#  Solution of the two-center Dirac equation with 20-digits precision using
the finite-element technique

O. Kullie  Theoretical Physics, Institute for Physics, Department of
Mathematics and Natural Science, Universität Kassel, 34132 Kassel, Germany
<EMAIL_ADDRESS>S. Schiller  Institut für Experimentalphysik, Heinrich-
Heine-Universität Düsseldorf, 40225 Düsseldorf, Germany<EMAIL_ADDRESS>
###### Abstract

We present a precise fully relativistic numerical solution of the two-center
Coulomb problem. The special case of unit nuclear charges is relevant for the
accurate description of the ${\rm H}_{2}^{+}$ molecular ion and its
isotopologues, systems that are an active experimental topic. The computation
utilizes the 2-spinor minmax approach and the finite-element method. The
computed total energies have estimated fractional uncertainties of a few times
$10^{-20}$ for unit charges and a bond length of 2 atomic units. The
fractional uncertainty of the purely relativistic contribution is $1\times
10^{-17}$. The result is relevant for future precision experiments, whereas at
present the uncertainties arising from the quantum electrodynamic treatment of
the rovibrational transition frequencies are dominant.

##  I Introduction

There is currently considerable interest in precisely measuring the rotational
and vibrational transitions in the molecular hydrogen ions and comparing the
values with ab initio theory predictions Korobov and Karr (2021). Such
comparisons allow a series of applications: determination of mass ratios, of
nuclear charge radii, tests of wave mechanics, and search for fifth forces
Alighanbari  _et al._ (2020); Kortunov  _et al._ (2021); Patra  _et al._
(2020). Disregarding hyperfine structure contributions, the relativistic
contributions are the largest ones beyond the Schrödinger energy. They make an
important, easily measurable, contribution to rotational and vibrational
transition frequencies of the molecular hydrogen ions. For example, for the
overtone vibrational transition
$(v=0,L=0)\rightarrow(v^{\prime}=5,L^{\prime}=1)$ of ${\rm HD}^{+}$ this
contribution is $\simeq 4.0\,$ GHz, or $1.5\times 10^{-5}$ relative to the
transition frequency. Here $v,\,L$ are the vibrational and rotational quantum
numbers of the level. Today’s experimental uncertainty of this transition
frequency is of the order of 1 kHz, i.e. $2.5\times 10^{-7}$ of the
relativistic contribution , and approximately $1\times 10^{-11}$ relative to
the transition frequency itself Alighanbari  _et al._ . There are excellent
prospects for further reduction of the experimental uncertainty in the near
future: this would be achieved using techniques already demonstrated for the
precision spectroscopy of single atomic ions, that have reached uncertainties
below the $1\times 10^{-17}$ level (see Ref. Ludlow  _et al._ (2015) for a
review). Two aspects can be mentioned in this respect. First, the controlled
trapping of single molecular hydrogen ions has recently been demonstrated
Wellers  _et al._ (2021) and, second, the systematic shifts of vibrational
transitions have been analyzed theoretically and found to allow an uncertainty
at the level below $1\times 10^{-16}$ Schiller  _et al._ (2014). Thus, it is
clearly desirable to perform a highly precise theoretical evaluation of the
relativistic contribution.

The currently employed approach to deal with the dominant relativistic effects
is the perturbative evaluation of the Breit-Pauli Hamiltonian, with respect to
the electronic wave function computed in the approximation of fixed nuclear
charge centers Tsogbayar and Korobov (2006). This gives the relativistic shift
of the order of $\alpha^{2}$ compared to the nonrelativistic energy, the
latter being close to $-1$ atomic unit (a.u.). Beyond this, the shift of
relative order $\alpha^{4}$ can also be computed perturbatively, with an
appropriate formalism Korobov and Tsogbayar (2007). Finally, computations of
the contribution of the order of $\alpha^{6}$ have been available since the
late 1980s.

In order to verify and extend the perturbation results, here we perform a
high-precision numerical solution of the Dirac equation for the electron in
the field of two static positive charges. We reduce the uncertainty of the
relativistic shifts by a factor $>1\times 10^{7}$ compared to the best
previously published finite-element method (FEM) calculation, from $1\times
10^{-14}$ a.u. Kullie and Kolb (2001) to below $1\times 10^{-21}$ a.u.

The paper begins in Sec. II A with a brief introduction of the minimax
approach for finding solutions of the Dirac equation, circumventing the issues
found in other approaches. We also explain the iteration procedure and the
nonrelativistic limit (Sec. II B). The implementation is discussed in Secs. II
C and D. Section III presents the computational aspects, including convergence
of the FEM calculations and the treatment of the limiting cases of the
hydrogen atom and of the nonrelativistic molecular hydrogen ion. The main
results are contained in Sec. IV, viz. the relativistic shift as a function of
the fine-structure constant and as a function of distance between the charge
centers. Finally, Sec. V evaluates the consequences of the present treatment
of the relativistic shifts on transition frequencies already measured for the
molecular hydrogen ion.

##  II Method

We apply the method previously developed in the works of one of us (O.K.)
Kullie and Kolb (2001, 2003); Kullie  _et al._ (2004). A minmax principle is
used that is based on the elimination of the small component from the Dirac
equation, leading to a non-linear eigenvalue problem that is solved
iteratively. The main extension implemented is this work is the use of larger
FEM order, larger number of grid points, and the use of quadruple precision
(32 digits) in order to achieve a high numerical accuracy. Also, stringent
tests are performed that allow one to determine the uncertainty of the energy
values obtained in the numerical solution. Specifically, we obtain a highly
accurate result for the ${\rm H}_{2}^{+}$ system. We also show that the chosen
relativistic treatment has an efficiency approaching that of the solution of
the non-relativistic Schrödinger equation.

###  II.1 Concept

A solution of the one particle 4-spinor Dirac equation can be obtained from a
stationarity principle for the functional
$I=\langle\psi|\hat{H}|\psi\rangle-\varepsilon\langle\psi|\psi\rangle$ but one
cannot apply a variational minimum principle as for the Schrödinger equation.
However, there exists a minimum principle in the space of bound electronic
states. The minmax principle, see Dolbeault  _et al._ (2000a) (and references
therein) Talman. (1986), applies to the construction of the eigenvalues of an
operator $\hat{H}$ that has a gap in its continuous spectrum (here from
$-mc^{2}$ to $mc^{2}$) and that is unbounded from above and below. The
principle considers the subspace of positronic states, $F_{-}$ and the
subspace of electronic states, $F_{+}$ and requires a two-step search for
extrema. The sequence of minmax level energies is given by

$\lambda_{k}=\inf_{\stackrel{{\scriptstyle{\rm
dim}G=k}}{{G\mbox{\footnotesize{ subspace of
}}F_{+}\,\,\,}}}\sup_{\stackrel{{\scriptstyle\psi\neq|0\rangle}}{{\,\,\,\psi\in(G\oplus
F_{-})}}}\frac{\langle\psi\mid\hat{H}\mid\psi\rangle}{\langle\psi\mid\psi\rangle}\
,$ (1)

where $F_{+}\oplus F_{-}$ is an orthogonal decomposition of a well-chosen
space of smooth square integrable functions and
${\langle\psi\mid\hat{H}\mid\psi\rangle}/{\langle\psi\mid\psi\rangle}$ is the
Rayleigh quotient. It has been proven Dolbeault  _et al._ (2000a) that the
sequence of the minmax energies $\lambda_{k}$ equals the sequence of
eigenvalues of $\hat{H}$ in the interval $(-mc^{2},+mc^{2})$. The minmax
principle transforms the problem of finding a solution of the Dirac equation
to a minimization (infimum) problem. It guarantees a solution of the Dirac
equation in the space of the large component $\phi^{+}$. The spectrum consists
only of positive eigenvalues, i.e. the negative eigenvalues are eliminated and
the spectrum is free from spurios states Dolbeault  _et al._ (2000a); Talman.
(1986).

###  II.2 The minmax eigenvalue equation

The Dirac eigenvalue equation of the electron in a scalar potential $V$,
$\widehat{H}_{D}\,\psi=\varepsilon\psi$, with the 4-spinor $\psi$, can be
written in the form

$\displaystyle\left(\begin{array}[]{cc}V&\hat{L}^{\dagger}\\\
\hat{L}&V-2mc^{2}\end{array}\right)\left(\begin{array}[]{c}\phi_{+}\\\
\phi_{-}\end{array}\right)=\varepsilon\left(\begin{array}[]{c}\phi_{+}\\\
\phi_{-}\end{array}\right)\ ,$ (8)

where we introduced the 2-spinors $\phi_{+}$ and $\phi_{-}$ for the large and
small component of $\psi$, respectively.
$\hat{L}=-i\,c\,\hbar\,{\bf{\sigma}}\cdot\nabla$,
${\bf\sigma}=\sum_{k=1}^{3}\sigma_{k}\,e_{k}$, where $\sigma_{k}$ are the
Pauli matrices. $\varepsilon$ is the eigenenergy that in the non-relativistic
limit corresponds to the eigenenergy of the Schrödinger equation. Rather than
solving Eq. (8) directly, or operating with the functional $I$ above, we
proceed to reduce the 4-spinor treatment to a 2-spinor treatment. First, we
eliminate the small component $\phi_{-}$ from Eq. (8), obtaining the
differential ("strong") form

$\hat{L}^{\dagger}\left(\frac{\hat{L}\,\phi_{+}}{\varepsilon+2mc^{2}-V}\right)=(\varepsilon-V)\,\phi_{+}\
.$ (9)

In addition, we turn to a "weak" (integral) formulation that provides a good
efficiency for FEM with large finite-element basis sets. It is obtained by
multiplying both sides in Eq. (9) with $\phi_{+}^{\dagger}$ and integrating
over the electron’s coordinate space Dolbeault  _et al._ (2000a):

$\int\frac{|\hat{L}\phi_{+}|^{2}}{\varepsilon+2mc^{2}-V}dr^{3}=\int(\varepsilon-V)\,|\phi_{+}|^{2}dr^{3}\
.$ (10)

We now apply the minmax principle and seek the minimum value of $\varepsilon$.
We expand the 2-spinor $\phi_{+}$ over a set of basis functions with unknown
coefficients. This set should be as large (complete) as possible. The vector
of expansion coefficients will be denoted by ${\overline{\phi}}_{+}$.
Variation of Eq. (10) with respect to the unknown coefficients in combination
with the requirement that $\varepsilon$ is stationary with respect to all
coefficients leads to a matrix equation that determines the coefficients. It
reads

${\bf
M}(\varepsilon)\,{\overline{\phi}}_{+}=\varepsilon\,{\overline{\phi}}_{+}\ .$
(11)

Thus, the original 4-spinor Dirac equation has been transformed into a
Schrödinger-like equation for the 2-spinor $\phi_{+}$, but where the effective
"Hamiltonian" ${\bf M}$ is eigenvalue-dependent. The equation is nonlinear in
the eigenvalue $\varepsilon$ and therefore has to be solved by iteration. It
can be shown that the solutions of this equation minimize the Rayleigh
quotient over all electronic bound states of $\hat{H}_{D}$ Dolbeault  _et al._
(2000b).

