# The Parallel-Transported (Quasi)-Diabatic Basis

Robert Littlejohn<EMAIL_ADDRESS>Department of Physics,
University of California, Berkeley, California 94720 USA  Jonathan Rawlinson
<EMAIL_ADDRESS>School of Mathematics, University of
Manchester, Manchester UK  Joseph Subotnik<EMAIL_ADDRESS>Department
of Chemistry, University of Pennsylvania, Philadelphia, PA, USA

###### Abstract

This article concerns the use of parallel transport to create a diabatic
basis. The advantages of the parallel-transported basis include the facility
with which Taylor series expansions can be carried out in the neighborhood of
a point or a manifold such as a seam (the locus of degeneracies of the
electronic Hamiltonian), and the close relationship between the derivative
couplings and the curvature in this basis. These are important for analytic
treatments of the nuclear Schrödinger equation in a neighborhood of
degeneracies. The parallel-transported basis bears a close relationship to the
singular-value basis; in this article both are expanded in power series about
a reference point and they are shown to agree through second order but not
beyond. Taylor series expansions are effected through the projection operator,
whose expansion does not involve energy denominators or any type of
singularity, and in terms of which both the singular-value basis and the
parallel-transported basis can be expressed. The parallel-transported basis is
a version of Poincaré gauge, well known in electromagnetism, which provides a
relationship between the derivative couplings and the curvature and which,
along with a formula due to Mead, affords an efficient method for calculating
Taylor series of the basis states and the derivative couplings. The case in
which fine structure effects are included in the electronic Hamiltonian is
covered.

††preprint: AIP/123-QED

##  I Introduction

This article concerns bases of electronic wave functions in polyatomic
molecules that span a chosen subspace of strongly coupled states. We are
especially interested in the parallel-transported basis and the closely
related singular-value basis. The context of this work is Born-Oppenheimer
theory, applied to multiple, strongly coupled electronic states, a subject
that is treated and reviewed in many places (Cederbaum (2004); Faraji, Gómez-
Carrasco, and Köppel (2012); Köppel (2004); Matsunaga and Yarkony (1998);
Yarkony (1996, 2004, 2001, 2012); Zhu and Yarkony (2016); Kendrick, Mead, and
Truhlar (2002); Mead (1988, 1992); Mead and Truhlar (1979, 1982); Richings and
Worth (2015); Richings and Habershon (2020)). The bases we refer to are really
fields of bases, that is, defined over some region of the nuclear
configuration space.

In this article we refer to any basis that is smooth over the region in
question as “diabatic,” without any assumption that the derivative couplings
vanish and without the common prefix “quasi.” We do this because as a
practical matter the derivative couplings in polyatomic molecules never
vanish; if we wish to refer to the (rare) cases in which they do, we will
refer to a “strictly diabatic” basis. In addition, the designation “quasi”
implies that a quasi-diabatic basis is close to a strictly diabatic basis,
that is, that the derivative couplings are small. There is much wishful
thinking in the literature that the derivative couplings can be made small
enough to be ignored, but without quantitative justification. A careful
analysis by Kendrick, Mead, and Truhlar (2002), however, shows that the
derivative couplings are generically of order unity over a range of the order
of one atomic unit, unless forced to be even larger by small energy
denominators. In addition numerical evidence (Choi and Vaniček (2021))
supports the conclusion that these couplings must be taken into account for
accurate results.

There is some subtlety in the question of the magnitude of the derivative
couplings, however, because these couplings are components of a vector
potential of a gauge theory (Pacher  _et al._ (1989); Pacher, Cederbaum, and
Köppel (1993); Wittig (2012); Kendrick and Mead (1995); Mead (1980b, 1987,
1992); Bohm, Boya, and Kendrick (1991); Bohm  _et al._ (1992); Bohm, Kendrick,
and Loewe (1992); Kendrick (2004); Berry (1984)), in which the gauge
transformations are changes of frame, specified by (ideally) smooth fields of
unitary matrices. We call this the Mead-Truhlar-Berry vector potential or
connection (Mead and Truhlar (1979); Berry (1984)). In this article, the
derivative couplings and the Mead-Truhlar-Berry connection or vector potential
(the only vector potential we consider) are almost the same thing (the small
differences are explained in Sec. IV.4).

Thus, diabatic bases are not unique, and different choices have different
advantages and disadvantages. Several choices of diabatic basis have been
reviewed by Pacher, Cederbaum, and Köppel (1993), including the singular-value
basis (Pacher, Cederbaum, and Köppel (1988)) and the Lorentz-gauge basis
(Pacher  _et al._ (1989)), which minimizes the mean square of the connection
over a region of nuclear configuration space.

In addition there is the basis of Werner and Meyer (1981), which diagonalizes
the dipole operator in a single dimension, and of Cave and Newton (1996, 1997)
and Subotnik  _et al._ (2008) who generalized this diagonalization to
multidimensional problems. Historically, within the chemistry literature, the
emphasis has been on finding a good set of diabatic states that serve as
initial and final states for electron transfer problems, e.g., Subotnik  _et
al._ (2009), so that one can apply Marcus theory; that being said, our
treatment here is more general dynamically (rather than serving as a platform
for a spin-boson model Hamiltonian (Xu and Schulten (1994); Rosso and Dupuis
(2006))). In such a case, there is no need to have an element of electronic
locality inherent within a smooth diabatic basis.

The advantages of the parallel-transported basis include the facility with
which it can be expanded in Taylor series about given points or even given
manifolds, such as seams or degeneracy manifolds, and the close relationship
it bears to the curvature. These features will be explained in detail in this
article, but the following is some of the basic ideas.

It turns out that by means of a gauge transformation it is possible to make
the vector potential vanish at any given point. Indeed, both the parallel-
transported basis and the singular-value basis do this, as will be explained
below. In fact, in some circumstances (triatomics in the electrostatic model)
there exists a parallel-transported frame that causes the vector potential to
vanish over the entire degeneracy manifold or seam, which is a one-dimensional
curve. In the neighborhood of such a point or such a manifold in such a basis,
the derivative couplings are indeed small. Then the question is how rapidly
the couplings grow as we move away from the chosen point or manifold.

This question draws attention to the curvature, a tensor that is a function of
the derivative couplings and their derivatives. Here the important fact is
that the curvature tensor, unlike the vector potential, is a tensor, so that
it transforms under a gauge transformation by simple conjugation by the
unitary matrix specifying the transformation, see (93). Thus, the magnitude of
the curvature tensor, whose square is the sum of squares of its components, is
invariant under a gauge transformation. In a sense the curvature represents
the part of the connection that is gauge-invariant, while the part that is not
is what is changed by a gauge transformation. We will not attempt to make this
notion precise but merely use it to motivate the idea that to minimize the
connection we should find a gauge that expresses the connection in terms of
the curvature. As we will show, the parallel-transported basis does this.

Our interest in the parallel-transported basis is part of a larger project in
which we aim to find analytical treatments of the nuclear Schrödinger equation
in the neighborhood of conical intersections. Part of that work involves
normal forms for Landau-Zener transitions in many dimensions; these are
illustrated by Littlejohn and Flynn (1992), in which a 2-dimensional Landau-
Zener problem is treated. A conical intersection has an effective range in the
nuclear configuration space, outside of which the potential energy surfaces
can be decoupled by adiabatic means (Weigert and Littlejohn (1993); Panati,
Spohn, and Teufel (2002); Teufel (2003)). That range depends on the nuclear
momentum and gets larger as that momentum grows, but even for fairly large
nuclear momenta by ordinary standards, that is, momenta such that the nuclear
kinetic energy is of the order of one atomic unit, the range of a conical
intersection is still quite small in atomic units. Thus, Taylor series
expansions are a viable approach for covering the strongly coupled region.

Expansions in Taylor series about a conical intersection have a long history,
including the works by Mead (1983); Thompson and Mead (1985); Yarkony (1997)
on derivative couplings and their singularities in the neighborhood of a
conical intersection and by Yarkony (2000) on the bifurcations of degeneracy
manifolds. This work focuses on the projection operator and its expansion, an
attractive intermediate goal since the projection operator is smooth and
diabatic bases such as the singular-value and parallel-transported basis can
be expressed in terms of it. The usual approaches work with the adiabatic
basis, which leads to small or vanishing energy denominators when one moves
tangent or nearly tangent to the degeneracy manifold or seam. In this article
we avoid the adiabatic basis as much as possible, mainly using the projection
operator as a substitute; this gives us expansions that are valid even when
one moves tangent to the degeneracy manifold.

In Sec. II we discuss aspects of the singular-value diabatic basis, which is
due to Pacher, Cederbaum, and Köppel (1988, 1993). We begin with a variational
principle for the singular-value diabatic basis that differs somewhat from the
one used in the original work on the subject, but which shows a closer
connection with the variational foundations of the parallel-transported basis,
which is our main topic. We emphasize the issue of the smoothness of the
solution; although the singular-value diabatic basis is smooth, some of the
the objects used in its derivation and construction are not. We express the
singular-value basis in terms of the projection operator, an important step
since the latter is smooth. In addition, the projection operator can be
expanded about a reference point, possibly on a conical intersection,
something that we carry out in Sec. III. We then use that to expand the
singular-value basis to third order about the reference point.

