time-dependent setting. In other words, the study of the electronic vector potential carried out here in a static setting is preparatory for understanding its role in a time-dependent reduced Schrödinger equation for the marginal amplitude $\chi_m(\mathbf{r}, t)$.

An additional observation on the vector potential concerns its dependence on the gauge. As we have discussed, the electronic vector potential can be transformed as in Eq. (23) and it is, thus, not an observable quantity (as in classical electromagnetism). However, its curl is a robust physical observable and is a gauge-independent quantity: the curl of such an effective vector potential is related to the curl of the (physical) paramagnetic current of the system. Therefore, in Sec. 3, we find it interesting to analyze the curl of the vector potential.

## 2.3. Equation for the Conditional Amplitude

We have seen in Sec. 2.1 that the original many-body Schrödinger Eq. (1) can be reduced to a one-body equation involving effective vector and scalar potentials [Eq. (11)], whose solution is the marginal amplitude, $\chi$. However, the scalar and vector potentials appearing in Eq. (11) are functionals of the conditional amplitude, $\Phi$. It seems then appropriate to look at the equation satisfied by the conditional amplitude. By simply substituting Eqs. (3) and (4) into Eq. (1), after some rearrangements, one gets:

$$ \left( \hat{H}^N - \frac{\nabla_r \chi}{\chi} \cdot \nabla_r \right) \Phi = \left( E^N + \frac{\nabla_r^2 \chi}{2\chi} \right) \Phi \qquad (32) $$

where the eigenvalue $(E^N + \frac{\nabla_r^2 \chi}{2\chi})$ on the RHS depends on the parameter $r$ and is sometimes referred to as the effective potential $v_{\text{eff}}$. [19,29] Clearly, Eq. (32) can be shifted by any constant, and usually $v_{\text{eff}}$ is defined so as to vanish at large $|r|$ (if possible), by adding the constant shift $-E^{N-1}$. The effective potential in this case equals the sum of the local scalar potentials in Eq. (11), i.e. $v_{\text{eff}} = (v + v_{\text{cond}} + v_{\text{kin}} + v_{N-1}) = (\frac{\nabla_r^2 \chi}{2\chi} - l_p)$. (For the reader who may wonder about the connection with KSDFT, the potential $v_{\text{eff}}$ for ground states differs from the KS potential, $v_s$, by what is sometimes called the Pauli potential, $v_p$, [105] i.e. $v_{\text{eff}} = v_s + v_p$.)

Inserting now the polar representation of the conditional amplitude of Eq. (9) into Eq. (32) and left-multiplying by $e^{-i\theta}$, we get, an equation for the modulus, $R$:

$$ \begin{aligned} & \left( \hat{V}_{\text{TOT}} - \frac{1}{2} \sum_{i=1}^{N} \left( \nabla_{r_i}^2 - (\nabla_{r_i} \theta)^2 \right) - \frac{\nabla_r \chi}{\chi} \cdot \nabla_r \right) R \\ & = \left( E^N + \frac{\nabla_r^2 \chi}{2\chi} \right) R \end{aligned} \qquad (33) $$

with $\hat{V}_{\text{TOT}} = \sum_{i=1}^{N} (\sum_{j>i}^{N} \frac{1}{|r_j - r_i|} + v(r_i))$, and an equation for the phase, $\theta$:

$$ \sum_{i=1}^{N} \left( \frac{\nabla_{r_i}^2}{2} + \frac{\nabla_{r_i} R}{R} \cdot \nabla_{r_i} \right) \theta = \left( - \frac{\nabla_r \chi}{\chi} \cdot \nabla_r \right) \theta \qquad (34) $$