It has been shown that an efficient approach consists of expanding the left-
hand side of Eq. (10) in a series Dolbeault  _et al._ (2000a), as follows. We
start with an approximate value $\varepsilon^{0}$ of an eigenvalue
$\varepsilon$. For the iteration $j+1$ $(j=0,...,j_{\rm max})$ we expand the
left-hand side as Kullie (2004):

$\displaystyle\int\frac{|\hat{L}\phi_{+}|^{2}}{\varepsilon^{j}+2mc^{2}-V}dr^{3}=\int\frac{|\hat{L}\phi{+}|^{2}}{\underbrace{(\varepsilon_{0}+2mc^{2}-V)}_{g(\varepsilon_{0})}+\Delta\varepsilon^{j}}dr^{3}=$
$\displaystyle\int\frac{|\hat{L}\phi_{+}|^{2}}{g(\varepsilon_{0})}dr^{3}+\sum_{k=1}^{k_{\rm
max}}(-\Delta\varepsilon^{j})^{k}\int\frac{|\hat{L}\phi_{+}|^{2}}{g(\varepsilon_{0})^{k+1}}dr^{3}\
,$

with $\Delta\varepsilon^{j}=\varepsilon^{j}-\varepsilon_{0}$. The series
expansion has the advantage that the iteration procedure reduces to solving
successive eigenvalue problems. At iteration ${j+1}$ one computes the updated
global matrix corresponding to the expression (II.2), ${\bf
M^{\prime}}(\varepsilon_{0},\Delta\varepsilon^{j})$. One then solves the
conventional eigenvalue problem ${\bf
M^{\prime}}(\varepsilon_{0},\Delta\varepsilon^{j}){\overline{\phi}}_{+}=\epsilon^{j+1}{\overline{\phi}}_{+}$.
This is the computationally heaviest and therefore longest part of the
numerical procedure. Another advantage of the series expansion is that the
matrix elements for the individual terms in (II.2) need to be computed just
once for a given grid, at the beginning of the iteration. For explicit
expressions of the elements of the matrix Hamiltonian ${\bf M^{\prime}}$ we
refer to Ref. Kullie (2004).

The matrix equation is solved by an iterative method with a Cholesky
decomposition Heinemann (1987). In our method only an open boundary condition
is implemented so far, which has been found to work well.

The iteration process is stopped at $j={j}_{\rm max}$ or when a required
accuracy between successive iterations is reached. The series converges
quickly and for atoms with small nuclear charge only $k_{\rm max}=3$ to $4$
terms are needed. Typically, a small number of iterations is sufficient
$j_{\rm max}=3$ for the $Z=1$ case. For large-$Z$ nuclei, usually $k_{\rm
max}=4$ to $9$ terms are sufficient and $j_{\rm max}=3-7$ remains small or
moderate. For the nonrelativistic case $c\longrightarrow\infty$ the $k$
expansion is unnecessary, see below.

In the FEM approach, one usually performs the computation for a series of
grids with increasing number of elements and thus increasing number of basis
functions. When one moves from one grid to the next finer grid, one uses as
new start value $\varepsilon^{0}$ the solution $\varepsilon^{j_{\rm max}+1}$
found in the previous grid.

The 2-spinor formulation exhibits major advantages compared to the 4-spinor
formulation: the number of matrix elements to be computed is a factor 3
smaller and the solution of the matrix (by inverse vector iteration) requires
a factor of 4 fewer operations. The reduced size of the problem enhances the
computational performance and allows one to tackle larger problems Kullie  _et
al._ (2004); Kullie (2004).

Note that Eq. (10) can be written in the form

$\int{\frac{c^{-2}|{\hat{L}}\phi^{+}|^{2}}{(2m+(\varepsilon-V)/c^{2})}}=\int(\varepsilon-V)\,|\phi_{+}|^{2}dr^{3}\
.$ (13)

Therefore, in the nonrelativistic limit ($c\to\infty$), Eq. (10) turns into
the Schrödinger equation, considering that $\hat{L}$ is proportional to $c$.
Thus, we recognize that Eq. (10) exhibits similar properties to the
Schrödinger equation. In practice, we calculate the nonrelativistic values by
setting $c$ to a large number, $c\geq 10^{15}(\alpha^{2}\leq 10^{-30})$. In
this limit the small component $\phi_{-}$ becomes zero and the two components
of $\phi_{+}$ are then identical. The possibility of computing the
nonrelativistic energy value in this way (i.e. using the same numerical
procedures) leads to an important advantage: by subtracting it from the value
for finite $c$ we can extract the relativistic shift with a better accuracy
than the accuracy of the total (nonrelativistic plus relativistic shift)
energy (see Sec. III).

We showed in previous work Kullie  _et al._ (2004); Zhang  _et al._ (2004)
that in the weak formulation the 2-spinor fully relativistic FEM approach to
the two-center Coulomb problem is numerically better behaved than the
numerical solution of the 4-spinor Dirac equation. One finds some very
desirable behaviors: the energy values converge from above with increasing
grid size (finer subdivisions) and do not show the typical convergence from
below or oscillatory convergence of the 4-spinor Dirac equation. This is a
consequence of the elimination of the small component $\phi_{-}$, which
effectively projects the problem onto electronic states and leads to a second-
order differential operator bounded from below.

###  II.3 Implementation

The Dirac Hamiltonian for a single particle of mass $m$ in a two-center
potential $V$ is

$\displaystyle\widehat{H}_{D}=c\,\widehat{\alpha}\cdot\widehat{\bf{p}}+mc^{2}\widehat{\beta}+V,$
(14)
$\displaystyle\mbox{}V=-\sum_{l=1}^{2}\frac{\hbar\,c\,\alpha\,Z_{l}}{|{\bf
r}-{\bf R}_{l}|}\ .$

$Z_{1},Z_{2}$ are the charges of the two nuclei in units of the elementary
charge, $\widehat{\bf\alpha}$ and $\widehat{\beta}$ the usual Dirac matrices,
${\bf r}$ is the position of the electron, $\widehat{\bf{p}}$ is the momentum
operator, and ${\bf R}_{l}$ are the positions of the nuclei. $\alpha$ is the
fine-structure constant. Alternatively, if atomic units are employed,

$\displaystyle\widehat{H}_{D}$ $\displaystyle=$ $\displaystyle
mc^{2}\alpha^{2}\left(\alpha^{-1}\widehat{\alpha}\cdot\widehat{\bf{p^{\prime}}}+\alpha^{-2}\widehat{\beta}+V^{\prime}\right)\
,$ (15) $\displaystyle\mbox{}V^{\prime}$ $\displaystyle=$
$\displaystyle-\sum_{l=1}^{2}\frac{Z_{l}}{|{\bf r}^{\prime}-{\bf
R^{\prime}}_{l}|}\ .$

The primed quantities correspond to the case when coordinates and momenta are
in atomic units. $m\alpha^{2}c^{2}$ is the atomic unit of energy.

The nonrelativistic energy is found from the difference between the total
energy and the rest-mass energy $m\,c^{2}$, in the limit $c\rightarrow\infty$
of Eq. (14), or equivalently, in the limit
$\alpha=e^{2}/(4\pi\epsilon_{0}\hbar\,c)\rightarrow 0$ of Eq. (15). In both
cases, the product $c\,\alpha$, i.e. the potential energy $V,V^{\prime}$ and
the atomic energy unit, are to be kept constant. In the following, often $c$
and $\alpha^{-1}$ are used synonomously; they are equal in an appropriate
system of units. For the atomic case ($Z_{2}=0$), we consider different values
of $Z_{1}$, in particular $Z_{1}\gg 1$; see below.

For the two-center case one has axial symmetry around the internuclear axis
(the $z$-axis) and favorably uses prolate spheroidal (elliptic spheroidal)
coordinates $\xi$ and $\eta$,

$\displaystyle
x=\frac{R}{2}u(\xi,\eta)\,\cos\varphi,\,y=\frac{R}{2}u(\xi,\eta)\,\sin\varphi,$
(16) $\displaystyle z=\frac{R}{2}\xi\cdot\eta,\mbox{ where
}u(\xi,\eta)=\sqrt{(\xi^{2}-1)(1-\eta^{2})}\,\ $

and $R$ is the inter-nuclear distance in atomic units. The electron’s angular
coordinate is $\varphi$. The distances between the electron and the nuclei are

$r_{1}=(\xi+\eta)\frac{R}{2},\quad r_{2}=(\xi-\eta)\frac{R}{2}\ .$ (17)

The Coulomb singularity of point nuclei causes a singular behavior of the
relativistic solutions at the position of the nuclei of the form
$r^{-1+\gamma_{l,\kappa}}_{l}$, with
$\gamma_{l,\kappa}=\sqrt{\kappa^{2}-{(\alpha Z_{l})^{2}}}$ and
$\kappa=|j_{z}|+\frac{1}{2},\,l=1,2$. This is well-known from atomic
calculations Yang  _et al._ (1993); Düsterhöft  _et al._ (1994). Thus, further
singular coordinate transformations (whose back transform is non-analytic) are
needed to take care of this issue Kullie and Kolb (2001); Kullie (2004);
Kullie and Kolb (2003). The transformation from $\xi,\eta$ to $s,t$ reads

$\displaystyle\xi$ $\displaystyle=$ $\displaystyle
c_{\nu}\sinh^{\nu}(s/2)\equiv
1+c_{1}\sinh^{\nu}(s/2)+c_{2}\sinh^{(\nu+2)}(s/2)\,\cdots,$ (18)
$\displaystyle\eta$ $\displaystyle=$ $\displaystyle
c_{\nu}\sinh^{\nu}(t/2)\equiv
1-c_{1}\sin^{\nu}(t/2)+c_{2}\sin^{(\nu+2)}(t/2)-\,\cdots;$ $\displaystyle
0\leq s<\infty\quad 0\leq t\leq\pi$ $\displaystyle\mbox{for
}\nu=2,4,6,8,10,\mbox{ with }c_{i}=0\quad\mbox{for }i>\frac{\nu}{2},$

which can be written in a differential closed form:

$\displaystyle\frac{d\xi}{ds}$ $\displaystyle=$ $\displaystyle
D_{n}\sinh(s/2),\,\frac{d\eta}{dt}=D_{n}\sin(t/2),\,D_{n}=\frac{(2n+1)!}{2^{2n}n!^{2}},n=\frac{\nu}{2}-1=0,1,2,\cdots.$

The transformation regularizes the singularities at the nuclei by increasing
the point density in the inner region. The higher $\nu$, the denser the points
near the nuclei to ensure a better approximation of the wave function. The
coefficients and the details are given in refs. Kullie and Kolb (2001), Kullie
and Kolb (2003), As a result of this transformation one can use a square grid
over $s$ and $t$.

A high value of $\nu$ (e.g. 6, 8) is needed for grids with a large number of
points, which in turn enable a higher convergence order $q$ for the energy and
the full utilization of a FEM approximation of the order $p$. See below and
Refs. Kullie and Kolb (2003); Kullie  _et al._ (2004); Kullie (2004).

Because of axial symmetry, the angular dependence is treated analytically by
the ansatz:

$\displaystyle\psi$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\phi_{+}(s,t,\varphi)\\\
\phi_{-}(s,t,\varphi)\end{array}\right)\\!=\\!\left(\begin{array}[]{c}\phi^{1}(s,t)\cdot
e^{i(j_{z}-1/2)\cdot\varphi}\\\ \phi^{2}(s,t)\cdot
e^{i(j_{z}+1/2)\cdot\varphi}\\\ i\phi^{3}(s,t)\cdot
e^{i(j_{z}-1/2)\cdot\varphi}\\\ i\phi^{4}(s,t)\cdot
e^{i(j_{z}+1/2)\cdot\varphi}\\\ \end{array}\right)\ .$ (25)

The wave function $\psi$ is an eigenstate of the total angular momentum, and
the good quantum number $j_{z}$ is the $z$-component of the total angular
momentum.