In Sec. IV we develop the theory of parallel transport and use it to construct
the parallel-transported basis and its expansion about a reference point,
which we show agrees with the singular-value basis through second order but
differs at third order. We then express the Hamiltonian, connection and
curvature in the parallel-transported bases as power series expansions about a
reference point. We point out that the parallel-transported frame is a version
of Poincaré gauge, which in electromagnetism is a gauge that is transverse in
real space. This allows us to express the connection in terms of an integral
involving the curvature, which provides an efficient method of calculating the
connection in the parallel-transported frame. Finally we point out that
constructions of diabatic bases are only needed in $3N-3$ directions of
nuclear configuration space, because in the remaining 3 directions the nuclear
configuration changes by a pure rotation, and in those directions the basis
should transform by a rotation operator. Thus the construction of diabatic
bases takes place on a $(3N-6)$-dimensional “section” of the nuclear
configuration space.

In Sec. V we discuss degeneracy manifolds, usually called seams, and the
construction of diabatic bases in the neighborhood of them. For example, in
triatomic molecules in the electrostatic model for the electronic Hamiltonian,
for which the degeneracy manifold is a 1-dimensional curve in the
3-dimensional section, we construct a diabatic basis in a tubular region
surrounding the curve. Again the connection is expressed in terms of the
curvature, but the components of the connection along the degeneracy manifold
and transverse to it have different expansions. We also point out that in the
case of triatomics with the electrostatic model, it is possible to choose a
gauge such that the derivative couplings (in all of their components) vanish
on the degeneracy manifold. Finally, in Sec. VI we present some conclusions.

In this article we denote the adiabatic basis by $|ax;k\rangle$, the singular-
value diabatic basis by $|dx;k\rangle$, the basis that is parallel transported
away from a reference point by $|px;k\rangle$ and the basis that is parallel
transported away from the degeneracy manifold by $|Dx;k\rangle$, where $x$ is
a point of nuclear configuration space and $k$ is a quantum number or
sequencing number for the basis state.

##  II The Singular-Value Diabatic Basis

In this section we cover some aspects of the singular-value diabatic basis,
which is due to Pacher, Cederbaum, and Köppel (1988, 1993). This basis is easy
to compute at a point $x$ of configuration space, and it does not require
integration along some path starting at another point $x_{0}$. For these
reasons it is popular in numerical work. It is also conceptually simple and
geometrically compelling. In this article we present some new results on the
singular-value basis, including a discussion of its smoothness properties, its
relation to the projection operator, and Taylor series expansions about a
reference point.

We begin by showing that the singular-value basis satisfies a certain
variational principle. This differs only slightly from the original
variational principle used by Pacher, Cederbaum, and Köppel (1993), but we
present it anyway because it reveals a close connection with parallel
transport which is the main topic of this paper. Both variational principles
lead to the same (singular-value) diabatic basis. Other variational principles
that have been considered, for example, the one by Cimiraglia  _et al._
(1985), are not equivalent and lead to different diabatic bases.

We also emphasize the issue of the smoothness of the singular-value diabatic
basis, something that takes some effort since the adiabatic basis with which
the construction begins is not smooth. Finally we show that the singular-value
diabatic basis can be expressed in terms of the projection operator onto the
coupled subspace of the electronic Hilbert space. Since the projection
operator is smooth, this shows that the singular-value diabatic basis is, too,
and it also provides a method for expanding the singular-value basis in a
Taylor series about a reference point.

###  II.1 Terminology and Notation

We let $x$ represent a point of the nuclear configuration space in the center-
of-mass frame, so that $x$ stands for the collection of $N-1$ Jacobi vectors
(Aquilanti and Cavalli (1986); Smirnov and Shitikova (1977)) or their $3(N-1)$
components, where $N\geq 3$ is the number of nuclei. We let $H(x)$ be the
electronic Hamiltonian, which initially we assume is taken in the
electrostatic model, although later we make some comments about the changes
necessary when fine structure effects are included. We label the energy levels
of $H(x)$ by a sequencing number $k$, and define a “coupled subspace” by a set
of adjacent energy levels,

${A}=\\{k_{0},k_{0}+1,\ldots,k_{0}+N_{l}-1\\},$ (1)

where $N_{l}$ is the number of coupled levels. Usually in practice $k_{0}$ is
the ground state and $N_{l}$ is small. We consider a region of nuclear
configuration space in which energy level $k_{0}$ is not degenerate with level
$k_{0}-1$ and level $k_{0}+N_{l}-1$ is not degenerate with level
$k_{0}+N_{l}$, that is, a region in which there are no degeneracies that cross
the boundaries of ${A}$. Degeneracies within ${A}$ (“internal degeneracies”)
are allowed, however (and these require $N_{l}\geq 2$). In practice, if the
region we are interested in does have degeneracies that cross the boundaries
of ${A}$ (caused by “intruder states” (Venghaus and Eisfeld (2016))), we can
either excise the locations where those degeneracies occur from the region in
question, or we can expand the definition of ${A}$ to make the problematic
degeneracies internal. If we choose the first option then the region after the
excision will often fail to be simply connected, which has topological
implications for the existence of smooth fields of frames.

For $k\in{A}$ we denote the energy eigenstates by $|ax;k\rangle$, so that

$H(x)|ax;k\rangle=\epsilon_{k}(x)|ax;k\rangle,\qquad k\in{A},$ (2)

where, as indicated, the energy eigenvalues are $\epsilon_{k}(x)$. The $a$ in
the notation $|ax;k\rangle$ indicates the adiabatic basis, and the
$x$-dependence of the eigenstate is separated by a semicolon from the quantum
number $k$.

We let ${\cal{S}}(x)$ be the subspace of the electronic Hilbert space spanned
by the set of vectors $|ax;k\rangle$ for $k\in{A}$, what we will call the
“coupled subspace.” The projection operator onto this subspace is

$P(x)=\sum_{k\in{A}}|ax;k\rangle\langle ax;k|.$ (3)

We denote the subspace of the Hilbert space orthogonal to ${\cal{S}}(x)$ by
${\cal{S}}^{\perp}(x)$, and denote the projection operator onto
${\cal{S}}^{\perp}(x)$ by

$Q(x)=1-P(x).$ (4)

For $k\notin{A}$ we write $|ax;k\rangle$ for a basis in ${\cal{S}}^{\perp}(x)$
that is discrete and orthonormal but otherwise arbitrary. We will call the set
of states $\\{|ax;k\rangle\\}$ for all $k$ the “adiabatic basis”, even though
the basis states for $k\notin{A}$ are not energy eigenstates. This is not
exactly standard terminology but in fact normal usage is often restricted to
the states $k\in{A}$. In addition there are reasons not to deal with energy
eigenstates when $k\notin{A}$, which present a number of difficulties, both
computational and theoretical. Computationally the high lying energy
eigenstates are seldom accessible, and theoretically they present problems
because their phase and frame conventions have singularities on surfaces that
proliferate at $k$ is increased. In addition there is the problem of the
continuum and the transition thereto, which changes as $x$ changes. Actually
one never needs to refer to energy eigenstates when $k\notin{A}$ and we shall
not do so. We must remember, however, that in our “adiabatic basis” the
vectors $|ax;k\rangle$ are energy eigenstates only when $k\in{A}$.

###  II.2 Phase Conventions and Smoothness

Energy eigenstates are defined only to within a phase convention (when
nondegenerate) or a frame convention inside the degenerate eigenspace (when
degenerate), and this must be kept in mind when using the notation
$|ax;k\rangle$ for $k\in{A}$. Although it is always possible to assign those
conventions at any given point $x$ of the region in question, in general this
assignment is not smooth on certain submanifolds of the region as $x$ is
varied. Notably this occurs when $x$ lies on an internal degeneracy (usually a
conical intersection). There may be additional singularities when $x$ is not
on an internal degeneracy; these occur on surfaces that can be moved about by
a gauge transformation like a branch cut in the complex plane or the strings
of monopoles but that cannot be eliminated. These singularities (of both
types) are the main drawback of the adiabatic basis.

In this article when we say that something is smooth we mean that it is a
continuous function of $x$ and that it has as many continuous derivatives as
needed for any applications we might make. A function is continuous at a point
$x_{0}$ if it is defined at $x_{0}$ and its limit as $x\to x_{0}$ is
independent of the direction of approach and equals the value of the function
at $x_{0}$. It is precisely in this sense that the adiabatic basis is not
continuous at an internal degeneracy (the limit exists as $x\to x_{0}$ but it
depends on the direction of approach), and the lack of continuity implies that
the derivatives of the adiabatic basis diverge at such points.

In fact the singularities of the adiabatic basis occur only on a subset of
measure zero, so elsewhere, on a subset of full measure, it is possible to
talk about the derivatives of the adiabatic basis vectors, as is commonly done
in the literature. Such derivatives occur in the derivative couplings and
other quantities of interest. Such quantities diverge as a degeneracy is
approached, and a great deal of literature is devoted to analyzing and
eliminating these divergences. In this article we will simply refrain from
ever assuming that the vectors $|ax;k\rangle$ are smooth functions of $x$,
and, in particular, we never talk about derivatives of these vectors with
respect to $x$. This applies both for $k\in{A}$ and $k\notin{A}$; at any given
$x$ we can choose $|ax;k\rangle$ for $k\notin{A}$ as a basis that spans
${\cal{S}}^{\perp}(x)$, but we will not assume that that assignment can be
made in a smooth manner as $x$ is varied.