Eqs. (33) and (34) are interesting in that they bring to light a local relationship between the modulus and the phase of the conditional amplitude. In particular, in the equation for the modulus [Eq. (33)], it is interesting to notice that the square of the gradient of the phase of the conditional amplitude acts as an extra scalar potential term in the N-electron Hamiltonian; while in the equation for the phase [Eq. (34)] there is a compensation between the Laplacian of the phase and the “coupling” between this latter and the modulus of the conditional wavefunction. This can also be seen directly from the total wavefunction, expressed in polar representation with its modulus $R_{\psi}$ and phase $\theta$ as done in Eq. (9)) for the conditional amplitude, i.e.:

$$ \left( \hat{V}_{\text{TOT},\theta} - \sum_{i=1}^{N} \frac{\nabla_{r_i}^2}{2} \right) R_{\psi} = E^N R_{\psi} \qquad (35a) $$

$$ \sum_{i=1}^{N} \left( \frac{\nabla_{r_i}^2 \theta}{2} + \nabla_{r_i} \theta \cdot \frac{\nabla_{r_i} R_{\psi}}{R_{\psi}} \right) = 0 \qquad (35b) $$

with $\hat{V}_{\text{TOT},\theta} = (\hat{V}_{\text{TOT}} + \sum_{i=1}^{N} (\nabla_{r_i} \theta)^2 / 2)$, and $R_{\psi} = \chi R$ such that $\Psi = e^{i\theta} R_{\psi}$.

## 3. N=2 Non-Interacting Fermions in the Coulomb Potential

In this section, we study a simple model of two non-interacting fermions in a Coulomb external potential that is exactly solvable such that we can perform an in-depth analysis of the properties of the electronic vector potential $\mathbf{A}$.

We consider a triplet state whose total spatial wavefunction is written as an antisymmetrized product of two hydrogenic functions $\psi_{nlm}$, a real one with $n=1, l=0$ and $m=0$ (the 1s orbital) and a complex one with $n=2, l=1$ and $m=1$:
<figure><img src="image_5.png" /></figure>

$$ \Psi(\mathbf{r}_1, \mathbf{r}_2) = \frac{1}{\sqrt{2}} (\psi_{100}(\mathbf{r}_1)\psi_{211}(\mathbf{r}_2) - \psi_{211}(\mathbf{r}_1)\psi_{100}(\mathbf{r}_2)) \quad (36) $$

with

$$ \psi_{100}(r) = \frac{2}{\sqrt{4\pi}} Z^{3/2} e^{-Zr} \quad (37a) $$

$$ \psi_{211}(r, \vartheta, \varphi) = \frac{1}{\sqrt{8\pi}} \left(\frac{Z}{2}\right)^{3/2} Zr e^{-\frac{Zr}{2}} \sin \vartheta e^{i\varphi} \quad (37b) $$

This wavefunction is solution to the Schrödinger equation with Hamiltonian:

$$ \hat{H}(Z, \mathbf{r}_1, \mathbf{r}_2) = -\frac{1}{2} (\nabla_{\mathbf{r}_1}^2 + \nabla_{\mathbf{r}_2}^2) - \left(\frac{Z}{\mathbf{r}_1} + \frac{Z}{\mathbf{r}_2}\right) \quad (38) $$

and energy $E = -\frac{5Zr}{8}$ in atomic units. The corresponding marginal and conditional amplitudes read, respectively:

$$ \chi(r, \vartheta) = \frac{\sqrt{Z^3 e^{-2Zr} (r^2 Z^2 \sin^2(\vartheta) e^{2Zr} + 64)}}{8\sqrt{2\pi}} \quad (39a) $$

$$ \Phi(r_2, \vartheta_2, \varphi_2; r, \vartheta, \varphi) = \frac{Z^{5/2} e^{-Zr_2} (r_2 \sin(\vartheta_2) e^{\frac{Zr_2}{2}+i\vartheta_2} - r \sin(\vartheta) e^{\frac{Zr}{2}+i\vartheta})}{\sqrt{\pi} \sqrt{r^2 Z^2 \sin^2(\vartheta) e^{2Zr} + 64}} \quad (39b) $$