In the present FEM treatment, the definition domain of $s,t$ is subdivided
into triangular elements $e$. Each component ${k}$ of the relativistic wave
function is approximated as

$\displaystyle\phi^{k}(s,t)=G^{{k}}(s,t)\sum_{{e}}\sum_{i}^{n}d^{k,e}_{i}N_{i}^{{k,e}}(s,t)\
,$ (26)

where $G^{k}(s,t)$ are global functions, and the sums run over all elements
$e$ of the grid and over all $n$ nodal points $(s_{i},t_{i})$ of each element.
$G^{k}(s_{i},t_{i})\,d^{k,e}_{i}$ is the value of the wave function at nodal
point $i$. The shape functions $N_{i}^{e}(s,t)$ are zero outside the element
$e$. Inside they are complete polynomials of the order of $p$ in $s,t$ [22] ,
implementing a Lagrange-form interpolation. The values of the coefficients of
the polynomials are determined by the conditions that
$N_{i}^{{k,e}}(s_{j},t_{j})=\delta_{ij}$ for all nodal points $i,j$ inside the
element. In our FEM implementation we use triangular Lagrangian-type elements
with equidistant point distribution. The functions $G^{k}(s,t)$ account for
the global behavior of the wave function, where $G_{1}^{k}(s,t)$ represents
the angular momentum dependence and $G_{2}^{k}(s,t)$ expresses the singular
behavior at the two nuclei. They are given by

$\displaystyle G^{{k}}(s,t)=G^{{k}}_{1}(s,t)\cdot G_{2}(s,t)\ ,$
$\displaystyle
G_{1}^{{k}}(s,t)=((\xi^{2}-1)(1-\eta^{2}))^{\frac{m_{k}}{2}}=R_{\perp}^{\frac{m_{k}}{2}}\
,$ (27) $\displaystyle\quad m_{1,3}=j_{z}-1/2,\quad m_{2,4}=j_{z}+1/2\ ,$
$\displaystyle G_{2}(s,t)=r_{1}^{-1+\gamma_{1,\kappa}}\cdot
r_{2}^{-1+\gamma_{2,\kappa}}\ .$ (28)

Here, $R_{\perp}$ is the (perpendicular) distance to the internuclear axis.
For larger $Z$, $\gamma_{l,k}$ becomes smaller and the singular behavior of
the wavefunction is stronger, hence the convergence is less efficient. Indeed,
as can be seen for the example in the hydrogenic atoms (Table 3), the
numerical precision decreases for higher atomic numbers. Still, introducing
the singular coordinate transformation given by Eq. (II.3) guarantees a high
convergence order, because it allows one to describe the singularity of the
wave function near the nuclei more accurately Yang  _et al._ (1993); Yang
(1991).

##  III Computational aspects

###  III.1 Generalities

We compute the lowest-energy state of energy $\varepsilon_{1(1/2)g}$, i.e.
with $j_{z}=1/2$, and gerade symmetry $g$. The notation of the corresponding
nonrelativistic state is ${1\sigma_{g}}$. We abbreviate the notation of the
(exact) energy in atomic units by the short-hand $E_{\rm rel}$.

In all calculations we use the FEM polynomial order $p=10$. We run the
calculation for different values of $\nu$ and size of the grid in order to
achieve the best convergence. The size is defined by the size of the largest
ellipse $\xi=\xi_{\rm max}=const.$ containing the grid elements. The size of
the grid can alternatively be defined by $D_{\rm max}(\xi_{\rm max})$, defined
as the distance between one of the centers to a point on the outermost ellipse
$\xi_{max}$, where this distance is perpendicular to the line between the two
centers Zhang  _et al._ (2004). $D_{\rm max}$ values (given in atomic units)
of approximately $30$ \- $50$ are used. This should be compared to the most
relevant value of the internuclear distance, $R=2$, the approximate
equilibrium bond length of the ${\rm H}_{2}^{+}$ molecule. From this
comparison we see that the space around the nuclei considered in the
calculation is large compared to the internuclear distance.

The largest number of grid points we were able to reasonably work with was
$N=32\,761$. As we show below, for the system ${\rm H}_{2}^{+}$ a fractional
uncertainty of the energy of the order of $10^{-20}$ is thereby achieved. For
the relativistic shift, the absolute uncertainty is of the order of $10^{-22}$
atomic units, where we profit from an error cancellation concerning the non-
relativistic energy $E_{\rm nrel}$. Thus, this is a high-performance
calculation. The computing time for the grid sequence up to $648/32761$ (see
Table 1) was 15 - 20 core-hours on a supercomputer. The code is not
parallelized and runs on one core.

$N_{e}/N$ |  Relativistic, $E_{\rm rel}$ |  Nonrelativistic, $E_{\rm nrel}$ |  rel. shift $\Delta E_{\rm rel}$ $(10^{-6})$
---|---|---|---
8/441 |  -1.102281044470406312209 |  -1.1022736006131289397101 |  -7.44385727737249928
32/1681 |  -1.102641575567778026666 |  -1.1026342090292764774058 |  -7.36653850154926045
72/3721 |  -1.102641581012739732920 |  -1.1026342144751026384076 |  -7.36653763709451296
128/6561 |  -1.10264158103240750588 |  -1.1026342144947767797576 |  -7.36653763072611806
200/10201 |  -1.102641581032576082616 |  -1.1026342144949453798398 |  -7.36653763070277600
288/14641 |  -1.102641581032577138209 |  -1.1026342144949464356152 |  -7.36653763070259364
392/19881 |  -1.102641581032577162741 |  -1.1026342144949464601360 |  -7.36653763070260506
512/25921 |  -1.102641581032577163917 |  -1.1026342144949464613087 |  -7.36653763070260827
648/32761 |  -1.102641581032577164097 |  -1.1026342144949464614889 |  -7.36653763070260893
extrpl1 |  -1.102641581032577164118 |  -1.1026342144949464615095 |  -7.36653763070260901
extrpl2 |  | -1.1026342144949464615089689454 |  -7.36653763070260903
Table 1: Energies of ${\rm H}_{2}^{+}$ at $R=2$ and for
$\alpha^{-1}=137.035999084$. All values in atomic units. The calculations
utilize $\nu=6$ and $D_{\rm max}=50$. $N_{e},N$ are the numbers of the
elements and grid points respectively. Superscript 1 indicates values
extrapolated over the sequence $N_{e}$. Bold digits are sgnificant. Superscrip
2 is with $\nu=2$, and $D_{\rm max}=50$; the relativistic shift is computed
using the last relativistic energy value of column 2.

###  III.2 Convergence for the fully relativistic and the non-relativistic
case

We present in Table 1 the relativistic and nonrelativistic energy values
$E_{\rm rel}(N)$, $E_{\rm nrel}(N)$ of ${\rm H}_{2}^{+}$ in grids of different
point number $N$, including their extrapolations to an infinite grid Zhang
_et al._ (2004). The relativistic shift $\Delta E_{\rm rel}$ (fourth column)
is the difference $E_{\rm rel}-E_{\rm nrel}$.

Performing an extrapolation is legitimate if the energy values exhibit a
regular dependence on the grid size. This is the case for this problem if
relevant parameters are chosen judiciously. For the computation reported in
Table 1 indeed one notices that with increasing number of grid points (finer
subdivision) the accuracy increases and convergence to the true value is from
above, not only for the nonrelativistic case, but also for the relativistic
case. This behavior is known from previous treatments of the 2-spinor approach
and is a major advantage of this method. As a guide to the eye, bold digits
show the significant digits, but the actual uncertainty may be smaller than
one unit of the last bold digit.

To test the convergence and confirm the accuracy of the result, we present in
Fig. 1 a log-log plot of the errors $\delta E(N)$ of the energies and of the
relativistic shift computed for a particular grid $N$, with respect to the
extrapolated value. As can be seen in the red and blue line of the left panel,
the convergence rate in the 2-spinor formulation is close to that of the
nonrelativistic Schrödinger equation Zhang  _et al._ (2004). This is the main
result of the minmax concept.

When we choose a suitably large value of $\nu$, the values converge extremely
rapidly. The mean convergence is approximately $\sim N^{-8}$ considering all
results from Table 1. The convergence rate even appears to increase for the
largest values of $N$ used.

The accuracy in the FEM using polynomials of the order of $p$ typically scales
as $\sim N^{-p}$ with grid size. In our case, $N$ can be made sufficiently
large, but the singular behavior near the centers significantly reduces the
efficiency of the FEM approximation. This can be controlled by the singular
coordinates transformation given by Eq. II.3, as already mentioned, but it is
necessary to adapt the value of $\nu$ to the grid size. This issue is not so
important in the nonrelativistic case because the wave function is finite at
the nuclei. To illustrate this we performed calculations for the same grids
given in table 1, but with a lower value $\nu=4$ and present the corresponding
log-log plot in Fig. 1 (right).

For small grids, the error evolution is similar for both the relativistic and
nonrelativistic values; here the distribution of the grid points is balanced
between inner and outer regions. Then, for a larger number of grid points, the
error caused by the relativistic singularity (Eq. 28) becomes larger. This is
because for small $\nu$ the points’ density in the inner region is not
sufficient to reproduce the singular behavior at the nuclei. The error of the
FEM solution then does not decrease any more strongly with increasing grid
size. In contrast, in the nonrelativistic case the error continues to decrease
strongly with increasing $N$.

Figure 1:  (Color online) Convergence behavior of the relativistic (red
square) and nonrelativistic (blue, left triangle) energies, and of the
relativistic shift (green circle), as a function of the number of grid points,
$N$. $\delta E=E(N)-E$ or $\delta E=\Delta E_{\rm rel}(N)-\Delta E_{\rm rel}$
is the deviation of the energy from the extrapolated value, as given in Table
1. Two cases are considered. Left: $D_{\rm max}=50$, $\nu=6$, good
convergence. Right: $D_{\rm max}=70$, $\nu=4$, unsatisfactory convergence for
the relativistic solution. Left, black triangles: nonrelativistic calculation
with $\nu=2$ and $D_{\rm max}=50$.

Returning to a larger value $\nu=6$ (Fig. 1, left), when the wave function is
better approximated near the charge centers, the error due to the approximate
treatment of the relativistic singularity (Eq. 28) becomes smaller also for
higher grid point numbers. Here, a suitable distribution of the grid points
between the inner and outer regions of the treated domain is achieved Kullie
and Kolb (2001); Kullie (2004).

In the relativistic shift, one profits from an error cancellation: the error
reduces by two or more orders, as the comparison of the green and red lines in
Fig. 1, left) evidences. The scaling of the error of the relativistic shift
with grid point number exhibits an exponent of approximately $q\simeq 8.5<p$.
Determining the reason for this is beyond the scope of this work.

We take as the uncertainty of the extrapolated value the difference between
the extrapolated value and the value computed for the most dense grid. Then,
the achieved uncertainties are as follows. For $E_{\rm rel}$: $\simeq 2\times
10^{-20}$ atomic units, or $\simeq 2\times 10^{-20}$ in fractional terms. The
same holds for the nonrelativistic energy (with the same $\nu$). For the
relativistic shift $\Delta E_{\rm rel}=E_{\rm rel}-E_{\rm nrel}$: $\simeq
1\times 10^{-22}$ atomic units or $\simeq 1\times 10^{-17}$ in fractional
terms.

To show that the precise value of $D_{\rm max}$ is not crucial we present in
Table 2 the values of the relativistic shifts for several grid extensions
$D_{\rm max}$, and where the grid point number is the largest one reported in
Table 1. As expected, the shifts stay stable over a large range of $D_{\rm
max}$. A larger number of grid points requires larger $D_{\rm max}$ to avoid
truncation errors. However, a larger value $\nu=6,8...$ condenses the points
in the inner region and dilutes them in the outer region. To counter this
effect one chooses a smaller $D_{\rm max}$ for larger $\nu$ for the same
system and grids sequence. Note that the variation of $D_{\rm max}$ mainly
affects the outer region; therefore, due to the error cancellation, the
relativistic shift is less sensitive to $D_{\rm max}$.

Finally, we consider the nonrelativstic calculation. Because the wave function
is not singular at the nuclei, $\nu=2$ is sufficient. This maintains the
uniform distribution of the grid points. The wave function error is determined
by the polynomial approximation. As Fig. 1 (black triangles) shows, the
convergence order is approximately $q=9.5$ for large grid point number, almost
equal to the polynomial order $p=10$. Test calculations similar to those in
Table 2 showed that the most accurate result is found for $D_{\rm max}\simeq
50$, yielding the result shown in Table 1 (last line).