The singularities of the adiabatic basis vectors $|ax;k\rangle$ for $k\in{A}$
are always due the impossibility of making phase and/or frame conventions in a
smooth manner as $x$ is varied, but the projection operator $P(x)$ is
independent of those conventions and is always smooth. (The projection
operator is not defined where degeneracies cross the boundaries of the set
$A$, but we are excluding such points from the region under consideration.)
This means that the derivatives of $P(x)$ exist and that $P(x)$ can be
expanded in a Taylor series about any point $x_{0}$, in which all the
derivatives are well defined. We will present such an expansion below. Since
$P(x)$ is smooth, $Q(x)=1-P(x)$ is too.

###  II.3 Diabatic Bases are Smooth

Now given a configuration $x_{0}$ we define a diabatic basis as one that is
smooth (free of singularities) in a neighborhood of $x_{0}$. In the literature
one is usually interested in the case that $x_{0}$ lies on a degeneracy
manifold (usually a seam or surface of conical intersection) but there are
applications for which this is not so and in the following we shall make no
assumptions about $x_{0}$. A diabatic basis so defined is not unique, since
any smooth field of unitary transformations on ${\cal{S}}(x)$ and another on
${\cal{S}}^{\perp}(x)$ will map one smooth basis into another. As is well
known (Pacher  _et al._ (1989); Bohm  _et al._ (1992); Mead (1992); Pacher,
Cederbaum, and Köppel (1993)), such changes of frame are gauge transformations
of the theory. Thus additional criteria must be imposed for the selection of a
unique diabatic basis.

To characterize a diabatic basis as one that is smooth does not completely
capture the practical requirements of such a basis. One could take a smooth
diabatic basis and subject it to a gauge transformation (a change of frame)
based on a unitary transformation that was a smooth function of $x$ in the
mathematical sense but rapidly varying on the atomic length scale. The
resulting basis would still be smooth in a mathematical sense but it would
have large derivative couplings and not be useful. So a diabatic basis must
not only be smooth it must be slowly varying, that is, it should have a
variation of order unity on the atomic length scale. One can formalize this in
terms of the dependence on the Born-Oppenheimer ordering parameter
$\kappa=(m/M)^{1/4}$; the diabatic basis should not depend on $\kappa$. In the
following, such understandings about the word “smooth” will be assumed.

For reference we will need a basis in the electronic Hilbert space that is
independent of $x$. For convenience we choose this basis to be
$|ax_{0};k\rangle$, which is the adiabatic basis at $x_{0}$. In the literature
when this basis is used to study dynamics at points $x\neq x_{0}$ it is
sometimes called the “crude adiabatic basis.”

###  II.4 A Variational Criterion

We now present a variational criterion for a diabatic basis, which leads to
the singular-value basis. Initially we focus on the part $k\in{A}$ of the
basis that spans the coupled subspace ${\cal{S}}(x)$. We define the diabatic
basis at $x$, denoted $|dx;k\rangle$ with a designation $d$ for “diabatic,” as
the frame at $x$ that is closest to the adiabatic frame at $x_{0}$, in the
sense that it minimizes the quantity

$\sum_{k\in{A}}\Big{|}|dx;k\rangle-|ax_{0};k\rangle\Big{|}^{2}=\sum_{k\in{A}}(2-\langle
ax_{0};k|dx;k\rangle-\langle dx;k|ax_{0};k\rangle).$ (5)

Both the diabatic basis $|dx;k\rangle$ and the adiabatic one $|ax;k\rangle$
for $k\in{A}$ span ${\cal{S}}(x)$, so we must have

$|dx;k\rangle=\sum_{l\in{A}}|ax;l\rangle\,U_{lk}(x),$ (6)

where $U_{lk}(x)$ is an $N_{l}\times N_{l}$ unitary matrix. Thus the quantity
(5) can be written,

$\sum_{k\in{A}}\left[2-\sum_{l\in{A}}(S_{kl}\,U_{lk}+S^{*}_{kl}U^{*}_{lk})\right],$
(7)

where

$S_{kl}=\langle ax_{0};k|ax;l\rangle.$ (8)

The notation $S$ for this matrix is the same as in Pacher, Cederbaum, and
Köppel (1993).

With no restrictions on $k$ and $l$, (8) defines $S$ as an infinite-
dimensional, unitary matrix, but for use in the variational principle we only
need the $N_{l}\times N_{l}$ block of this matrix, $S_{kl}$ for $k,l\in{A}$.
We find it convenient to partition infinite dimensional matrices like this
according to what we will call the $a$ and $b$ subsets of indices, which are
those for which $k\in{A}$ and $k\notin{A}$, respectively. For example, we
partition the infinite dimensional matrix $S$ as follows:

$S=\left(\begin{array}[]{cc}S^{aa}&S^{ab}\\\ S^{ba}&S^{bb}\end{array}\right),$
(9)

so that $S^{aa}$ is an $N_{l}\times N_{l}$ matrix, etc. In terms of this
notation, the quantity (5) or (7) can be written

$2N_{l}-[{\mathop{\textrm{tr}}}(S^{aa}U)+\hbox{\rm c.c.}].$ (10)

We assume that the adiabatic basis is given as a function of $x$ so that the
matrix $S^{aa}$ is known (but it is not assumed to be a smooth function of
$x$). We wish to minimize the quantity (7) with respect to the choice of the
unitary matrix $U_{kl}$ (where $k,l\in{A}$). We enforce the unitarity of $U$
by adding the term,

${\mathop{\textrm{tr}}}[\Lambda(UU^{\dagger}-I)]=\sum_{k,l\in{A}}\Lambda_{lk}\left(\sum_{m\in{A}}U_{km}U^{*}_{lm}-\delta_{kl}\right),$
(11)

where $\Lambda_{kl}$ is a Hermitian matrix of Lagrange multipliers. Now
varying with respect to either $U$ or $U^{\dagger}$ we find

$S^{aa}=U^{\dagger}\Lambda,$ (12)

in which $S^{aa}$ is given and unitary $U$ and Hermitian $\Lambda$ are to be
determined. The theory of the polar decomposition tells us that if $S^{aa}$ is
nonsingular then (12) has a unique solution for a unitary matrix $U$ and a
positive definite, Hermitian matrix $\Lambda$. We will show below that
$S^{aa}$ is, in fact, nonsingular in a neighborhood of $x_{0}$ and that
$\Lambda$ must be positive definite for a meaningful solution; thus we obtain
unique results for $U$ and $\Lambda$ over the neighborhood in question. That
is, since $(S^{aa})^{\dagger}S^{aa}=\Lambda^{2}$,
$\Lambda=[(S^{aa})^{\dagger}S^{aa}]^{1/2}$, where here and below by the square
root of a positive definite, Hermitian matrix we mean the unique positive
definite, Hermitian square root; and given $\Lambda$,
$U=\Lambda(S^{aa})^{-1}$. Then (6) gives us the diabatic basis at $x$ in terms
of a finite linear combination of adiabatic basis vectors at $x$. This is
identical to the singular-value diabatic basis of Pacher, Cederbaum, and
Köppel (1988).

###  II.5 Smoothness of the Solution

This construction does not, however, make it evident that the resulting
diabatic basis is a smooth function of $x$. In fact, it is smooth, but neither
$S^{aa}$, $U$ or $\Lambda$ are. Since singularities are due to the phase and
frame conventions for the adiabatic basis states $|ax;k\rangle$ (frame
conventions applying inside a degenerate subspace), let us change these
conventions for $k\in{A}$ by means of an $N_{l}\times N_{l}$ unitary matrix
$V$,

$|a^{\prime}x;k\rangle=\sum_{l\in{A}}|ax;l\rangle V_{lk},$ (13)

and study how various quantities transform under such a change. In (13) $V$ is
allowed to be a nonsmooth function of $x$. Combining (13) with (6), the
definition of $U$, and with (8), the definition of $S^{aa}$, we find

$S^{aa\prime}=S^{aa}V,\qquad
U^{\prime}=V^{\dagger}U,\qquad\Lambda^{\prime}=V^{\dagger}\Lambda V.$ (14)

Thus neither $S^{aa}$, $U$ nor $\Lambda$ is smooth, in general.

On the other hand, the products $S^{aa}U=U^{\dagger}\Lambda U$ and
$S^{aa}(S^{aa})^{\dagger}=U\Lambda^{2}U$ are invariant under the change (13)
and are therefore smooth (for example, $S^{aa\,\prime}U^{\prime}=S^{aa}U$), as
is the quantity $|\det S^{aa}|$. As for the latter, we note that at $x=x_{0}$,
$S^{aa}=I$ so $|\det S^{aa}|=1$. Therefore by continuity $\det S^{aa}$ is
bounded away from 0 in some finite neighborhood of $x_{0}$ (and note that we
can use continuity arguments only on smooth functions), and therefore $S^{aa}$
is nonsingular in this neighborhood.

Let us define

$\Sigma=S^{aa}U=U^{\dagger}\Lambda U=\Sigma^{\dagger},$ (15)

which, as noted, is a smooth function of $x$. To get its value at $x_{0}$ we
require of our diabatic basis that $|dx_{0};k\rangle=|ax_{0};k\rangle$, that
is, it agree with the adiabatic basis at $x_{0}$. This means that $U=I$ at
$x=x_{0}$, and thus also $\Lambda=\Sigma=I$ at $x_{0}$. But since $\Sigma$ is
a smooth function of $x$, it must be positive definite in a neighborhood of
$x_{0}$. This means that $\Lambda$, which is not smooth, is also positive
definite in the same neighborhood.