First of all, we notice that $\chi(r, \vartheta, \varphi) = \chi(r, \vartheta)$, independent of $\varphi$, as obtained by integrating over $r_2, \vartheta_2, \varphi_2$ the squared modulus of Eq. (36), namely:

$$ \chi(r, \vartheta) = \sqrt{\int |\Psi(r, \vartheta, \varphi, r_2, \vartheta_2, \varphi_2)|^2 r_2^2 \sin \vartheta_2 dr_2 d\vartheta_2 d\varphi_2} \quad (40) $$

The modulus squared (times $r_2^2 \sin \vartheta_2$) of the conditional amplitude [Eq. (39b)] is plotted in Figure 1 as a function of the coordinates of the remaining electron $r_2$ and $\vartheta_2$, with $\varphi_2$ fixed at $\pi$ rad. The reference electron is placed at polar angle $\vartheta = \pi/2$

and azimuthal angle $\varphi = \pi$, while the radial distance is varied from $r=0.1$ bohr in the left panel to 20.0 bohr in the right panel, passing through $r=2.5$ bohr in the center panel. The modulus squared of the conditional amplitude represents the probability density of finding the remaining electron in a certain region of space, depending on where the reference electron is placed.

As visible in Figure 1, when the reference electron is in the region dominated by $\psi_{100}$ ($r=0.1$ bohr), the probability density to find the remaining electron essentially mirrors the shape of the $\psi_{211}$ function (left panel), viceversa when the reference electron is in the region dominated by $\psi_{211}$, the conditional probability amplitude resembles the $\psi_{100}$ function (right). For intermediate distances, the conditional amplitude is a mixture of the two functions (center). In other words, the behaviour of the (modulus squared of the) conditional amplitude reflects the common understanding that the remaining electron "occupies" one or the other orbital according to where the reference electron is placed and clearly exemplifies the so-called Fermi correlation. This is the type of correlation that stems from the indistinguishability of electrons and manifests itself in the spatial part of the wavefunction for systems with parallel spins.[106,107]

Having the basic quantities in analytical form, it is possible to calculate the electronic vector potential explicitly. Because the full electronic wavefunction and, therefore, the conditional amplitude are imaginary only along the $\varphi$ component, the radial and polar components of the vector potential, $A_r$ and $A_\varphi$, are exactly zero and the only non-zero component is the azimuthal one:

$$ A(r, \vartheta, \varphi) = \begin{pmatrix} A_r(r, \vartheta, \varphi) \\ A_\varphi(r, \vartheta, \varphi) \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \quad (41) $$

where

$$ A_{\varphi}(r, \vartheta) = \frac{Z^2 r \sin(\vartheta) e^{2r}}{Z^2 r^2 \sin^2(\vartheta) e^{2r} + 64} \quad (42) $$

<figure><img src="image_6.png" /><figcaption>Figure 1. Plot of the modulus squared of the conditional amplitude for $\varphi = \varphi_2 = \pi$ and $\vartheta = \pi/2$ as a function of the radial distance $r_2$ and the polar angle $\theta_2$ for different values of the radial distance of the reference electron: $r=0.1$ (left), $r=2.5$ (center) and $r=20.0$ (right).</figcaption></figure>
In Figure 2, we schematically represent the vector potential in Cartesian coordinates as a vector field that somehow “rotates” around the z-axis. The magnitude of the vector potential appears to increase as the distance from the nucleus (a proton, in Figure 2, since Z = 1) which is in agreement with the interpretation of A as the current density divided by the density in our gauge. Moreover, since A<sub>φ</sub> only depends on the radial distance r and the polar angle ϑ, the left hand side of Eq. (17) is itself zero, implying that, in this case, not only the divergence of the current density is zero, but the electronic vector potential (or the related current density) is orthogonal to the gradient of the density:

<figure><img src="image_7.png" /><figcaption>Figure 2. Electronic vector potential A [Eq. (42)] in Cartesian coordinates x, y, z for nuclear charge Z = 1. Color coding reflects the magnitude of the vector field at each point.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 3. Curl of the vector potential A [Eq. (44)] in Cartesian coordinates x, y, z for nuclear charge Z = 1. Color coding reflects the magnitude of the vector field at each point.</figcaption></figure>

$$
A \cdot \nabla_r \chi = \begin{pmatrix} 0 \\ 0 \\ A_\psi(r, \vartheta) \end{pmatrix} \cdot \begin{pmatrix} \partial_r \chi \\ \frac{1}{r} \partial_\vartheta \chi \\ 0 \end{pmatrix} = 0 \qquad (43)
$$

The curl of the electronic vector potential, shown in Figure 3, is however non-zero and reads:

$$
\begin{align}
\nabla \times \mathbf{A} &= \left( \frac{1}{r \sin(\vartheta)} \frac{\partial A_{\varphi} \sin(\vartheta)}{\partial \vartheta} \right) \hat{\mathbf{r}} + \left( -\frac{1}{r} \frac{\partial (r A_{\varphi})}{\partial r} \right) \hat{\mathbf{r}} \tag{44/45} \\
&= \frac{64 Z^2 e^{2r}}{(64 + Z^2 e^{2r} r^2 \sin^2(\vartheta))^2} \left( 2 \cos(\vartheta) \hat{\mathbf{r}} - (2 + Zr) \sin(\vartheta) \right) \nonumber
\end{align}
$$

Note that, if we choose as eigenstate of Eq. (38) Ψ(r₁, r₂) = 1/√2 (ψ₁₀₀(r₁)ψ'₂₁⁻₁(r₂) - ψ'₂₁⁻₁(r₁)ψ₁₀₀(r₂)), i.e. using the function ψ₂₁⁻₁(r, ϑ) = 1/√8π (Z/2)³/² Zr e⁻²r sin ϑ e⁻⁴q instead of ψ₂₁₁, the resulting vector potential is opposite in sign.

Since under a gauge transformation the vector potential transforms with the gradient of a scalar function, the curl of the vector potential is a gauge-independent quantity. We have shown here, for the particular case of two non-interacting electrons in an external Coulomb potential, that the vector potential has non-vanishing curl. Therefore, it cannot be gauged away in this situation. Whether this property holds valid in more realistic cases, as for instance in the presence of electron-electron interaction, will be investigated in future work.

4. Geometric Phase Related to the Electronic
Vector Potential

The formal analogy between the reduced Schrödinger equation
for the square root of the electron density [Eqs. (11) or (20)] and
that used in the electro-nuclear formulation of the exact
factorization, together with the appearance of a not-irrotational
(i.e. not curl-free) vector potential suggests the possibility of
observing a geometric phase in this purely electronic system.

A local gauge potential such as **A**(r) and the associated curl,
or gauge field, are identified as a connection and a curvature,
similarly, but not exactly equivalently, to the Berry correction
and the Berry curvature introduced in Ref. [108] for an adiabatic
evolution. Therefore, a geometric phase can be defined in
relation to our electronic vector potential as the line integral of
the connection along a path **C**, that we consider closed in this
case, namely:

$$
\gamma(C) = i \oint_C \langle \psi(\xi) | \nabla_{\xi} \psi(\xi) \rangle \cdot d\xi \qquad (46)
$$

where ξ identifies the variable(s) which the connection depends
on. If γ(C) depends on the geometry of the path, it is called a
geometric phase. Usually, however, the concept of geometric
<figure><img src="image_9.png" /></figure>

phase has been discussed in relation to the adiabatic approximation (not that of time-dependent DFT), namely when considering that the Hamiltonian of the system depends on some parameter that varies slowly, i.e. adiabatically. In this case, when an eigenstate of the Hamiltonian is transported along a closed path in the parameter space, the eigenstate can pick up a geometric phase factor in addition to the standard dynamical phase factor that comes from the Schrödinger equation.