$\begin{array}[]{ccc}D_{\rm max}&{\rm value\ for\ densest\ grid}&{\rm
extrapolated\ value}\\\ \hline\cr
30&-7.3665376307026089560&{\bf-7.36653763070260}899\\\
40&-7.3665376307026089666&{\bf-7.3665376307026090}4\\\
50&-7.3665376307026089298&{\bf-7.3665376307026090}1\\\
60&-7.3665376307026088816&{\bf-7.36653763070260}898\\\
70&-7.3665376307026088322&{\bf-7.36653763070260}894\\\ \hline\cr\end{array}$
---
Table 2: Dependence of the relativistic shift $\Delta E_{\rm rel}$ (in
$10^{-6}$ atomic units) at $R=2$ on the domain size $D_{\rm max}$. The second
column is the result for the densest grid, with $648/32761$ elements/points.
The third column is the extrapolated value for infinite point number. All
evaluations were performed with $\nu=6$. Bold digits are significant

###  III.3 Test of the numerical procedure on the hydrogen-like ions

We can test our FEM procedures on the hydrogen-like ions, since the exact
solution of the Dirac equation is available. The energy levels are given by
(in explicit units, excluding the rest-mass energy)

$E_{jn}=-m_{\rm
e}c^{2}\left[1-\left(1+\left(\frac{Z\alpha}{(n-j-1/2)+\sqrt{(j+1/2)^{2}-(Z\alpha)^{2}}}\right)^{2}\right)^{-1/2}\right]\
.$ (29)

The relativistic energy shift in atomic units is $\Delta E_{\rm
rel}=E_{jn}/(1\,{\rm a.u.})-E_{\rm nrel}=(m_{\rm
e}c^{2}\alpha^{2})^{-1}E_{jn}+Z^{2}/(2\,n^{2})$. Here, we consider the ground
state, $n=1,j=1/2$.

The FEM values were computed by setting the second center to be a dummy center
with $Z_{2}=0$. In Table 3, we show the results for various values of $Z_{1}$,
each obtained with 10th-order FEM, with the maximum number of grid points
computed being 32761, and extrapolated.

The agreement with the exact results is excellent. For the case of nuclear
charge $Z_{1}=1$ the difference of the relativistic shifts is of the order of
$10^{-24}\,$a.u., or $5\times 10^{-19}$ fractionally. This value confirms our
uncertainty estimate given above for the two-center system $\hbox{\rm
H}_{2}^{+}$. The last two entries in the table compare the result of the
energies for the relativistic and nonrelativistic cases. We see that our FEM
is almost as accurate in the relativistic case as in the nonrelativistic;
there is only a factor 2 loss in accuracy.

Note that unlike what is obviously the case in Eq. (29), in the FEM
computations of both the hydrogenic ions and the two-center problem, varying
$Z_{1}$ or $Z_{1},Z_{2}$ simultaneously is not equivalent to varying $\alpha$
(such a variation is reported in Table 11 below). This is because our
computations of the $\alpha$-dependence are based on Eq. (15) with fixed
atomic energy unit and atomic distance unit.

$Z$ |  $\Delta E_{\rm rel}$, exact (a.u.) |  $\Delta E_{\rm rel}$, FEM (a.u.) | difference (a.u.) | note
---|---|---|---|---
$\alpha^{-1}=137.0359895$ |
1 | -6.65659748374605054203 $\times 10^{-6}$ |  -6.656597483746050539 $\times 10^{-6}$ | $-3.0\times 10^{-24}$ | 1
2 | -1.06514068278487728906 $\times 10^{-4}$ |  -1.0651406827848772881 $\times 10^{-4}$ | $-1.0\times 10^{-22}$ | 1
$\alpha^{-1}=137.035999084$ |
1 | -6.6565965526253642790$\times 10^{-6}$ |  -6.656596552625364281$\times 10^{-6}$ | $-2.0\times 10^{-24}$ | 2
2 | -1.0651405337817627608$\times 10^{-4}$ |  -1.065140533781762761$\times 10^{-4}$ | $2.2\times 10^{-22}$ | 3
10 | -6.674201689468916918 $\times 10^{-2}$ |  -6.674201689468916913 $\times 10^{-2}$ | $-4.8\times 10^{-20}$ | 3
20 | -1.076523210794734716 |  -1.076523210794734713 | $-3.1\times 10^{-18}$ | 3
30 | -5.524906318343685119 |  -5.52490631834368509 | $-3.0\times 10^{-17}$ | 3
|  $E_{\rm rel}$, exact |  $E_{\rm rel}$, FEM | difference |
30 | $-455.52490631834368512$ |  -455.5249063183436834 | $-1.7\times 10^{-15}$ | 3
|  $E_{\rm\bf nrel}$, exact |  $E_{\rm\bf nrel}$, FEM | difference |
30 |  $-450$ |  -449.999999999999998309 | $-6.9\times 10^{-16}$ | 3
Table 3: Comparison of the extrapolated FEM numerical results for the
relativistic shift of hydrogen-like ions with the exact result. $Z$ is the
nuclear charge. Exact values are computed using Eq. (29). The cases $Z=1,2$
are computed with two different values of $c$. The first value used is for
ease of comparison with other values reported in the literature. The next-to-
last entry shows the actual relativistic energies. The last entry reports a
nonrelativistic calculation. FEM parameters used: Superscript 1: $D_{\rm
max}=50$, $\nu=4$; Superscript 2: $D_{\rm max}=40$, $\nu=6$; Superscript 3:
$D_{\rm max}=30$, $\nu=6$.

###  III.4 Test of the numerical procedure on an exactly known excited
nonrelativistic molecular state

It is little-known that there exist exact solutions of the nonrelativistic
two-center problem for particular combinations of charge values
$Z_{1},\,Z_{2}$ and distances $R$ Demkov (1968). These solutions are of great
interest, because performing the corresponding FEM calculation allows one to
verify the extrapolation procedure and the uncertainty estimate.
Unfortunately, these exact solutions are not the electronic ground states but
excited states. We have treated the case ($Z_{1}=3$, $Z_{2}=2$,
$R=\sqrt{15}$), whose $4s\sigma$ state has the exact nonrelativistic
electronic energy $E_{\rm nrel}=-(1/2)$ a.u. The FEM computation is very
cumbersome, because this state is the 21st $j_{z}=1/2$ state in order of
increasing energy, and all intermediate states have to be calculated before
treating the state of interest. As for other calculations in the present
study, the energy values are iterated until they remain stable at the level of
$\sim 10^{-27}-10^{-30}$ a.u.

The result is shown in Table 4. The difference of the extrapolated value
relative to the exact value is $5\times 10^{-20}$ a.u. This agreement obtained
for a system having substantially larger squared charge than the ${\rm
H}_{2}^{+}$ system, comparatively large ground state energy
$-5.01691106677841796618$ a.u. (significant digits, extrapolated), and using a
moderate value of $D_{\rm max}$, gives us confidence that our uncertainty
estimates for the relativistic solution of ${\rm H}_{2}^{+}$ are reasonable.

$N_{e}/N$ | $E_{\rm nrel}$, FEM (a.u.) | difference (a.u.)
---|---|---
8/441 | -0.493598112780690785516 | $6.4\times 10^{-4}$
32/1681 | -0.499999685189264323528 | $3.2\times 10^{-7}$
72/3721 | -0.499999998831658333453 | $1.2\times 10^{-9}$
128/6561 | 0.499999999995025573402 | $5.\times 10^{-12}$
200/10201 | -0.499999999999921236179 | $7.9\times 10^{-14}$
288/14641 | -0.499999999999998240336 | $1.8\times 10^{-15}$
392/19881 | -0.499999999999999914507 | $8.5\times 10^{-17}$
512/25921 | -0.499999999999999994099 | $5.9\times 10^{-18}$
648/32761 | -0.499999999999999999158 | $8.4\times 10^{-19}$
extrpl | -0.4999999999999999999483 | $5.1\times 10^{-20}$
Table 4: FEM nonrelativistic energy of a particular excited state of the two-
center problem with $Z_{1}=3,Z_{2}=2$ and distance $R=\sqrt{15}$, that has the
exact energy $-(1/2)$ a.u. The third column shows the differences FEM minus
exact. The FEM computations were performed with $D_{\rm max}=50,\nu=4$.
$N_{e}/N$ is the number of the grid elements/points. The values are truncated.

##  IV Results

###  IV.1 Series expansion of the relativistic shift

We have computed the FEM relativistic energy shift at $R=2\,$a.u. for a set of
values of $c=\alpha^{-1}$, ranging from 5 to 1200. The shifts are reported in
Table 11. We fitted a series $\Delta E_{\rm rel}=\sum_{s=1}^{s_{\rm
max}}{d_{2s}\alpha^{2s}}$ to the FEM data. The best-fitting series is with
$s_{\rm max}=6$ and is reported in Table 5, together with the standard errors
of the best-fit coefficients. The resulting fractional deviations of the
fitted values from the FEM values are approximately $(1-10)\times 10^{-16}$,
substantially larger than the uncertainties of the FEM values. Therefore these
FEM uncertainties were not taken into account in the fit.

We also show in Table 5 the comparison of our best-fit series expansion with
what we believe is the most precise perturbation series. The coefficient for
the order $\alpha^{2}$ was recently recomputed by Korobov Korobov (2022). It
differs by $7\times 10^{-12}\,\hbox{\rm a.u.}$ from the value in the
supplemental material of Korobov (2018) Korobov (2018) (file
"tmph-2017-0313-File001.dat"). It is now in agreement with the FEM-computed
coefficient. We recognize that a main difference between the perturbation
result and FEM result is the coefficient of the order of $\alpha^{4}$. The
uncertainty of the perturbation coefficient value as calculated by Korobov is
not known. However, our result for the $\alpha^{6}$-order coefficient confirms
the value and uncertainty estimate of Mark and Becker Mark and Becker (1987).
In total, at $R=2$ a.u. the FEM result is approximately 0.3 kHz smaller than
the perturbation result.

Table 5:  Numerical series expansion of the FEM relativistic shift $\Delta
E_{\rm rel}$ and comparison with perturbation and variational theory results.
Column 2 contains the coefficients $d_{2s}$, $s=1,\ldots,6$, for $\hbox{\rm
H}_{2}^{+}$ at $R=2$ a.u. Columns 3 and 4 list the available best results of
perturbation and variational theory, respectively. Column 5 shows rounded
values of the coefficients for the exact solution of the hydrogen atom. The
uncertainties in column 2 are for the 95% confidence intervals. The
uncertainty in column 3, line 3 is one unit of the last digit. [a] Ref.
Korobov (2022); [b] Ref. Korobov and Tsogbayar (2007); [c] Ref. Mark and
Becker (1987).

###  IV.2 Electronic binding energy curve

The $R$-dependence of the Dirac energy for $\hbox{\rm H}_{2}^{+}$ has received
little attention after the early works by Luke et al. Luke  _et al._ (1969)
and Bishop Bishop (1977) because, after the experiment of Wing et al. in 1976
Wing  _et al._ (1976), for several decades no precision spectroscopy
experiments were performed that could challenge the theory. The work of
Howells and Kennedy is one of the few exceptions Howells and Kennedy (1990).
In the early 2000s, Korobov computed the $R$-dependent precise perturbation
theory in connection with the new generation of experiments that had started.
The result consists of the $\alpha^{2}$-coefficient Tsogbayar and Korobov
(2006), whose updated values have been provided by Korobov for this work, and
the $\alpha^{4}$-coefficient Korobov and Tsogbayar (2007); Korobov (2018). We
denote the relativistic shift computed from this data by $\Delta E_{\rm
rel,Korobov}$.

We have computed by FEM the relativistic shift for $R$-values from $R=0.05$ to
$R=5.0$ in steps of $0.05$, see Table 6. From the smallest to the largest $R$,
the absolute uncertainties vary from $7\times 10^{-20}$ to $1.2\times
10^{-23}$, i.e. from $7\times 10^{-16}$ to $3\times 10^{-18}$ in fractional
terms.