These considerations suggest that rather than factoring $S^{aa}$ via the polar
decomposition as in (12), we factor it the other way using $S^{aa}=\Sigma
U^{\dagger}$ to obtain $\Sigma$ and $U$. That is, we compute
$\Sigma=[S^{aa}(S^{aa})^{\dagger}]^{1/2}$, and then compute
$U=(S^{aa})^{-1}\Sigma$. The advantage of this approach is that $\Sigma$, a
smooth function of $x$, takes the place of the nonsmooth $\Lambda$, which is
never needed.

###  II.6 Relation to the Projection Operator

We still have not shown that the diabatic basis $|dx;k\rangle$ is a smooth
function of $x$, but if it is we guess that it should be possible to express
it purely in terms of the projection operator $P(x)$, which is known to be
smooth. This turns out to be the case, as we now show.

For $k\in{A}$ the procedure above gives the diabatic basis $|dx;k\rangle$ as a
finite linear combination of the adiabatic basis $|ax;k\rangle$, also for
$k\in{A}$. For some purposes, however, it is convenient to have the diabatic
basis expressed as a linear combination of the reference basis
$|ax_{0};k\rangle$, in spite of the fact that this involves an infinite number
of terms. We obtain a useful form of this expansion by writing, for $k\in{A}$,

$\displaystyle|dx;k\rangle$ $\displaystyle=$ $\displaystyle
P(x)|dx;k\rangle=\sum_{{\rm all}\;l}|ax_{0};l\rangle\langle
ax_{0};l|\sum_{m\in{A}}|ax;m\rangle\langle ax;m|dx;k\rangle$ (16)
$\displaystyle=$ $\displaystyle\sum_{l,m\in{A}}|ax_{0};l\rangle
S^{aa}_{lm}U_{mk}+\sum_{l\notin{A},m\in{A}}|ax_{0};l\rangle S^{ba}_{lm}U_{mk}$
$\displaystyle=$
$\displaystyle\sum_{l\in{A}}|ax_{0};l\rangle(S^{aa}U)_{lk}+\sum_{l\notin{A}}|ax_{0};l\rangle(S^{ab}U)_{lk}.$

Evidently, for this expansion we need the matrices $S^{aa}U=\Sigma$ and
$S^{ba}U$.

We define the matrix representation of the projection operator $P(x)$ in the
reference basis,

$P_{kl}(x)=\langle ax_{0};k|P(x)|ax_{0};l\rangle=\sum_{m\in{A}}\langle
ax_{0};k|dx;m\rangle\langle dx;m|ax_{0};l\rangle,$ (17)

for all $k$, $l$. Now substituting (16) into (17), we obtain

$\displaystyle P^{aa}$ $\displaystyle=$
$\displaystyle(S^{aa}U)(S^{aa}U)^{\dagger}=S^{aa}(S^{aa})^{\dagger}=\Sigma^{2},$
$\displaystyle P^{ba}$ $\displaystyle=$
$\displaystyle(S^{ba}U)(S^{aa}U)^{\dagger}=S^{ba}(S^{aa})^{\dagger},$ (18)

where we partition $P(x)$ as in (9). Thus the matrix $S^{aa}U$ that appears in
(16) is the positive definite matrix $\Sigma=(P^{aa})^{1/2}$. As for the
matrix $S^{ba}U$, we have

$P^{ba}=S^{ba}U(S^{aa}U)^{\dagger}=S^{ba}U\Sigma^{\dagger}=S^{ba}U\Sigma,$
(19)

or,

$S^{ba}U=P^{ba}(P^{aa})^{-1/2}.$ (20)

Thus, both matrices needed in the expansion (16) can be expressed in terms of
the projection operator.

Let us write $T$ for the infinite-dimensional unitary matrix that connects the
diabatic basis $|dx;k\rangle$ with the reference basis $|ax_{0};k\rangle$,

$|dx;k\rangle=\sum_{{\rm all}\;l}|ax_{0};l\rangle T_{lk},$ (21)

which holds for all $k$, $l$. Again, this notation follows Pacher, Cederbaum,
and Köppel (1988, 1993), and we note that their variational principle
minimizes the quantity

$\big{|}T-I\big{|}^{2}.$ (22)

So far we have worked on the case $k\in{A}$ and found $T^{aa}=S^{aa}U$ and
$T^{ba}=S^{ba}U$ in terms of the projection operator. We can find the other
blocks, $T^{ab}$ and $T^{bb}$, by repeating the same procedure we have just
been through, but now taking $k\notin{A}$ and swapping subspaces
$a\leftrightarrow b$ and projectors $P\leftrightarrow Q$. This gives
altogether,

$\displaystyle T$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}(P^{aa})^{1/2}&Q^{ab}(Q^{bb})^{-1/2}\\\
P^{ba}(P^{aa})^{-1/2}&(Q^{bb})^{1/2}\end{array}\right)$ (25) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}P^{aa}&Q^{ab}\\\
P^{ba}&Q^{bb}\end{array}\right)\left(\begin{array}[]{cc}(P^{aa})^{-1/2}&0\\\
0&(Q^{bb})^{-1/2}\end{array}\right),$ (30)

where we note that $Q^{ab}=-P^{ab}$ and $Q^{bb}=I^{bb}-P^{bb}$, so that the
entire result can be expressed in terms of $P$. It can be directly checked
that $TT^{\dagger}=T^{\dagger}T=I$; for this one must use the facts that
$P=P^{\dagger}$ and $P^{2}=P$.

Now (21) and (30) show that the transformation from the reference basis to the
singular-value diabatic basis can be expressed purely in terms of the
projection operator. This not only shows that the diabatic basis is smooth in
a neighborhood of $x_{0}$ but it also allows us to expand the diabatic basis
in a Taylor series about $x_{0}$.

##  III Expansion of the Projection Operator

In the following we will use the symbol $P$ or $P(x)$ for both the projection
operator and its matrix in the reference basis, as in (17), with hopefully
little danger of confusion. We wish to expand $P$ in a power series about the
reference point $x_{0}$. We write $x=x_{0}+\xi$, where $\xi$ is a displacement
vector in nuclear configuration space. We think of all the symbols, $x$,
$x_{0}$ and $\xi$, as standing for the $N-1$ Jacobi vectors or their $3(N-1)$
components, effectively using these as coordinates on the nuclear
configuration space in the center-of-mass system. We will write these
coordinates explicitly as $x^{\mu}$, $x_{0}^{\mu}$, $\xi^{\mu}$, etc., where
$\mu=1,\ldots,3N-3$. We use the summation convention on such coordinate
indices $\mu$, $\nu$, etc.

The expansion of the projection operator is a part of degenerate or
quasidegenerate perturbation theory, see for example Kato (1949); Bloch
(1958); des Cloizeaux (1960); Klein (1974), which discuss, among other
techniques, the use of Cauchy’s theorem and the resolvent operator to obtain
expansions of the projection operator. In the following we summarize a
straightforward approach, which is suitable to the order to which we carry the
expansion. We simply note that the usual goal of perturbation theory is to
expand the energy eigenstates (which in our case would be the adiabatic basis)
in a power series, something which however must cope with degeneracies, near
degeneracies and small or vanishing energy denominators, none of which is an
issue in the expansion of the projection operator. These complications are due
to the singularity of the adiabatic basis at a degeneracy, which we are
avoiding by using the projection operator.

We expand the projection operator in a power series in $\xi$,

$P(x)=P_{0}+P_{1}+P_{2}+\ldots,$ (31)

where

$P_{0}=P(x_{0}),\;P_{1}=\xi^{\mu}\frac{\partial P}{\partial
x^{\mu}}(x_{0}),\;P_{2}=\frac{1}{2}\xi^{\mu}\xi^{\nu}\frac{\partial^{2}P}{\partial
x^{\mu}\partial x^{\nu}}(x_{0}),$ (32)

etc., and similarly we expand the Hamiltonian,

$H(x)=H_{0}+H_{1}+H_{2}+\ldots.$ (33)

We carry out the perturbation expansion by requiring
$P(x)^{\dagger}=P(x)^{2}=P(x)$ and $[P(x),H(x)]=0$. At zeroth order, that is,
at $x=x_{0}$, we have

$P_{0}=\left(\begin{array}[]{cc}I^{aa}&0\\\ 0&0\end{array}\right),\qquad
H_{0}=\left(\begin{array}[]{cc}H_{0}^{aa}&0\\\
0&H_{0}^{bb}\end{array}\right),$ (34)

where matrices are partitioned as in (9). Since the reference basis is an
energy eigenbasis at $x_{0}$ for $k\in{A}$, we have

$H^{aa}_{0,kl}=\epsilon_{0k}\,\delta_{kl},$ (35)

where $\epsilon_{0k}=\epsilon_{k}(x_{0})$ for $k\in{A}$. Equation (35) applies
when $k,l\in{A}$; in equations like this we shall not indicate the ranges of
indices when they are evident from the superscripts (for example, $aa$ in this
case).