Beyond the adiabatic approximation, geometric phases have been discussed as well in the context of the exact factorization using both a time-independent[102,109] and a time-dependent[97,99] formalism but only in the electro-nuclear formulation. We find, therefore, interesting here to point out the property of the exact electron factorization of a complex wavefunction: that of giving rise to a non-vanishing geometric phase in relation to the vector potential of the theory.

Similarly to the time-dependent electro-nuclear case, such a phase is not quantized, thus it is not topological (at least in the portion of "parameter" space that we analyzed), but it is related to an observable property of the system which is the curl of the electronic paramagnetic current density of the system (see Section 2.2 on the relation between the vector potential and the current density).

To calculate $\gamma(C)$, we first recall that the electronic vector potential in Eq. (42) has cylindrical symmetry. Switching to cylindrical coordinates by introducing the cylindrical radius $r_{cyl} = (r \sin \vartheta)$, the $\varphi$-component of A reads:

$$A_{\varphi, \text{cyl}}(r_{\text{cyl}}, z) = \frac{Z^2 r_{\text{cyl}} e^{Z \sqrt{r_{\text{cyl}}^2 + z^2}}}{Z^2 r_{\text{cyl}}^2 e^{Z \sqrt{r_{\text{cyl}}^2 + z^2}} + 64} \quad (47)$$

Then, we choose to calculate the loop integral along a path C that encloses the z axis bearing in mind the "rotating" behaviour of the vector potential around the z-axis that we have pointed out when discussing Figure 2. We obtain:

$$\begin{aligned} \gamma_Z(r_{cyl}, z) &= \oint_C \mathbf{A} \cdot d\mathbf{r} = \int_0^{2\pi} A_{\varphi, \text{cyl}} r_{cyl} d\varphi \\ &= \frac{Z^2 r_{cyl}^2 e^{Z \sqrt{r_{cyl}^2 + z^2}}}{Z^2 r_{cyl}^2 e^{Z \sqrt{r_{cyl}^2 + z^2}} + 64} 2\pi \end{aligned} \quad (48)$$

Note that in Eq. (48) we have introduced a minus sign compared to Eq. (46), due to the difference in sign in the conventional definition of the connection, as done in Ref. [102].

The geometric phase $\gamma_Z(r_{cyl}, z)$ in Eq. (48) is path-dependent: when the radius or the nuclear charge go to zero, $\gamma$ vanishes. In turn, at large distances, $z \to \infty$ and/or $r_{cyl} \to \infty$, or for large nuclear charge, $Z \to \infty$, $\gamma$ approaches $2\pi$ [see Figures 4–6]. By virtue of Stokes' theorem, the value of $\gamma_Z(r_{cyl}, z)$ corresponds to the flux of the effective magnetic field $B = \nabla \times A$ [Eq. (44) and Figure 3], i.e. the curvature, through the surface bounded by C. Thus, it reaches an asymptotic value at large distances because then all flux is enclosed. Similarly, increasing the nuclear charge corresponds to concentrating all

the flux at the origin leading to a path-independent value. The limit $Z \to \infty$ is vaguely reminiscent of the $M \to \infty$ limit of the nuclear mass(es) in the electro-nuclear problem when invoking the Born–Oppenheimer adiabatic approximation,[102,109] typically $\pi$.