$R$ |  $\Delta E_{\rm rel}(R)$ | $R$ |  $\Delta E_{\rm rel}(R)$ | $R$ |  $\Delta E_{\rm rel}(R)$ | $R$ |  $\Delta E_{\rm rel}(R)$
---|---|---|---|---|---|---|---
0.05 | -102.205220874297353 | 1.30 | -12.448456768274240 | 2.55 | -5.97244880905323561 | 3.80 | -5.28072339944993510
0.10 | -94.2070961513458068 | 1.35 | -11.840587433312817 | 2.60 | -5.89682181016418189 | 3.85 | -5.28498620517162691
0.15 | -85.3131813342957982 | 1.40 | -11.287362392410170 | 2.65 | -5.82730252634455702 | 3.90 | -5.29060881471213234
0.20 | -76.6282196861760657 | 1.45 | -10.783014143550542 | 2.70 | -5.76349219079285695 | 3.95 | -5.29751088104089074
0.25 | -68.6084308516448949 | 1.50 | -10.322491660645794 | 2.75 | -5.70502398629843855 | 4.00 | -5.30561552816506309
0.30 | -61.4076071297781014 | 1.55 | -9.9013599726656259 | 2.80 | -5.65155994593625421 | 4.05 | -5.31484911635246100
0.35 | -55.0368465872734128 | 1.60 | -9.5157152038954672 | 2.85 | -5.60278818603095446 | 4.10 | -5.32514103071051427
0.40 | -49.4432261483497075 | 1.65 | -9.1621125180967957 | 2.90 | -5.55842043205063507 | 4.15 | -5.33642349106801577
0.45 | -44.5490384746501491 | 1.70 | -8.8375048531073833 | 2.95 | -5.51818980322643993 | 4.20 | -5.34863138128002584
0.50 | -40.2710819788872559 | 1.75 | -8.5391906977447463 | 3.00 | -5.48184882610666476 | 4.25 | -5.36170209623029187
0.55 | -36.5297231045446188 | 1.80 | -8.2647694632039667 | 3.05 | -5.44916765105193533 | 4.30 | -5.37557540494242218
0.60 | -33.2527185045808470 | 1.85 | -8.0121032479359465 | 3.10 | -5.41993244895295341 | 4.35 | -5.39019332833313161
0.65 | -30.3763935052245251 | 1.90 | -7.7792839978640807 | 3.15 | -5.39394396828117659 | 4.40 | -5.40550003025016080
0.70 | -27.8455359036944480 | 1.95 | -7.5646052307100206 | 3.20 | -5.37101623503049416 | 4.45 | -5.42144172053575057
0.75 | -25.6127144763335920 | 2.00 | -7.3665376307026090 | 3.25 | -5.35097538022926582 | 4.50 | -5.43796656894539199
0.80 | -23.6373874480319195 | 2.05 | -7.1837079333992589 | 3.30 | -5.33365858154336776 | 4.55 | -5.45502462883235896
0.85 | -21.8849832650076221 | 2.10 | -7.0148806141308055 | 3.35 | -5.31891310709145427 | 4.60 | -5.47256776958247562
0.90 | -20.3260390746755839 | 2.15 | -6.8589419712531499 | 3.40 | -5.30659545098689629 | 4.65 | -5.49054961685173189
0.95 | -18.9354315392755268 | 2.20 | -6.7148862598532203 | 3.45 | -5.29657055133523823 | 4.70 | -5.50892549972265814
1.00 | -17.6917086864986668 | 2.25 | -6.5818035851749282 | 3.50 | -5.28871108247578041 | 4.75 | -5.52765240395459001
1.05 | -16.5765189239048591 | 2.30 | -6.4588693097269201 | 3.55 | -5.28289681418178407 | 4.80 | -5.54668893055877092
1.10 | -15.5741278636581426 | 2.35 | -6.3453347653767748 | 3.60 | -5.27901403134357336 | 4.85 | -5.56599525898221531
1.15 | -14.6710118162609399 | 2.40 | -6.2405190930066166 | 3.65 | -5.27695500836772192 | 4.90 | -5.58553311423485584
1.20 | -13.8555168671494631 | 2.45 | -6.1438020585500913 | 3.70 | -5.27661753314668177 | 4.95 | -5.60526573734309895
1.25 | -13.1175733519842682 | 2.50 | -6.0546177163079877 | 3.75 | -5.27790447599794821 | 5.00 | -5.62515785855980919
Table 6: Relativistic shift $\Delta E_{\rm rel}$, in units $10^{-6}$ a.u., as
a function of internuclear distance. $\alpha^{-1}=137.035999084$. The
calculation is performed with $D_{\rm max}=35,\,\nu=8$ in the range $R=0.05$
\- $0.25$, $D_{\rm max}=40,\,\nu=8$ in the range $R=0.30$ \- $1.95$ and with
$D_{\rm max}=40,\,\nu=6$ for $R=2.0$ \- $5.0$, and in addition extrapolated to
infinite grid point density.

##  V Discussion and Conclusion

###  V.1 Comparison with other work

Historically, the first numerical solution of the Dirac equation for the two-
center problem was undertaken by Pavlik and Blinder Pavlik and Blinder (1967),
followed by Luke et al. Luke  _et al._ (1969) and by Müller et al. Müller  _et
al._ (1973). Two decades later, the precision had improved by five orders with
the work of Yang et al. Yang  _et al._ (1991). Another one order of
improvement followed in the next decade, reported by Kullie and Kolb Kullie
and Kolb (2001). In the two decades from that work until the present work no
other independent result was published with comparable or lower uncertainty,
to the best of our knowledge. Several studies of the Dirac equation were
undertaken, but they were not aimed towards record precision for the
$\hbox{\rm H}_{2}^{+}$ system. Table 7 presents some results on the total
energy, that were published in the last 25 years. For earlier listings of
results on the Dirac equation for $\hbox{\rm H}_{2}^{+}$, see e.g. Rutkowski
and Rutkowska Rutkowski and Rutkowska (1987), Sundholm Sundholm  _et al._
(1987), and Mironova et al. Mironova  _et al._ (2015). Table 8 instead focuses
on the relativistic shift, going back to the earliest works. We see that the
present result is not or only marginally in agreement with Refs. Kullie and
Kolb (2001); Korobov and Tsogbayar (2007); Mironova  _et al._ (2015), perhaps
because their uncertainties were underestimated.

Reference |  $E_{\rm rel}$
---|---
This work | -1.10264158103360758005
Mironova et al. 2015 Mironova  _et al._ (2015) | -1.1026415810330
Tupitsyn et al. 2014 Tupitsyn and Mironova (2014) | -1.1026415810330
Fillion-Gourdeau et al. 2012 Fillion-Gourdeau  _et al._ (2012) | -1.102641580782
Artemyev et al. 2010 Artemyev  _et al._ (2010) | -1.1026409
Ishikawa et al. 2008 Ishikawa  _et al._ (2008) | -1.102641581033598
Kullie and Kolb 2001 Kullie and Kolb (2001) | -1.10264158103358
Parpia and Mohanty 1995 Parpia and Mohanty (1995) | -1.1026415801
Sundholm 1994 Sundholm (1994) | -1.102641581
Yang et al. 1991 Yang  _et al._ (1991) | -1.1026415810336
Table 7: Comparison of present and literature values for the ground-state
energy of the ${\rm H}_{2}^{+}$ molecular ion at $R=2$. This work: FEM result
computed with $\nu=6,D_{\rm max}=40\,$. All results were obtained for
$\alpha^{-1}=137.0359895$.  Table 8:  Some results on the relativistic shift
at $R=2\,$a.u. The value of $\alpha$ used for the computation is indicated.
"Rescaled" means that the original computation was made for a different value
of $\alpha$. We rescaled the original value of the shift by
$(\alpha/\alpha_{\hbox{\rm\tiny orig}})^{2}$ in order to allow comparison of
the shift values. This rescaling is adequate, given the moderate number of
significant digits in the quoted works. The uncertainty indicated in
parentheses is either taken from a statement of the authors, or, in absence,
the unit of the last digit of the reported shift value is assumed. Notation:
"7.38(3)" means an uncertainty of 0.03.

Figure 2 plots in red the difference between the FEM result and $E_{\rm
rel,Korobov}$ for some of the available values of $R$. The same value of $c$
is used. The difference is less than 1.5 kHz in magnitude for the range of $R$
values where the considered nuclear wavefunctions have significant
probability. Note that the uncertainty of the FEM values is estimated to be of
the order of $10^{-22}$ a.u. $\simeq 10^{-9}$ kHz, completely negligible on
the scale of the plot. We have not analyzed the origin of the visible small
deviations of the values from a smooth dependence, especially near $R=1$, as
they do not impact the following discussion. For concreteness, Table 9
compares our values of the relativistic shift with the previous result, for
selected values of internuclear distance.

Figure 2:  (Color online) Black triangles: FEM relativistic energy shift $\Delta E_{\rm rel}$ for the two-center problem as a function of nuclear distance $R$. Red circles: difference between FEM and perturbation theory. The units of these two quantities are stated in the labels. The values of a selection of red circle points is given in column 3 of Table 9. Other lines: the nuclear radial probability densities and density differences for some relevant rotational-vibrational levels and transitions of the ${\rm HD}^{+}$ molecule. $R$ |  $\Delta E_{\rm rel,Korobov}\ (10^{-6}$ a.u.) |  $\Delta E_{\rm rel}-\Delta E_{\rm rel,Korobov}$ ($10^{-14}$ a.u.)
---|---|---
0.5 | -40.2710800643 | -193.577
1.0 | -17.6917083247 | -36.1806
2.0 | -7.36653763008 | -4.9999
3.0 | -5.48184882611 | -1.77565
4.0 | -5.30561552817 | -1.21509
5.0 | -5.62515785856 | -0.522041
Table 9:  Relativistic shifts: column 2 is the perturbation results $\Delta
E_{\rm rel,Korobov}$; column 3 is the deviations of the FEM results, $\Delta
E_{\rm rel}$, from the perturbation results. For the FEM values, see Table 6.
Note: $1\times 10^{-14}$ a.u. corresponds to 66 Hz.

###  V.2 Relativistic shift of transition frequencies

A key question is whether the highly accurate results obtained in this work
affect the interpretation of the experiments performed so far. Those
interpretations were based on the theoretical treatment of Korobov Korobov and
Karr (2021) and coworkers.

We compute the relativistic shift to any rotational-vibrational level energy
as the average over the nuclear rotational-vibrational probability density,

$\Delta E_{\rm rel}(v,L)=\int_{R_{\rm min}}^{R_{\rm max}}{\Delta E_{\rm
rel}(R)(\psi_{v,L}(R))^{2}R^{2}dR}$ (30)

Here, $\psi_{v,L}(R))$ is the nuclear wave function, and $v,\,L$ are the
vibrational and rotational quantum numbers of the level. The relativistic
shift of a transition frequency between an upper level
$(v^{\prime},L^{\prime})$ and a lower level $(v,L)$ is $\Delta\nu_{\rm
rel}(v,L,v^{\prime},L^{\prime})=\Delta E_{\rm
rel}(v^{\prime},L^{\prime})-\Delta E_{\rm rel}(v,L)$. This relativistic shift
is not complete: there are finite-nuclear-mass corrections to it. However, the
fixed-nuclei shifts that we treat here are the dominant ones. As they turn out
to be small (see below), we do not need to consider their corrections.

We apply this expression to the molecule HD+, since precision spectroscopic
results on rovibrational transitions are so far available only for it. The
range $\\{R_{\rm min},R_{\rm max}\\}$ over which we computed the relativistic
energy shift is sufficiently large for the levels investigated experimentally
so far. The HD+ wave functions $\psi_{v,L}(R)$ were obtained by averaging over
the electronic degrees of freedom the full nonrelativistic three-body wave
function computed by Korobov.

We show in Fig. 2 the differences between the squared wave functions of a few
levels that have recently been studied experimentally. Because these
differences oscillate as a function of $R$, and because the theory difference
$\Delta E_{\rm rel}-\Delta E_{\rm rel,Korobov}$ is a slowly varying function
over the range of $R$ values where the levels in question have a substantial
probability density, the differences (corrections) $\delta\nu=\Delta\nu_{\rm
rel}-\nu_{\rm rel,Korobov}$ turn out to be small. The values are reported in
Table 10.

We conclude that the corrections $\delta\nu$ are negligible compared to
today’s uncertainty of the QED contributions, which amount to $\simeq 1\times
10^{-11}$ relative to the transition frequencies. Nevertheless, we expect that
in the not-too-distant future the precise results obtained here will become
relevant, given that the QED calculations may improve and that experiments
definitely have the potential to improve their precision by several orders.
Another application of the present results is to use the obtained
wavefunctions to compute quantities of relevance for a more precise treatment
of the QED corrections.