At first order we must satisfy $P_{0}P_{1}+P_{1}P_{0}=P_{1}$, which gives
$P^{aa}_{1}=0$ and $P^{bb}_{1}$=0. In addition, when we require
$[P_{0},H_{1}]+[P_{1},H_{0}]=0$, we find

$P^{ab}_{1,kl}=\sum_{m\notin{A}}H^{ab}_{1,km}R(\epsilon_{0k})_{ml},$ (36)

where the resolvent $R(\epsilon)$ is a $bb$-type matrix defined by

$\sum_{m\notin{A}}R(\epsilon)_{km}\big{(}\epsilon\,\delta_{ml}-H^{bb}_{0,ml}\big{)}=\delta_{kl},$
(37)

that is,

$R(\epsilon)=(\epsilon I^{bb}-H^{bb}_{0})^{-1}.$ (38)

The resolvent $R(\epsilon)$ is defined when $\epsilon=\epsilon_{0k}$ for
$k\in{A}$, which is outside the range of eigenvalues of $H^{bb}_{0}$. Thus we
can write

$P_{1}=\left(\begin{array}[]{cc}0&P^{ab}_{1}\\\
P^{ba}_{1}&0\end{array}\right),$ (39)

where $P^{ba}_{1}=(P^{ab}_{1})^{\dagger}$.

At second order we require $P_{0}P_{2}+P_{1}^{2}+P_{2}P_{0}=P_{2}$, which
gives $P^{aa}_{2}=-P^{ab}_{1}\,P^{ba}_{1}$ and
$P^{bb}_{2}=+P^{ba}_{1}\,P^{ab}_{1}$. We also require
$[P_{0},H_{2}]+[P_{1},H_{1}]+[P_{2},H_{0}]=0$, which gives

$P^{ab}_{2,kl}=\sum_{m\notin{A}}\big{(}H^{ab}_{2}+P^{ab}_{1}\,H^{bb}_{1}-H^{aa}_{1}\,P^{ab}_{1}\big{)}_{km}\,R(\epsilon_{0k})_{ml}.$
(40)

Thus we can write

$P_{2}=\left(\begin{array}[]{cc}-P^{ab}_{1}P^{ba}_{1}&P^{ab}_{2}\\\
P^{ba}_{2}&P^{ba}_{1}P^{ab}_{1}\end{array}\right).$ (41)

Finally, at third order we find

$P_{3}=\left(\begin{array}[]{cc}-P^{ab}_{1}P^{ba}_{2}-P^{ab}_{2}P^{ba}_{1}&P^{ab}_{3}\\\
P^{ba}_{3}&P^{ba}_{1}P^{ab}_{2}+P^{ba}_{2}P^{ab}_{1}\end{array}\right)$ (42)

where

$\displaystyle P^{ab}_{3,kl}$ $\displaystyle=$
$\displaystyle\sum_{m\notin{A}}(H^{ab}_{3}+P^{ab}_{1}H^{bb}_{2}-H^{aa}_{2}P^{ab}_{1}-P^{ab}_{1}P^{ba}_{1}H^{ab}_{1}$
(43) $\displaystyle+$ $\displaystyle
P^{ab}_{2}H^{bb}_{1}-H^{aa}_{1}P^{ab}_{2}-H^{ab}_{1}P^{ba}_{1}P^{ab}_{1})_{km}R(\epsilon_{0k})_{ml},$

and $P^{ba}_{3}=(P^{ab}_{3})^{\dagger}$.

Now we can take the square roots indicated in (30), which to third order are
given by

$\displaystyle(P^{aa})^{1/2}$ $\displaystyle=$ $\displaystyle
I^{aa}-\frac{1}{2}P^{ab}_{1}P^{ba}_{1}-\frac{1}{2}(P^{ab}_{1}P^{ba}_{2}+P^{ab}_{2}P^{ba}_{1}),$
(44a) $\displaystyle(Q^{bb})^{1/2}$ $\displaystyle=$ $\displaystyle
I^{bb}-\frac{1}{2}P^{ba}_{1}P^{ab}_{1}-\frac{1}{2}(P^{ba}_{1}P^{ab}_{2}+P^{ba}_{2}P^{ab}_{1}).$
(44b)

These allow us to write out the expansion of $T$,

$\displaystyle T$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}I^{aa}&0\\\
0&I^{bb}\end{array}\right)+\left(\begin{array}[]{cc}0&-P^{ab}_{1}\\\
P^{ba}_{1}&0\end{array}\right)+\left(\begin{array}[]{cc}-(1/2)P^{ab}_{1}P^{ba}_{1}&-P^{ab}_{2}\\\
P^{ba}_{2}&-(1/2)P^{ba}_{1}P^{ab}_{1}\end{array}\right)$ (51) $\displaystyle+$
$\displaystyle\left(\begin{array}[]{cc}-(1/2)(P^{ab}_{1}P^{ba}_{2}+P^{ab}_{2}P^{ba}_{1})&-P^{ab}_{3}-(1/2)P^{ab}_{1}P^{ba}_{1}P^{ab}_{1}\\\
P^{ba}_{3}+(1/2)P^{ba}_{1}P^{ab}_{1}P^{ba}_{1}&-(1/2)(P^{ba}_{1}P^{ab}_{2}+P^{ba}_{2}P^{ab}_{1})\end{array}\right),$
(54)

which gives the expansion of the singular-value diabatic basis about an
arbitrary point $x_{0}$ to third order.

##  IV A Diabatic Basis via Parallel Transport

Parallel transport is a standard topic in gauge theories (Kobayashi and Nomizu
(1963); Nakahara (2003)) and is an important part of the theory of Berry’s
phases (Berry (1984); Simon (1983)). Its role in Born-Oppenheimer theory is
also well known (Mead (1992); Bohm, Boya, and Kendrick (1991)). Parallel
transport can be used to create a diabatic basis; one simply takes a basis at
a reference point $x_{0}$, possibly on a conical intersection, and parallel
transports it along radial lines emanating from $x_{0}$ to fill in a basis
defined over a neighborhood of $x_{0}$. Such bases have a long history in the
chemical literature, going back at least to Smith (1969), who used parallel
transport in diatomics to create a strictly diabatic basis, possible in his
case because his parameter space was 1-dimensional. Strictly diabatic bases
(by any means of construction) do not exist in the polyatomic molecules
considered in this article (Mead and Truhlar (1982)), but for a contrary
opinion see Baer (1975, 1976, 1980); Baer and Englman (1992); Baer and Alijah
(2000); Baer (2000a, b, 2001, 2002).

###  IV.1 Parallel Transport

We review some aspects of parallel transport that we will need; for a slightly
different perspective see Mead (1992). Given a vector
$\psi(x)\in{\cal{S}}(x)$, we wish to find the vector
$\psi(x+dx)=\psi(x)+d\psi$ at a nearby point such that
$\psi(x+dx)\in{\cal{S}}(x+dx)$ and such the square distance in Hilbert space

$\big{|}\psi(x+dx)-\psi(x)\big{|}^{2}$ (55)

is minimum. Writing simply $\psi$ for $\psi(x)$ and likewise $P$ for $P(x)$,
we have $P\psi=\psi$ and $(P+dP)(\psi+d\psi)=\psi+d\psi$, or,

$d\psi=Pd\psi+(dP)\psi.$ (56)

We regard $\psi$ as given and $d\psi$ as unknown. Since $d\psi=Pd\psi+Qd\psi$
we see that $(dP)\psi=Qd\psi$ and that the two terms in (56) are orthogonal.
Also, the term $(dP)\psi$, the component of $d\psi$ orthogonal to
${\cal{S}}(x)$, is fixed; therefore to minimize the square distance (55) we
choose $d\psi$ so that it is orthogonal to ${\cal{S}}(x)$, that is, so that
$Pd\psi=0$.

This gives $d\psi=(dP)\psi$. If now the step from $x$ to $x+dx$ is taken along
a curve $x(\lambda)$, then by dividing by $d\lambda$ we obtain an equation of
parallel transport along the curve,

$\psi^{\prime}=P^{\prime}\psi,$ (57)

where the prime means $d/d\lambda$. This equation is to be used only for
vectors $\psi(x)$ that lie in ${\cal{S}}(x)$, that is, that satisfy $Q\psi=0$.
It can be shown (see Appendix A) that (57) implies

$\left(\frac{d}{d\lambda}+P^{\prime}\right)(Q\psi)=0,$ (58)

so that if $Q\psi=0$ at $\lambda=0$ then $Q\psi=0$ for all $\lambda$ (this
follows from the uniqueness of the solution of the differential equation).
Similarly, if we wish to transport a vector that remains in
${\cal{S}}^{\perp}(x)$ if it starts out in that space, then we use the
transport equation

$\psi^{\prime}=Q^{\prime}\psi=-P^{\prime}\psi.$ (59)

We can combine both types of parallel transport into a single equation by
writing

$\psi^{\prime}=(P^{\prime}P-PP^{\prime})\psi=[P^{\prime},P]\psi.$ (60)

It can be shown that if $\psi$ satisfies this equation then $Q\psi$ satisfies
(58), so that if $\psi\in{\cal{S}}(x)$ initially then it remains in
${\cal{S}}(x)$; and it can be shown that if $\psi$ satisfies this equation and
$Q\psi=0$ then $\psi$ satisfies the simpler equation (57). (See Appendix A.)
Thus, (57) and (60) are equivalent if $\psi\in{\cal{S}}(x)$. Similarly it can
be shown that (59) and (60) are equivalent if $\psi\in{\cal{S}}^{\perp}(x)$.