The value of $2\pi$ that we get here is mathematically a consequence of $A_\varphi$ being independent of $\varphi$. Physically, this means that the quantum system does not acquire a phase shift upon transportation around the z-axis, indicating that the path does not enclose non-analiticities or abrupt changes in the conditional wavefunction and that the limiting behaviour can be reached smoothly. In the case of the Z-dependence this should not surprise since our Hamiltonian is non-interacting and can therefore be trivially rescaled as $\hat{H}(Z, r_1, r_2) = Z^2 \hat{H}(Z = 1, Zr_1, Zr_2)$. The Z-dependence is in this sense fictitious and quantities such as the vector potential and the derived magnetic field can be trivially rescaled as $X(Z, r) = Z^2 X(Z = 1, Zr)$. As an illustration, we plot in Figure 7 the z-

<figure><img src="image_10.png" /><figcaption>Figure 4. Geometric phase [Eq. (48)] over $\pi$ as a function of the cylindrical radius, $r_{cyl}$, for $z = 0$ and nuclear charge $Z = 1$.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Figure 5. Geometric phase [Eq. (48)] over $\pi$ as a function of the z-axis, for $r_{cyl} = 1$ and nuclear charge $Z = 1$.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Figure 6. Geometric phase [Eq. (48)] over $\pi$ as a function of the nuclear charge $Z$, for $r_{cyl} = 1$ and $z = 0$.</figcaption></figure>
component of the effective magnetic field on the plane $z = 0$, which reads:

$$B_z(r_{cyl}, 0) = \frac{64 Z^2 e^{Zr_{cyl}} (2 + Zr_{cyl})}{(64 + e^{Zr_{cyl}} Z^2 r_{cyl}^2)^2} \quad (49)$$

and is the only non-vanishing component of the field on this plane. Note that the maximum intensity of $B_z$ is not on the nucleus, conferring a ring shape to the field.

In the case of an interacting Hamiltonian with a Z-dependence such as the He isoelectronic series, increasing the nuclear charge corresponds to gradually turning off the interaction.[29,110-112] It seems plausible that approaching the $Z \to \infty$ there would inherently change the nature of the quantum system and lead to non-analiticities in analogy to the $M \to \infty$ limit in the electro-nuclear case. This conjecture needs to be tested and will be the subject of future study.

## 5. Conclusions

To conclude, in this work, we have discussed the exact factorization formalism for the static Schrödinger equation of a purely electronic system and for a general eigenstate, introducing generalized local potentials $v_{kin}$ and $v_{N-1}$ which could in principle serve the description of excited states within density-functional theorie(s). Our definitions presume the gauge choice of real positive marginal wavefunction, which however has the remarkable advantage that the electronic vector potential A, stemming from the fact that the wavefunction is complex [Eq. (8)], is related in a straightforward way to the current density [Eq. (28)].

Previous work on conditional amplitudes and on the exact electron factorization already discussed the features of various contributions of the total effective scalar potential that are important to retain in the design of approximations in relation to orbital-free or Kohn-Sham density-functional theory. In this work, we focused on the vector potential of the theory, which is essential when describing current-carrying states. Since, so far, not much work has been devoted to the electronic vector potential, we found it interesting to analyze it in a model case

<figure><img src="image_13.png" /><figcaption>Figure 7. $B_z$ component of the effective magnetic field $B = \nabla \times A$ divided by $Z^2$ as a function of the scaled coordinate $Zr_{cyl}$ on the plane $z = 0$ (cylindrical coordinates). $B_z$ is the only non-vanishing component on this plane. In the inset, contour plot of $B_z(x, y, z = 0)$ to highlight the ring-shape of the field around the z-axis (Cartesian coordinates).</figcaption></figure>

of non-interacting electrons, highlighting its behavior and that of its curl (related to an effective magnetic field generated by the conditional amplitude on the marginal wavefunction).

As a next step, it seems interesting to study the electronic vector potential in the case of interacting electrons to see, for instance, whether the asymptotic value of $2\pi$ found here for the related geometric phase is a result of the lack of Coulomb correlation.

## Acknowledgements

This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 101104947.

## Conflict of Interests

The authors declare no conflict of interest.

## Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Keywords: exact factorization · density functional theory · scalar potential · vector potential · complex electronic states

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Manuscript received: February 5, 2024

Revised manuscript received: June 1, 2024

Accepted manuscript online: June 5, 2024

Version of record online: July 25, 2024