$\begin{array}[]{ccc}{\rm transition}&{\rm correction}\
\delta\nu\,{\rm(Hz)}&{\rm fractional\ correction}\\\
\hline\cr(v=0,L=0)\rightarrow(v^{\prime}=0,L^{\prime}=1)&0.9&7\times
10^{-13}\\\ (v=0,L=0)\rightarrow(v^{\prime}=5,L^{\prime}=1)&60&2\times
10^{-13}\\\ (v=0,L=3)\rightarrow(v^{\prime}=9,L^{\prime}=3)&110&3\times
10^{-13}\\\ \hline\cr\end{array}$
---
Table 10: Correction $\delta\nu$ to the transition frequencies of ${\rm HD}^{+}$ arising from the present treatment of the relativistic shift, to be added to the perturbation result of Korobov. The fractional correction is the contribution normalized to the respective transition frequency. $c$ |  $E_{\rm rel}$ (a.u.) |  $\Delta E_{\rm rel}\,(10^{-6}$ a.u.) |  $u$ (a.u.) | $c$ |  $E_{\rm rel}$ (a.u.) |  $\Delta E_{\rm rel}\,(10^{-6}$ a.u.) |  $u$ (a.u.)
---|---|---|---|---|---|---|---
5 | -1.10823616628281245725 | -5601.95178786599669 | 1(-20) | 160 | -1.10263961819119284065 | -5.40369624637914234 | 5(-23)
10 | -1.10402174575200841030 | -1387.53125706194975 | 2(-21) | 170 | -1.10263900115369296245 | -4.78665874650093787 | 4(-23)
15 | -1.10324985455012380333 | -615.640055177342779 | 9(-22) | 180 | -1.10263848407154501463 | -4.26957659855311778 | 3(-23)
20 | -1.10298030822228538131 | -346.093727338920758 | 5(-22) | 190 | -1.10263804646585469507 | -3.83197090823356170 | 3(-23)
25 | -1.10285565422468955126 | -221.439729743090708 | 3(-22) | 200 | -1.10263767284587354132 | -3.45835092707980879 | 3(-23)
30 | -1.10278796937627037902 | -153.754881323918467 | 2(-22) | 250 | -1.10263642783353080771 | -2.21333858434620445 | 1(-23)
35 | -1.10274716721032824078 | -112.952715381780223 | 1(-22) | 300 | -1.10263575153336329279 | -1.53703841683127793 | 1(-23)
40 | -1.10272068892299195324 | -86.4744280454926898 | 1(-22) | 350 | -1.10263534374665671285 | -1.129251710251342481 | 8(-24)
45 | -1.10270253726390794456 | -68.3227689614840043 | 8(-23) | 400 | -1.10263507907778813941 | -0.864582841677900829 | 6(-24)
50 | -1.10268955437077065101 | -55.3398758241904579 | 7(-23) | 450 | -1.10263489762185970781 | -0.683126913246303846 | 4(-24)
55 | -1.10267994897141281291 | -45.7344764663523530 | 5(-23) | 500 | -1.10263476782758958165 | -0.553332643120139307 | 3(-24)
60 | -1.10267264354692895289 | -38.4290519824923411 | 4(-23) | 550 | -1.10263467179455575659 | -0.457299609295080670 | 2(-24)
65 | -1.10266695836994784029 | -32.7438750013797330 | 3(-23) | 600 | -1.10263459875358473660 | -0.384258638275091429 | 2(-24)
70 | -1.10266244745636394391 | -28.2329614174833596 | 3(-23) | 650 | -1.10263454191055032518 | -0.327415603863668284 | 1(-24)
75 | -1.10265880834615650990 | -24.5938512100493472 | 2(-23) | 700 | -1.10263449680735263922 | -0.282312406177710643 | 1(-24)
80 | -1.10265583004409394271 | -21.6155491474821583 | 2(-23) | 750 | -1.10263446042040077440 | -0.245925454312894882 | 1(-24)
85 | -1.10265336172953448692 | -19.1472345880263714 | 2(-23) | 800 | -1.10263443064035125509 | -0.216145404793583294 | 1(-24)
90 | -1.10265129327838995379 | -17.0787834434932376 | 1(-23) | 850 | -1.10263440595937777350 | -0.191464431311990959 | 8(-25)
95 | -1.10264954276284737702 | -15.3282679009164686 | 1(-23) | 900 | -1.10263438527648397256 | -0.170781537511048560 | 7(-25)
100 | -1.10264804821164058439 | -13.8337166941238396 | 1(-23) | 950 | -1.10263436777255322485 | -0.153277606763340967 | 5(-25)
110 | -1.10264564725880168723 | -11.4327638552257233 | 1(-22) | 1000 | -1.10263435282798205741 | -0.138333035595908438 | 5(-25)
120 | -1.10264382115409753260 | -9.60665915107108865 | 9(-23) | 1050 | -1.10263433996708407279 | -0.125472137611282305 | 4(-25)
130 | -1.10264240002530585058 | -8.18553035938906732 | 7(-23) | 1100 | -1.10263432881976328289 | -0.114324816821377799 | 3(-25)
C18 | -1.10264158103257716412 | -7.36653763070260900 | 7(-23) | 1150 | -1.10263431909458761106 | -0.104599641149552752 | 3(-25)
140 | -1.10264127240938096226 | -7.05791443450075182 | 6(-23) | 1200 | -1.10263431055954565992 | -0.096064599198404129 | 2(-25)
150 | -1.10264036271043522934 | -6.14821548876783325 | 5(-23) |  |  |  |
Table 11: Energy and relativistic shift at $R=2\,$a.u. for different values of
$c=\alpha^{-1}$. Values $E_{\rm rel}$, $\Delta E_{\rm rel}$ are extrapolations
to an infinitely dense grid. The values are significant and the last digit is
rounded. Column 4 lists the estimated uncertainty of the relativistic shift,
$u(\Delta E_{\rm rel})$, where the notation ($X$) stands for $\times 10^{X}$.
${\rm C18}=137.035999084$ is the CODATA 2018 value. For $\alpha^{-1}=5$ \-
$100$ the calculations were performed with $D_{\rm max}=40$, $\nu=8$; for the
remaining values $D=50,\,\nu=6$ was used. The uncertainty is estimated as
$u=|\Delta E_{\rm rel}-\Delta E_{\rm rel}({\rm densest\,grid})|$.

Note added:  Extending Table 1, we obtained the energies for an even larger
grid, $Ne/N=800/40401$, using $D=50,\,\nu=6$, $\alpha=137.035999084$. The more
precise extrapolated values are $E_{rel}=-1.10264158103257716411725$,
$E_{nrel}=-1.10263421449494646150814$, $\Delta
E_{rel}=-7.36653763070260911\times 10^{-6}$. The value $\Delta E_{rel}$ has an
estimated uncertainty of $2.3\times 10^{-23}$ a.u., fractionally $3\times
10^{-18}$.

##  VI Acknowledgments.

We are very grateful to V.I. Korobov for motivating the subject of this work,
providing precise nonrelativistic results, updated values of his earlier
relativistic calculations, and the vibrational wave functions. One of us
(O.K.) thanks Prof. D. Kolb for discussions and Prof. M. Garcia at the
Universität Kassel for his support. We also thank the computing centers at
Universität Kassel and Universität Düsseldorf for providing resources and
advice. This work was supported the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation program (grant no.
786306, "PREMOL").

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# Homotopies of constant Cuntz class

Andrew S. Toms  Department of Mathematics, Purdue University, 150 N University
St, West Lafayette IN 47907, USA<EMAIL_ADDRESS>
###### Abstract.

Let $A$ be a unital simple separable exact C∗-algebra which is approximately
divisible and of real rank zero. We prove that the set of positive elements in
$A$ with a fixed Cuntz class is path connected. This result applies in
particular to irrational rotation algebras and AF algebras.

##  1\. Introduction

The Cuntz semigroup of a C∗-algebra was introduced by J. Cuntz in 1978 with a
view to examining rank functions on C∗-algebras [7]. It has enjoyed renewed
relevance in recent years owing both to its role in the classification theory
of simple separable amenable C∗-algebras and to its capacity for reflecting
C∗-algebraic properties in a purely algebraic setting. A salient example is
the recent progress on the Toms-Winter conjecture and its confirmation in both
the unique trace and stable rank one cases. This has been a concerted effort
by many authors and an exhaustive list of credits would be prohibitively long,
but some important works include [1], [5], [8], and [13]. It has firmly
established the breadth and importance of C∗-algebras which absorb the Jiang-
Su algebra $\mathcal{Z}$ tensorially, a class that includes the approximately
divisible C∗-algebras considered here ([3], [12]).

A steady theme in applications of the Cuntz semigroup is that what is true of
projections and Murray-von Neumann equivalence is often true also of positive
elements and Cuntz equivalence, albeit with greater effort. We continue this
theme with an investigation of the homotopy properties of positive elements in
a class of C∗-algebras which includes, notably, the irrational rotation
algebras. When we say that two projections are homotopic, it is implicit that
the homotopy consists of projections whose Murray-von Neumann equivalence
classes are necessarily constant. Naïvely, homotopies of positive elements are
less interesting: any two positive elements of a C∗-algebra are homotopic via
the line segment between them, and this segment consists of positive elements.
A better question, however, one that extends the spirit of homotopy for
projections, is to ask whether two positive elements are homotopic via a path
of positive elements whose Cuntz classes are constant. It is this question
that we address here. The chief difficulty is that unlike Murray-von Neumann
classes, Cuntz classes are not robust under small perturbations in norm.
Indeed, every positive element in a unital C∗-algebra is arbitrarily close in
norm to a representative of the Cuntz class of the unit. Constructing
homotopies of positive elements with constant Cuntz class is therefore a
delicate matter.

######  Theorem 1.1.

Let $A$ be a unital simple separable exact C∗-algebra which is approximately
divisible and of real rank zero. For each $a\in A_{+}$, let $\langle a\rangle$
denote its Cuntz class. It follows that the set

$S_{a}=\\{b\in A_{+}\ |\ \langle b\rangle=\langle a\rangle\\}$

is path connected.

$\mathrm{K}$-theory is rooted in the classification of vector bundles over a
topological space $X$ up to homotopy. Assuming that $X$ is compact and
Hausdorff, these can be thought of as homotopy classes of projections in
$\mathrm{M}_{n}(\mathrm{C}(X))$ for large enough $n$. Positive elements in
this algebra with the same Cuntz class must at least have the same rank at
every point of $X$, and so homotopies as in Theorem 1.1 would have to preserve
rank pointwise. Such homotopies are obstructed by the $\mathrm{K}$-theory of
closed subsets of $X$. Indeed, this is the fundamental observation of [11].
Our result says that for positive elements as in Theorem 1.1 which are not
homotopic to projections, this obstruction vanishes—these elements behave like
trivial vector bundles, with their homotopy class determined only by their
pointwise rank with respect to lower semicontinuous dimension functions. We
expect that the conditions of real rank zero and approximate divisibility can
be relaxed to the property of $\mathcal{Z}$-stability.

The paper is organized as follows: Section 2 recalls briefly some facts and
tools involving the Cuntz semigroup and the notion of approximate
divisibility; Section 3 reduces the proof of Theorem 1.1 to the special case
of certain positive elements with completely disconnected spectrum; and
Section 4 appeals to a result of Rørdam on $\mathrm{K}_{1}$-surjectivity to
prove Theorem 1.1 for the restricted class of elements found in Section 3.

Acknowledgement.. The author would like to thank N. C. Phillips for several
helpful conversations and A. Asadi-Vasfi for a careful reading of the
manuscript.