The form (60) is convenient because it applies to the transport of any vector,
one in ${\cal{S}}(x)$, one in ${\cal{S}}^{\perp}(x)$, or any linear
combination thereof. Also, since the commutator $[P^{\prime},P]$ is anti-
Hermitian, it follows that parallel transport via (60) is unitary, and
preserves the scalar product of vectors. In particular, the parallel transport
of an orthonormal frame is orthonormal. Furthermore, if we parallel transport
a frame that block diagonalizes the Hamiltonian, it continues to do so as we
evolve along the curve. Finally, we note that another useful form for (60) is

$\psi^{\prime}=(P^{\prime}P+Q^{\prime}Q)\psi.$ (61)

We mention this because it makes it evident how parallel transport is
generalized to the case in which there are multiple subspaces of the
electronic Hilbert spaee under consideration (not just two).

###  IV.2 The Parallel Transported Basis

Now let us replace the displacement $\xi$ by $\lambda\xi$, so that

$x^{\mu}(\lambda)=x_{0}^{\mu}+\lambda\xi^{\mu},$ (62)

which represents a radial line parameterized by $\lambda$ starting at $x_{0}$
when $\lambda=0$. We expand the Hamiltonian $H(x)$ and projector $P(x)$ as
before, except now with a $\lambda$ dependence, so that for example

$H(x(\lambda))=H_{0}+\lambda H_{1}+\lambda^{2}H_{2}+\lambda^{3}H_{3}+\ldots,$
(63)

instead of (33). We expand $P(x(\lambda))$ and a parallel-transported wave
function $\psi(\lambda)$ similarly. The wave function $\psi$ is initially
defined only along a single curve, so it is a function of $\lambda$ but not of
$x$.

Then we have

$\displaystyle P(x)$ $\displaystyle=$ $\displaystyle P_{0}+\lambda
P_{1}+\lambda^{2}P_{2}+\lambda^{3}P_{3}+\ldots,$ (64a) $\displaystyle
P^{\prime}(x)$ $\displaystyle=$ $\displaystyle P_{1}+2\lambda
P_{2}+3\lambda^{2}P_{3}+\ldots,$ (64b) $\displaystyle[P^{\prime},P]$
$\displaystyle=$ $\displaystyle[P_{1},P_{0}]+2\lambda[P_{2},P_{0}]$ (64c)
$\displaystyle+$
$\displaystyle\lambda^{2}\big{(}3[P_{3},P_{0}]+[P_{2},P_{1}]\big{)}+\ldots$

and with
$\psi=\psi_{0}+\lambda\psi_{1}+\lambda^{2}\psi_{2}+\lambda^{3}\psi_{3}+\ldots$,
(60) becomes

$\displaystyle\psi_{1}$ $\displaystyle=$ $\displaystyle[P_{1},P_{0}]\psi_{0},$
(65a) $\displaystyle 2\psi_{2}$ $\displaystyle=$
$\displaystyle[P_{1},P_{0}]\psi_{1}+2[P_{2},P_{0}]\psi_{0},$ (65b)
$\displaystyle 3\psi_{3}$ $\displaystyle=$
$\displaystyle[P_{1},P_{0}]\psi_{2}+2[P_{2},P_{0}]\psi_{1}$ (65c)
$\displaystyle+$
$\displaystyle\big{(}3[P_{3},P_{0}]+[P_{2},P_{1}]\big{)}\psi_{0},$

etc. These allow us to express $\psi_{1}$, $\psi_{2}$, etc., in terms of
$\psi_{0}$.

We write the result in the following way. Define $\psi_{k}(\lambda)$ as the
components of $\psi(\lambda)$ with respect to the reference basis
$|ax_{0};k\rangle$, that is, $\psi_{k}(\lambda)=\langle
ax_{0};k|\psi(\lambda)\rangle$. Then $\psi_{k}(\lambda)$ is a linear function
of $\psi_{k}(0)$, which is expressed by

$\psi_{k}(\lambda)=\sum_{{\rm all}\;l}T_{p;kl}\,\psi_{l}(0),$ (66)

where $T_{p}$ is an infinite-dimensional matrix that depends on $\lambda$
along a single curve, or, with $\lambda=1$ and $\xi$ regarded as a variable,
on $x=x_{0}+\xi$. This matrix plays a similar role to $T$ for the singular-
value diabatic basis and, in particular, if we define $|px;k\rangle$ as the
basis that is parallel transported along radial lines from $x_{0}$, with the
initial values being the reference basis $|ax_{0};k\rangle$ at $x_{0}$, then

$|px;k\rangle=\sum_{{\rm all}\;l}|ax_{0};l\rangle[T_{p}(x)]_{lk},$ (67)

which may be compared to (21). The subscript on $T_{p}$ refers to the
parallel-transported basis.

Now we expand $T_{p}=T_{p0}+\lambda T_{p1}+\lambda^{2}T_{p2}+\ldots$ and use
(65) to obtain

$\displaystyle T_{p1}$ $\displaystyle=$ $\displaystyle[P_{1},P_{0}],$ (68a)
$\displaystyle T_{p2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[P_{1},P_{0}]T_{p1}+[P_{2},P_{0}],$ (68b)
$\displaystyle T_{p3}$ $\displaystyle=$
$\displaystyle\frac{1}{3}[P_{1},P_{0}]T_{p2}+\frac{2}{3}[P_{2},P_{0}]T_{p1}$
(68c) $\displaystyle+$ $\displaystyle[P_{3},P_{0}]+\frac{1}{3}[P_{2},P_{1}].$

Then we set $\lambda=1$ and work out the commutators to obtain $T_{p}$ through
third order,

$\displaystyle T_{p}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}I^{aa}&0\\\
0&I^{bb}\end{array}\right)+\left(\begin{array}[]{cc}0&-P^{ab}_{1}\\\
P^{ba}_{1}&0\end{array}\right)+\left(\begin{array}[]{cc}-(1/2)P^{ab}_{1}P^{ba}_{1}&-P^{ab}_{2}\\\
P^{ba}_{2}&-(1/2)P^{ba}_{1}P^{ab}_{1}\end{array}\right)$ (75) $\displaystyle+$
$\displaystyle\left(\begin{array}[]{cc}-(2/3)P^{ab}_{1}P^{ba}_{2}-(1/3)P^{ab}_{2}P^{ba}_{1}&-P^{ab}_{3}-(1/2)P^{ab}_{1}P^{ba}_{1}P^{ab}_{1}\\\
P^{ba}_{3}+(1/2)P^{ba}_{1}P^{ab}_{1}P^{ba}_{1}&-(2/3)P^{ba}_{1}P^{ab}_{2}-(1/3)P^{ba}_{2}P^{ab}_{1}\end{array}\right).$
(78)

Comparing this with (54) we see that the singular-value diabatic basis and the
parallel-transported diabatic basis are identical through second order in an
expansion about $x_{0}$ but differ at third order. The two expansions cannot
be identical to all orders, since in effect each small step of the parallel
transport involves an infinitesimal version of the singular-value basis, but
one that compares frames between $x$ and $x+dx$ rather than between $x_{0}$
and $x$.

###  IV.3 Hamiltonian in Parallel-Transported Basis

Let us write

$H_{p,kl}(x)=\langle px;k|H(x)|px;l\rangle$ (79)

for the Hamiltonian matrix in the parallel-transported basis, while other
matrices ($H_{1}$, $P_{2}$, etc.) are understood to be in the reference basis
$|ax_{0};k\rangle$, as previously. Then

$H_{p}(x)=T_{p}(x)^{\dagger}H(x)T_{p}(x).$ (80)

Now we expand $H(x)$ as in (33) and $T_{p}(x)$ as in (68) and multiply series
to compute $H_{p}(x)$ to second order. We find that the off-diagonal elements
vanish, $H^{ab}_{p}(x)=H^{ba}_{p}(x)=0$, as they should, while through first
order the diagonal elements of $H_{p}(x)$ are the same as those of $H(x)$,
that is, $H^{aa}_{p0}=H^{aa}_{0}$, $H^{aa}_{p1}=H^{aa}_{1}$,
$H^{bb}_{p0}=H^{bb}_{0}$ and $H^{bb}_{p1}=H^{bb}_{1}$. As for the diagonal
elements at second order, we find

$\displaystyle H^{aa}_{p2}$ $\displaystyle=$ $\displaystyle
H^{aa}_{2}+\frac{1}{2}(H^{ab}_{1}P^{ba}_{1}+P^{ab}_{1}H^{ba}_{1}),$ (81)
$\displaystyle H^{bb}_{p2}$ $\displaystyle=$ $\displaystyle
H^{bb}_{2}-\frac{1}{2}(H^{ba}_{1}P^{ab}_{1}+P^{ba}_{1}H^{ab}_{1}).$ (82)

We note that if the electronic Hamiltonian at $x_{0}$ is completely degenerate
in the coupled subspace, then we can use (127) to obtain

$H^{aa}_{p2}=H^{aa}_{2}+H^{ab}_{1}R(\epsilon_{0})H^{ba}_{1},$ (83)

the usual result from degenerate perturbation theory.

Now by diagonalizing $H^{aa}_{p}$ we obtain the adiabatic basis
$|ax;k\rangle$, if we want it. If $x_{0}$ is on an internal degeneracy (a seam
or conical intersection), then this will reveal the usual singularities of the
adiabatic basis, for example, the dependence of $|ax;k\rangle$ on the
direction of approach as $x\to x_{0}$ and the $\pi$ phase discontinuity as we
encircle the degeneracy manifold in the case of a 2-fold degeneracy in the
electrostatic model (Herzberg and Longuet-Higgins (1963); Longuet-Higgins
(1975); Mead and Truhlar (1979)). In the case of a triatomic with an odd
number of electrons and spin-orbit effects included it will reveal the string
of a Dirac monopole (Mead (1980a, 1987)).

###  IV.4 The Derivative Couplings and Connection

If $|x;k\rangle$ is any smooth basis that block diagonalizes the Hamiltonian
then we define the derivative couplings by

$F_{\mu,kl}=\langle x;k|\partial_{\mu}|x;l\rangle=-(\partial_{\mu}\langle
x;k|)|x;l\rangle=-F_{\mu,lk}^{*},$ (84)

where $\partial_{\mu}=\partial/\partial x^{\mu}$ and where the alternative
forms follow from the orthonormality of the basis vectors. For example, the
basis $|x;k\rangle$ could be $|dx;k\rangle$ or $|px;k\rangle$ or the adiabatic
basis $|ax;k\rangle$ in regions where the latter is smooth. If we just write
$F_{\mu}$ we mean the infinite dimensional matrix whose components are
$F_{\mu,kl}$; this can be decomposed into $aa$-, $ab$-, etc., blocks. As is
well known, the derivative couplings appear when the molecular Schrödinger
equation is transformed to the Born-Oppenheimer representation. The matrix of
derivative couplings $F_{\mu}$, as well as its diagonal blocks, $F^{aa}_{\mu}$
and $F^{bb}_{\mu}$, are anti-Hermitian and thus belong to the Lie algebra of
the unitary group.

Any process of parallel transport gives rise to a connection, which in the
general case is a rule for associating a unique object at a point $x+dx$,
given that object at point $x$. The object in question could be a vector such
as $\psi$, a frame, or other things. In the case of transporting the vector
$\psi$ as in Sec. IV.1, the components of the connection, which we denote by
$\Gamma_{\mu,kl}$, are defined by

$\frac{d\psi_{k}}{d\lambda}=-\sum_{{\rm
all}\,l}\Gamma_{\mu,kl}\frac{dx^{\mu}}{d\lambda}\,\psi_{l},$ (85)

where the minus sign is conventional. If we now substitute

$|\psi(\lambda)\rangle=\sum_{{\rm
all}\,k}|x(\lambda);k\rangle\,\psi_{k}(\lambda)$ (86)

into the equation of parallel transport (60) or (61) we find (see Appendix A),

$\Gamma_{\mu}=\left(\begin{array}[]{cc}F^{aa}_{\mu}&0\\\
0&F^{bb}_{\mu}\end{array}\right),$ (87)

that is, the diagonal blocks of the connection are the same as the derivative
couplings. The off-diagonal components of the connection vanish because of the
way we designed the process of parallel transport in Sec. IV.1, that is, so
that a vector that begins in one of the subspaces ${\cal{S}}(x)$ or
${\cal{S}}^{\perp}(x)$ remains in that subspace.

If $|bx;k\rangle$ is any smooth basis that block-diagonalizes the Hamiltonian,
where $b$ just means “basis,” then another such basis is given by

$|b^{\prime}x;k\rangle=\sum_{l}|bx;l\rangle\,U_{lk}(x),$ (88)

where $U(x)$ is unitary, a smooth function of $x$ and also block-diagonal.
Then the derivative couplings in the new basis are given by

$F^{\prime}_{\mu}=U^{\dagger}F_{\mu}U+U^{\dagger}\partial_{\mu}U,$ (89)

which, as is well known, is the transformation law for a (generally) non-
Abelian gauge potential. Because of the inhomogeneous term the derivative
couplings can be made to take on any value at any given point by means of such
a gauge transformation. Indeed, in both the singular-value and parallel-
transported basis the derivative couplings vanish at the reference point
$x_{0}$. Since a common goal in practice is to minimize the derivative
couplings over some region, naturally the question arises as to how fast these
couplings grow (in some basis) as we move away from a point where they are
known to vanish. This question leads us to the curvature, which we take up
next.

###  IV.5 Curvature

Let $|bx;k\rangle$ be an arbitrary, smooth basis that block-diagonalizes the
Hamiltonian, as above, and let $\xi^{\mu}$ and $\eta^{\mu}$ be two
infinitesimal displacements from a point $x$. Then let us carry out a parallel
transport around the parallelogram, $x\to x+\xi\to x+\xi+\eta\to x+\eta\to x$.
Initially we parallel transport the $a$-part of the frame, that is, the
vectors $|bx;k\rangle$ for $k\in{A}$. This is a standard calculation in gauge
theories (Kobayashi and Nomizu (1963); Nakahara (2003); Mead (1992)) and we
just quote the results. Since parallel transport preserves orthonormality, the
parallel-transported frame, once again back at the starting point $x$, must be
related to the original frame at that point by a unitary matrix. Moreover, the
unitary matrix must be infinitesimal, that is, the identity plus an
infinitesimal, anti-Hermitian correction. If we denote the parallel-
transported frame, back at $x$, by $|{\tilde{b}}x;k\rangle$, then we have

$|{\tilde{b}}x;k\rangle=\sum_{l\in{A}}|bx;l\rangle\,[\delta_{lk}-\xi^{\mu}\eta^{\nu}\,G^{aa}_{\mu\nu,lk}(x)],$
(90)

which defines the $aa$-block of the curvature tensor $G_{\mu\nu;kl}$ at $x$.
In this formula, $G$ is understood to be expressed in the basis
$|bx;k\rangle$; if we omit the $k,l$ indices, we refer to the matrix
$G^{aa}_{\mu\nu}$. The calculation shows that

$G^{aa}_{\mu\nu}=\partial_{\mu}F^{aa}_{\nu}-\partial_{\nu}F^{aa}_{\mu}+[F^{aa}_{\mu},F^{aa}_{\nu}].$
(91)

If we carry out this calculation for the $b$-block, we find the same equation
but with $aa$ replaced by $bb$; and the off-diagonal blocks of $G$ vanish,
$G^{ab}_{\mu\nu}=G^{ba}_{\mu\nu}=0$, for the same reason that the off-diagonal
blocks of the connection do (see (87)).

Given the fact that $F^{aa}_{\mu}$ and $F^{bb}_{\mu}$ are anti-Hermitian, it
follows from (91) that $G^{aa}_{\mu\nu}$ and $G^{bb}_{\mu\nu}$ are, too, that
is, $G^{aa}_{\mu\nu,kl}=-G^{aa*}_{\mu\nu,lk}$, etc.

The nuclear Hamiltonian involves the vector potential $F$, not the curvature
$G$, but the curvature appears in the commutators of the velocity operators
(Littlejohn and Reinsch (1997)) and in the classical equations of motion,
which are useful for wave packet evolution (Berry and Robbins (1993); Wu,
Miao, and Subotnik (2020); Bian  _et al._ (2021)), especially when nuclear
dynamics depend on spin.

We now present another formula for the curvature, which is a simplification of
Eq. (3.42) of Mead (1992) and which connects the curvature with the projection
operator. It is

$G^{aa}_{\mu\nu,kl}(x)=\langle bx;k|[P_{\mu},P_{\nu}]|bx;l\rangle,$ (92)

where $P_{\mu}=\partial_{\mu}P$. The proof of this is straightforward. In our
applications this turned out to be the most convenient means of calculating
the curvature (much more convenient than (91)).

Under the change of basis (88) the curvature transforms as a tensor, that is,

$G^{\prime}_{\mu\nu}=UG_{\mu\nu}U^{\dagger},$ (93)

(Mead (1992)), where $U$ is the same as in (89). This means that the
quantities

$\sum_{kl}|G_{\mu\nu,kl}|^{2}$ (94)

are independent of basis. These quantities, which constrain the derivatives of
the derivative couplings, cannot be transformed away, not even at a single
point. For this reason, there is some interest in finding a gauge that
expresses $F_{\mu}$ in terms of $G_{\mu\nu}$.

Finally, we quote the Bianchi identity, which is

$\displaystyle\partial_{\sigma}G^{aa}_{\mu\nu}+\partial_{\mu}G^{aa}_{\nu\sigma}+\partial_{\nu}G^{aa}_{\sigma\mu}$
(95) $\displaystyle+$
$\displaystyle[F^{aa}_{\mu},G^{aa}_{\nu\sigma}]+[F^{aa}_{\nu},G^{aa}_{\sigma\mu}]+[F^{aa}_{\sigma},G^{aa}_{\mu\nu}]=0,$

with a similar formula for the $bb$-block. This is the analog of Maxwell’s
equation $\nabla\cdot{\mathbf{B}}=0$ in electromagnetism.

###  IV.6 Non-Abelian Poincaré Gauge

In electromagnetism, Poincaré gauge (Cohen-Tannoudji, Dupont-Roc, and Grynberg
(1989)) is transverse in real space, that is,
${\mathbf{x}}\cdot{\mathbf{A}}({\mathbf{x}})=0$. In a molecule, if $x_{0}$ is
a reference point in nuclear configuration space and
$x(\lambda)=x_{0}+\lambda\xi$ is a radial line, as above, then the analog of
Poincaré gauge for the Mead-Truhlar-Berry connection is a gauge such that

$\xi^{\mu}F^{aa}_{\mu}\big{(}x(\lambda)\big{)}=\xi^{\mu}F^{bb}_{\mu}\big{(}x(\lambda)\big{)}=0,$
(96)

that is, the radial component of the connection vanishes.