##  2\. Preliminaries

A full treatment of the basic theory of the Cuntz semigroup can be found in
[2]. We make a brief recollection here of the details required for the sequel.
Let $A$ be a $C^{*}$-algebra and let $A_{+}$ denote the subset of positive
elements. If $a,b\in A_{+}$, we write $a\precsim b$ if there is a sequence
$(v_{n})$ in $A$ such that $v_{n}bv_{n}^{*}\to a$. If $a\precsim b$ and
$b\precsim a$ then we write $a\sim b$ and say that $a$ and $b$ are Cuntz
equivalent. Set $\mathrm{Cu}(A)=(A\otimes\mathcal{K}_{+})/\sim$, and let
$\langle a\rangle$ denote the equivalence class of $a$ in $\mathrm{Cu}(A)$.
Now $\langle a\rangle+\langle b\rangle=\langle a\oplus b\rangle$ defines
addition on $\mathrm{Cu}(A)$ and $\langle a\rangle\leq\langle
b\rangle\Leftrightarrow a\precsim b$ defines a partial order. These make
$\mathrm{Cu}(A)$ into an ordered semigroup, the Cuntz semigroup.

It was shown in [6] that $\mathrm{Cu}(A)$ admits suprema for increasing
sequences. We say that $\langle a\rangle$ is compactly contained in $\langle
b\rangle$, written $\langle a\rangle\ll\langle b\rangle$, if whenever
$(\langle b_{i}\rangle)$ is increasing and $\sup\langle b_{i}\rangle=\langle
b\rangle$ we have $\langle a\rangle\leq\langle b_{i_{0}}\rangle$ for some
$i_{0}\in\mathbb{N}$. An element $\langle a\rangle$ of $\mathrm{Cu}(A)$ is
called compact if $\langle a\rangle\ll\langle a\rangle$. If $A$ is simple or
of stable rank one, then $\langle a\langle$ is compact if and only if $a\sim
p$ for some projection $p$ ([2], [4]). Recall that $A$ has real rank zero if
every self-adjoint element in $A$ can be approximated in norm by self-adjoint
elements with finite spectrum. Consequently, for such algebras, each $\langle
a\rangle\in\mathrm{Cu}(A)$ is seen to be the supremum of a sequence of compact
elements by exploiting (i) below.

Let $\delta>0$, and define a continuous map
$g_{\epsilon}:\mathbb{R}\to[0,\infty)$ by

$g_{\epsilon}(t)=\max\\{0,t-\epsilon\\}.$

Set $(a-\epsilon)_{+}=g_{\epsilon}(a)$. We record three useful facts related
to this construction (see [2]):

  1. (i)

$a\precsim b$ if and only if $(a-\epsilon)_{+}\precsim b$, for every
$\epsilon>0$;

  2. (ii)

$\langle(a-\epsilon)_{+}\rangle\ll\langle a\rangle$ for every $\epsilon>0$;

  3. (iiii)

If $\|a-b\|<\epsilon$, then $(a-\epsilon)_{+}\precsim b$ for every
$\epsilon>0$.

The next Lemma is Proposition 2.7 (i) of [9].

######  Lemma 2.1.

Let $A$ be a C∗-algebra with $a,b\in A_{+}$. If $b\in\overline{aAa}$, then
$b\precsim a$.

We record one more technical Lemma for use in the sequel.

######  Lemma 2.2.

Let $A$ be a C∗-algebra and let $0\neq a\in A_{+}$. Suppose that zero is an
isolated point of the spectrum of $a$. It follows that there are a homotopy
$h:[0,1]\to\mathrm{C}^{*}(a)$ and a projection $p\in\mathrm{C}^{*}(a)$ such
that $h(0)=a$, $h(1)=p$, and $\langle h(t)\rangle=\langle a\rangle$ for every
$t\in[0,1]$.

###### Proof.

Let $\epsilon>0$ be the smallest nonzero element of the spectrum $\sigma(a)$.
Define $h(t)=a^{\frac{1}{1-t}}$ for $t\in[0,1)$ and
$p=h(1)=\chi_{[\epsilon,\infty)}(a)$. This is clearly a homotopy inside
$\mathrm{C}^{*}(a)$, and $\langle h(t)\rangle=\langle a\rangle$ by Proposition
2.5 of [2]. ∎

Let $A$ be unital and exact, and let $\mathrm{T}(A)$ denote the space of
tracial states on $A$. For $\tau\in\mathrm{T}(A)$ and
$a\in(A\otimes\mathcal{K})_{+}$ we define

$d_{\tau}(a)=\lim_{n\to\infty}\tau(a^{1/n}).$

This defines a lower semicontinuous dimension function on $A$. We say that $A$
has strict comparison of positive elements if $a\precsim b$ whenever

$d_{\tau}(a)<d_{\tau}(b),\ \forall\tau\in\mathrm{T}(A).$

A unital separable C∗-algebra $A$ is said to be approximately divisible if
there exists, for each $N\in\mathbb{N}$, an approximately central sequence of
unital $*$-homomorphisms
$\phi_{n}:\mathrm{M}_{N}(\mathbb{C})\oplus\mathrm{M}_{N+1}(\mathbb{C})\to A$
([3]). It is shown in [3] that the irrational rotation algebras are
approximately divisible. Approximately divisible C∗-algebras are
$\mathcal{Z}$-stable, and simple $\mathcal{Z}$-stable C∗-algebras have strict
comparison of positive elements ([10], [12]).

##  3\. Totally disconnected spectra

In this section we reduce the proof of our main result to the case of positive
elements constructed from orthogonal projections. While the statement here is
different, the essential idea for the following Lemma is contained in Lemma
2.19 of [2].

######  Lemma 3.1.

Let $A$ be a unital simple separable exact C∗-algebra with strict comparison
of positive elements. Suppose further that $A$ is stably finite. Let $a\in A$
be positive, and let $q\in A\otimes\mathcal{K}$ be a projection such that
$q\precsim a$. It follows that there are a projection $p\sim q$ and a positive
element $b$ in $\overline{aAa}$ such that $bp=pb=0$ and

$d_{\tau}(a)=d_{\tau}(b)+d_{\tau}(p),\ \forall\tau\in\mathrm{T}(A).$

###### Proof.

Let $0<\epsilon<1/2$ be given and find $\delta>0$ such that

$(q-\epsilon)_{+}\precsim(a-\delta)_{+}.$

It follows that for any $0<\gamma<1/2$ there is $v\in A\otimes\mathcal{K}$
such that

$v(a-\delta)_{+}v^{*}=(q-\epsilon-\gamma)_{+}\sim q.$

Set

$s=(a-\delta)_{+}^{1/2}v^{*}v(a-\delta)_{+}^{1/2},$

so that $q\sim s$ and $\langle s\rangle$ is compact in $\mathrm{Cu}(A)$ ([2],
Corollary 2.6). There is then a projection $p$ in C${}^{*}(s)$ with $s\sim p$
by Lemma 2.2. Clearly,

$p\in\overline{(a-\delta)_{+}^{1/2}A(a-\delta)_{+}^{1/2}}.$

Set $f_{\delta}(t)=\min\\{t/\delta,1\\}$, $t\geq 0$, so that
$f_{\delta}(a)p=pf_{\delta}(a)=p$. Note that $a\sim f_{\delta}(a)$ as these
elements both have the same support in C${}^{*}(a)=\mathrm{C}(\sigma(a))$.
Now,

$\displaystyle f_{\delta}(a)$ $\displaystyle=$ $\displaystyle
f_{\delta}(a)(p+(1_{A}-p))$ $\displaystyle=$ $\displaystyle
f_{\delta}(a)p+f_{\delta}(a)(1_{A}-p)$ $\displaystyle=$ $\displaystyle
p+(1_{A}-p)f_{\delta}(a)(1_{A}-p)$ $\displaystyle=$ $\displaystyle
p\oplus(f_{\delta}(a)-p)$

Setting $b=f_{\delta}(a)-p$ and observing that $a\sim f_{\delta}(a)$, we have

$d_{\tau}(a)=d_{\tau}(f_{\delta}(a))=d_{\tau}(p\oplus
b)=d_{\tau}(p)+d_{\tau}(b),$

as desired. ∎

######  Proposition 3.2.

Let $A$ be a unital simple separable exact C∗-algebra which is approximately
divisible and of real rank zero. Suppose further that $A$ is stably finite,
and let $a\in A$ be positive. It follows that there is a sequence
$p_{1},p_{2},\ldots$ of mutually orthogonal projections in $\overline{aAa}$
such that with

$b=\bigoplus_{i=1}^{\infty}\frac{1}{2^{i}}p_{i},$

we have a homotopy $h:[0,1]\to(\overline{aAa})_{+}$ satisfying $h(0)=a$,
$h(1)=b$, and $\langle h(t)\rangle=\langle a\rangle\in\mathrm{Cu}(A)$ for each
$t\in[0,1]$.

###### Proof.

If $\langle a\rangle\in\mathrm{Cu}(A)$ is compact, then zero is an isolated
point in the spectrum of $a$ and the Proposition follows from Lemma 2.2.

If $a$ is not compact, then real rank zero implies the existence of a sequence
$q_{1},q_{2},\ldots$ of projections in $A\otimes\mathcal{K}$ such that
$q_{i}\neq q_{i+1}$, $\langle q_{i}\rangle\ll\langle q_{i+1}\rangle$, and

$\sup_{i}\langle q_{i}\rangle=\langle a\rangle$

Since $A$ is approximately divisible, the $\mathrm{K}_{0}$-group of $A$ is
weakly unperforated. It follows that $\langle q_{i}\rangle-\langle
q_{i-1}\rangle=\langle r_{i}\rangle$ for each $i>1$ and some projection $0\neq
r_{i}\in A\otimes\mathcal{K}$. Set $r_{1}=q_{1}$. Apply Lemma 3.1 with
$q=r_{1}$ and $a$ as in the present Proposition to find a projection
$p_{1}\sim r_{1}$ and positive element $b_{1}$ in $\overline{aAa}$ such that
$p_{1}b_{1}=0$ and

$d_{\tau}(a)=d_{\tau}(p_{1})+d_{\tau}(b_{1}),\ \forall\tau\in\mathrm{T}(A).$

Assume, inductively, that we have found in $\overline{aAa}$ mutually
orthogonal projections $p_{i}\sim r_{i}$, $1\leq i\leq n$, and a positive
element $b_{n}$ orthogonal to each $p_{i}$ such that

(1)
$d_{\tau}(a)=d_{\tau}\left(\oplus_{i=1}^{n}p_{i}\right)+d_{\tau}(b_{n})=\sum_{i=1}^{n}d_{\tau}(p_{i})+d_{\tau}(b_{n}).$

Applying Lemma 3.1 with $a=b_{n}$ and $q=r_{n+1}$ and labeling the resulting
projection and positive element as $p_{n+1}$ and $b_{n+1}$, respectively,
yields (1) above with $n$ replaced by $n+1$. Continuing inductively in this
manner produces a sequence of mutually orthogonal projections
$p_{1},p_{2},\ldots$ in $\overline{aAa}$.

Set

$b=\bigoplus_{i=1}^{\infty}\frac{1}{2^{i}}p_{i}.$

Note that zero is an accumulation point of the spectrum of $b$, so that $b$ is
not compact in $\mathrm{Cu}(A)$. Now

$q_{n}\sim\oplus_{i=1}^{n}p_{i}\precsim b\precsim a$

by Lemma 2.1 so that

$d_{\tau}(q_{n})\leq d_{\tau}(b)\leq d_{\tau}(a),\
\forall\tau\in\mathrm{T}(A).$

Since $\langle a\rangle=\sup_{n}\langle q_{n}\rangle$ in $\mathrm{Cu}(A)$ we
then have

$d_{\tau}(a)=\sup_{n}d_{\tau}(q_{n})\leq d_{\tau}(b)\leq d_{\tau}(a),\
\forall\tau\in\mathrm{T}(A)$

and so

$d_{\tau}(b)=d_{\tau}(a),\ \forall\tau\in\mathrm{T}(A).$

$A$ is approximately divisible, hence $\mathcal{Z}$-stable, and so has strict
comparison of positive elements ([10], [12]). Since neither $a$ nor $b$ is
compact it follows that $\langle a\rangle=\langle b\rangle$ in
$\mathrm{Cu}(A)$ ([4]).

Set

$h(t)=(1-t)a+tb,\ t\in[0,1].$

Evidently, $\langle h(0)\rangle=\langle h(1)\rangle=\langle a\rangle$. Since
$h(t)\in\overline{aAa}$ we have $\langle h(t)\rangle\leq\langle a\rangle$. On
the other hand, for $t\in(0,1)$,

$a\sim(1-t)a\precsim(1-t)a+tb=h(t),$

whence $\langle h(t)\rangle\geq\langle a\rangle$ also. ∎

##  4\. Path connected Cuntz classes

The following Lemma is due to Rørdam ([10], Lemma 6.3), although we have
expanded its statement slightly. Recall that a unital C∗-algebra $A$ is
strongly $\mathrm{K}_{1}$-surjective if the canonical map

$\mathcal{U}(B+\mathbb{C}(1_{A}))\to\mathrm{K}_{1}(A)$

is surjective for every full hereditary subalgebra $B$ of $A$.

######  Lemma 4.1.

Let $A$ be a unital approximately divisible C∗-algebra. Then $A$ is strongly
$\mathrm{K}_{1}$-surjective. In particular, for any full projection $p\in A$
and $g\in\mathrm{K}_{1}(A)$, there is a unitary $v\in pAp$ such that
$g=[v+(1_{A}-p)]_{1}$ in $\mathrm{K}_{1}(A)$.

######  Lemma 4.2.

Let $A$ be a unital simple approximately divisible C∗-algebra and let $p,q$ be
full projections in $A$ with $[p]=[q]$ in $\mathrm{K}_{0}(A)$. It follows that
there is a continuous path of unitaries $(u_{t})_{t\in[0,1]}$ in $A$ such that
$u_{0}=1_{A}$ and $u_{1}pu_{1}^{*}=q$. In particular, $p$ and $q$ are
homotopic via projections in $A$.

###### Proof.

Since $[p]=[q]$ in $\mathrm{K}_{0}(A)$, we have also that
$[1_{A}-p]=[1_{A}-q]$. The approximate divisibility of $A$ implies that $A$
has stable rank one, hence enjoys cancellation of projections. Thus, $p\sim q$
and $1_{A}-p\sim 1_{A}-q$ in the Murray-von Neumann semigroup. It follows that
there is a unitary $v\in\mathcal{U}(A)$ such that $vpv^{*}=q$. We now modify
$v$ to ensure that it is in the connected component of
$1_{A}\in\mathcal{U}(A)$.

If $p=1_{A}$ there is nothing to prove, so we may assume that $1_{A}-p$ is
full. Set $g=[v]_{1}$ in $\mathrm{K}_{1}(A)$. Apply Lemma 4.1 with $1_{A}-p$
in place of $p$ to find a unitary $w\in\mathcal{U}((1_{A}-p)A(1_{A}-p)+p)$
such that $[w]_{1}=-g$ in $\mathrm{K}_{1}(A)$. Now $[vw]_{1}=0$ in
$\mathrm{K}_{1}(A)$ and $A$ has stable rank one, so there is a continuous path
$(u_{t})_{t\in[0,1]}$ in $\mathcal{U}(A)$ such that $u_{0}=1_{A}$ and
$u_{1}=vw$. Observing that

$u_{1}pu_{1}^{*}=(vw)p(vw)^{*}=vwpw^{*}v^{*}=vpv^{*}=q$

completes the proof. ∎

We are now ready to prove our main result.

###### Proof.

(Theorem 1.1) Let $A$ be a simple unital separable exact C∗-algebra which is
approximately divisible and of real rank zero, and let $a,b\in A$ be positive
with $\langle a\rangle=\langle b\rangle$ in $\mathrm{Cu}(A)$. Since $A$ is
$\mathcal{Z}$-stable by Theorem 2.3 of [12], it is either stably finite (when
it has a trace) or purely infinite (when it is traceless) ([10]). In the
purely infinite case, the Cuntz classes of any two nonzero positive elements
are the same, so the line segment $(1-t)a+tb,\ t\in[0,1]$, defines a homotopy
between $a$ and $b$. This segment consists entirely of nonzero positive
elements and the Theorem follows. We may thus assume that $A$ is stably
finite, whence $\mathrm{T}(A)\neq\emptyset$. In fact, $A$ has stable rank one
by Theorem 6.7 of [10].

If $\langle a\rangle=\langle b\rangle$ is compact in $\mathrm{Cu}(A)$ then
zero is not an accumulation point of the spectrum of $a$ or $b$ by Proposition
2.23 of [2]. By Lemma 2.2 we may assume that $a$ and $b$ are projections and
the desired result follows from Proposition 4.2.

It remains to address the case where $\langle a\rangle=\langle b\rangle$ is
not compact, so that $a$ and $b$ have zero as an accumulation point of their
spectrum. By Proposition 3.2, we may assume that there are sequences of
projections $p_{1},p_{2},\ldots$ and $q_{1},q_{2},\ldots$ in $\overline{aAa}$
and $\overline{bAb}$, respectively, with the following properties:

  1. (i)

$p_{i}p_{j}=0=q_{i}q_{j}$ for every pair $i\neq j$;

  2. (ii)

$p_{i}\sim q_{i}$ for each $i\in\mathbb{N}$;

  3. (iii)

$a=\bigoplus_{i=1}^{\infty}\frac{1}{2^{i}}p_{i}\ \mathrm{and}\
b=\bigoplus_{i=1}^{\infty}\frac{1}{2^{i}}q_{i}.$

For convenience we set $s_{k}=\oplus_{i=1}^{k}1/2^{i}$. Assume that we have
found a continuous path $u_{t}:[0,s_{k}]\to\mathcal{U}(A)$ with the property
that $u_{0}=1_{A}$ and

(2)
$u_{s_{k}}\left(\oplus_{i=1}^{k}p_{i}\right)u_{s_{k}}^{*}=\oplus_{i=1}^{k}q_{i}.$

(The existence of this path for $k=1$ follows from (ii) above and Lemma 4.2.)
Let us show that $u_{t}$ can be extended to $[0,s_{k+1}]$ so that (2) holds
with $k+1$ in place of $k$ and moreover that

(3) $u_{t}\left(\oplus_{i=1}^{k}p_{i}\right)u_{t}^{*}=\oplus_{i=1}^{k}q_{i},\
\forall t\in[s_{k},s_{k+1}].$

Set

$Q_{k}=1_{A}-\left(\oplus_{i=1}^{k}q_{i}\right).$

Note that $Q_{k}AQ_{k}$ is approximately divisible by [3][Corollary 2.9], and
that $u_{s_{k}}p_{k+1}u_{s_{k}}^{*}\sim q_{k+1}$ in $Q_{k}AQ_{k}$. It follows
from Lemma 4.2 that there is a continuous path

$\tilde{v}_{t}:[s_{k},s_{k+1}]\to\mathcal{U}(Q_{k}AQ_{k})$

such that $\tilde{v}_{s_{k}}=Q_{k}$ and

$\tilde{v}_{s_{k+1}}(u_{s_{k}}p_{k+1}u_{s_{k}}^{*})\tilde{v}_{s_{k+1}}^{*}=q_{k+1}.$

Set $v_{t}=\tilde{v}_{t}+(1_{A}-Q_{k}),\ t\in[s_{k},s_{k+1}]$, so that
$v_{t}\in\mathcal{U}(A)$ and $v_{s_{k}}=1_{A}$. We now define our extension of
$u_{t}:[0,s_{k+1}]\to\mathcal{U}(A)$ via

(4) $u_{t}=\left\\{\begin{array}[]{rcl}u_{t}&=&t\in[0,s_{k})\\\
v_{t}u_{s_{k}}&=&t\in[s_{k},s_{k+1}]\end{array}\right..$

This extension is continuous since $v_{s_{k}}=1_{A}$. Also,

(5) $\displaystyle
u_{s_{k+1}}\left(\oplus_{i=1}^{k+1}p_{i}\right)u_{s_{k+1}}^{*}$
$\displaystyle=v_{s_{k+1}}\left(u_{s_{k}}\left(\oplus_{i=1}^{k+1}p_{i}\right)u_{s_{k}}^{*}\right)v_{s_{k+1}}^{*}$
(6)
$\displaystyle=v_{s_{k+1}}\left(u_{s_{k}}\left(\oplus_{i=1}^{k}p_{i}\right)u_{s_{k}}^{*}\right)v_{s_{k+1}}^{*}$
(7) $\displaystyle\hskip 28.45274pt+\
v_{s_{k+1}}u_{s_{k}}p_{i+1}u_{s_{k}}^{*}v_{s_{k+1}}^{*}$ (8)
$\displaystyle=v_{s_{k+1}}\left(\oplus_{i=1}^{k}q_{i}\right)v_{s_{k+1}}^{*}$
(9) $\displaystyle\hskip 28.45274pt+\ \
\tilde{v}_{s_{k+1}}u_{s_{k}}p_{i+1}u_{s_{k}}^{*}\tilde{v}_{s_{k+1}}^{*}$ (10)
$\displaystyle=\left(\oplus_{i=1}^{k}q_{i}\right)+q_{k+1}$ (11)
$\displaystyle=\oplus_{i=1}^{k+1}q_{i}$

satisfying (2) with $k+1$ in place of $k$, as required. Note also that for
$t\in[s_{k},s_{k+1}]$ we have

(12) $\displaystyle u_{t}\left(\oplus_{i=1}^{k}p_{i}\right)u_{t}^{*}$
$\displaystyle=v_{t}\left(u_{s_{k}}\left(\oplus_{i=1}^{k}p_{i}\right)u_{s_{k}}^{*}\right)v_{t}^{*}$
(13) $\displaystyle=v_{t}\left(\oplus_{i=1}^{k}q_{i}\right)v_{t}^{*}$ (14)
$\displaystyle=\oplus_{i=1}^{k}q_{i}$

as required by (3). This completes our extension of $u_{t}$ to $[0,s_{k+1}]$.
Beginning at $t=0$ and continuing these extensions inductively yields a
continuous path $u_{t}:[0,1)\to\mathcal{U}(A)$ such that

(15) $u_{t}\left(\oplus_{i=1}^{l}p_{i}\right)u_{t}^{*}=\oplus_{i=1}^{l}q_{i},\
\forall t\geq s_{k},\ \forall l\leq k.$

Define $h:[0,1]\to A$ as follows:

(16) $h(t)=\left\\{\begin{array}[]{rl}u_{t}au_{t}^{*},&t\in[0,1)\\\
b,&t=1\end{array}\right.$

Clearly, $\langle h(t)\rangle=\langle a\rangle$ in $\mathrm{Cu}(A)$ for every
$t$ in $[0,1]$, and $h(t)$ is continuous at every $t$ in $[0,1)$ since $u_{t}$
is. It remains to prove that $h(t)$ is continuous at $t=1$. Recall that
$s_{k}\to 1$ as $k\to\infty$. For $t\geq s_{k}$ we have

(17) $\displaystyle\|h(1)-h(t)\|$ $\displaystyle=\|b-u_{t}au_{t}*\|$ (18)
$\displaystyle=\left\|\bigoplus_{i=1}^{\infty}\frac{1}{2^{i}}q_{i}-u_{t}\left(\bigoplus_{i=1}^{\infty}\frac{1}{2^{i}}p_{i}\right)u_{t}^{*}\right\|$
(19)
$\displaystyle\stackrel{{\scriptstyle(\ref{conjequal2})}}{{=}}\left\|\bigoplus_{i=k+1}^{\infty}\frac{1}{2^{i}}q_{i}-u_{t}\left(\bigoplus_{i=k+1}^{\infty}\frac{1}{2^{i}}p_{i}\right)u_{t}^{*}\right\|$
(20)
$\displaystyle\leq\left\|\bigoplus_{i=k+1}^{\infty}\frac{1}{2^{i}}q_{i}\right\|+\left\|\bigoplus_{i=k+1}^{\infty}\frac{1}{2^{i}}p_{i}\right\|$
(21) $\displaystyle=\frac{1}{2^{k+1}}+\frac{1}{2^{k+1}}$ (22)
$\displaystyle=\frac{1}{2^{k}}$

This shows that $h(t)$ is continuous at $t=1$, completing the proof. ∎

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