It turns out that parallel-transported gauge is Poincaré gauge, in this sense.
To prove this we start with $k,l\in{A}$ and note that in the parallel-
transported frame we have

$\displaystyle\xi^{\mu}F^{aa}_{\mu,kl}$ $\displaystyle=$
$\displaystyle\xi^{\mu}\langle px;k|\partial_{\mu}|px;l\rangle=\langle
px;k|\frac{d}{d\lambda}|px;l\rangle$ (97) $\displaystyle=$
$\displaystyle\langle px;k|[P^{\prime},P]|px;l\rangle=\langle
px;k|P(P^{\prime}P-PP^{\prime})P|px;l\rangle$ $\displaystyle=$ $\displaystyle
0,$

where we use $d/d\lambda=\xi^{\mu}\partial_{\mu}$ and (60). A similar proof
holds for $F^{bb}_{\mu}$, but radial components of the off-diagonal blocks of
the derivative couplings, $F^{ab}_{\mu}$ and $F^{ba}_{\mu}$, do not vanish.

Poincaré gauge has the interesting property that the vector potential can be
expressed in terms of the magnetic field. Thus one obtains Hamiltonians that
are not expressed in terms of some arbitrary vector potential, but rather
directly in terms of the magnetic field, which is fully physical. This is
mainly useful in some neighborhood of a given point, by means of Taylor series
expansions.

This property carries over to the non-Abelian case. Specifically, if
$F^{aa}_{\mu}$ is in parallel-transported gauge, then

$F^{aa}_{\mu}(x)=\int_{0}^{1}d\lambda\,\lambda\xi^{\nu}G^{aa}_{\nu\mu}\big{(}x(\lambda)\big{)},$
(98)

where $x$ on the left-hand side means $x=x_{0}+\xi$ and $x(\lambda)$ under the
integral means $x_{0}+\lambda\xi$, as above. The integral is carried along a
radial line from $x_{0}$ to $x=x_{0}+\xi$. A similar equation holds for the
$bb$-block of the connection and curvature.

This is a rather remarkable formula, because it appears to give a linear
relationship between $F$ and $G$, which in the non-Abelian case are in fact
connected by the nonlinear relationship (91). For us (98) is merely an
identity connecting the curvature and connection in parallel-transported
gauge, but if one were to interpret it as a map from a given curvature to a
corresponding vector potential, then it must be understood that the formula
only works if $G$ satisfies the Bianchi identity (95).

To prove (98) we assume that $F$ is the parallel-transported connection and
$G$ is the curvature and we examine the integral on the right-hand side, but
without assuming that it equals the left-hand side. Writing $G$ in terms of
$F$, the integral becomes

$\int_{0}^{1}d\lambda\,\lambda\xi^{\nu}\big{(}\partial_{\nu}F^{aa}_{\mu}-\partial_{\mu}F^{aa}_{\nu}+[F^{aa}_{\nu},F^{aa}_{\mu}]\big{)},$
(99)

where it is understood that everything under the integral is evaluated at
$x(\lambda)=x_{0}+\lambda\xi$. Because
$\xi^{\nu}F^{aa}_{\nu}\big{(}x(\lambda)\big{)}=0$ the commutator term
vanishes. The first term in (99) can be written

$\int_{0}^{1}d\lambda\,\lambda\frac{dF^{aa}_{\mu}}{d\lambda},$ (100)

where we use $d/d\lambda=\xi^{\nu}\partial_{\nu}$. As for the second term, it
is

$\displaystyle-\int_{0}^{1}d\lambda\,\xi^{\nu}\frac{\partial
F^{aa}_{\nu}}{\partial\xi^{\mu}}$ $\displaystyle=$
$\displaystyle-\int_{0}^{1}d\lambda\left[\frac{\partial}{\partial\xi^{\mu}}(\xi^{\nu}F^{aa}_{\nu})-F^{aa}_{\mu}\right]$
(101) $\displaystyle=$ $\displaystyle\int_{0}^{1}d\lambda\,F^{aa}_{\mu},$

where we use $\partial/\partial\xi^{\mu}=\lambda\partial_{\mu}$. Now adding
(100) and (101) we obtain

$\int_{0}^{1}d\lambda\,\frac{d}{d\lambda}[\lambda
F^{aa}_{\mu}\big{(}x(\lambda)\big{)}]=F^{aa}_{\mu}\big{(}x(1)\big{)}=F^{aa}_{\mu}(x),$
(102)

which is the given left-hand side of (98).

Now we may expand the integrand in (98) in a power series in $\lambda$ and do
the integral, which through order $\lambda^{2}$ gives us

$F^{aa}_{\mu}(x_{0}+\xi)=\frac{1}{2}\xi^{\nu}G^{aa}_{\nu\mu}(x_{0})+\frac{1}{3}\xi^{\nu}\xi^{\sigma}(\partial_{\sigma}G^{aa}_{\nu\mu})(x_{0})+\ldots$
(103)

This can be used to express $F$ and its derivatives at $x_{0}$ in terms of the
curvature,

$\displaystyle F^{aa}_{\mu}$ $\displaystyle=$ $\displaystyle 0,$ (104a)
$\displaystyle\partial_{\nu}F^{aa}_{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}G^{aa}_{\nu\mu},$ (104b)
$\displaystyle\partial_{\nu}\partial_{\sigma}F^{aa}_{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{3}[\partial_{\sigma}G^{aa}_{\nu\mu}+\partial_{\nu}G^{aa}_{\sigma\mu}],$
(104c)

where it is understood that everything is evaluated at $x_{0}$. As we see, the
Mead-Truhlar-Berry connection in the parallel-transported frame vanishes at
$x_{0}$ and the curvature governs the rate at which it grows as we move away
from $x_{0}$.

The facility of expressing the connection in terms of the curvature is one of
the advantages of the parallel-transported frame over the singular-value
frame.

###  IV.7 Connection and Curvature in Parallel-Transported Basis

We can now work out the components of the connection and curvature in the
parallel-transported basis. It is straightforward to do this with the
expansion of the parallel-transported basis $|px;k\rangle$ in terms of the
reference basis $|ax_{0};k\rangle$, given by (67), with coefficients given by
(78). First we substitute this into the definition (84) of the derivative
couplings to compute $F_{\mu}$, and then use (91) to compute $G_{\mu\nu}$.
This will give expressions for $F_{\mu}$ and $G_{\mu\nu}$ in terms of the
components of the projection operator, and, ultimately, in terms of the
Hamiltonian and its derivatives.

It turns out to be much easier, however, to use first Mead’s formula (92) in
the parallel-transported basis to find $G$, and then to use (98) to find $F$.
In this work we must be careful with notation. We will henceforth write
${\mathcal{P}}$ for the projection operator (previously simply denoted $P$),
$P$ for its matrix with respect to the reference basis $|ax_{0};k\rangle$ and
${\bar{P}}$ for its matrix with respect to the parallel-transported basis.
Also, we will append $(x)$ to a quantity if we mean it to be evaluated at
$x=x_{0}+\xi$, and we will omit the $(x)$ if we mean it to be evaluated at
$x_{0}$. Also, we define
${\mathcal{P}}_{\mu}(x)=(\partial_{\mu}{\mathcal{P}})(x)$, etc.

We are interested in the curvature tensor in the parallel-transported basis,
which according to Mead’s formula (92) in the case $k,l\in{A}$ is

$G^{aa}_{\mu\nu;kl}(x)=\langle
px;k|[{\mathcal{P}}_{\mu}(x),{\mathcal{P}}_{\nu}(x)]|px;l\rangle=[{\bar{P}}_{\mu}(x),{\bar{P}}_{\nu}(x)]_{kl},$
(105)

since the matrix of the commutator is the commutator of the matrices. A
similar formula applies to the $bb$-block. For this we need the matrix,

${\bar{P}}_{\mu}(x)=T_{p}(x)^{\dagger}P_{\mu}(x)T_{p}(x),$ (106)

which is a change of basis from $|ax_{0};k\rangle$ to $|px;k\rangle$. This
matrix vanishes on the diagonal blocks, for example, if $k,l\in{A}$ then

$\displaystyle[{\bar{P}}_{\mu}(x)]_{kl}$ $\displaystyle=$
$\displaystyle\langle px;k|{\mathcal{P}}_{\mu}(x)|px;l\rangle$ (107)
$\displaystyle=$ $\displaystyle\langle
px;k|{\mathcal{P}}(x){\mathcal{P}}_{\mu}(x){\mathcal{P}}(x)|px;l\rangle=0,$

as follows since
${\mathcal{P}}(x)[\partial_{\mu}{\mathcal{P}}(x)]{\mathcal{P}}(x)=0$.

The expansion of $P(x)=P(x_{0}+\xi)$ is given by (34), (39), (41) and (42),
which to second order can be written

$\displaystyle P(x)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}I^{aa}&0\\\
0&0\end{array}\right)+\left(\begin{array}[]{cc}0&P^{ab}_{\mu}\\\
P^{ba}_{\mu}&0\end{array}\right)\xi^{\mu}$ (112) $\displaystyle+$
$\displaystyle\left(\begin{array}[]{cc}-P^{ab}_{\mu}P^{ba}_{\nu}&(1/2)P^{ab}_{\mu\nu}\\\
(1/2)P^{ba}_{\mu\nu}&P^{ba}_{\mu}P^{ab}_{\nu}\end{array}\right)\xi^{\mu}\xi^{\nu}+\ldots,$
(115)

where $P^{ab}_{\mu}$ is the $ab$-block of $(\partial_{\mu}P)(x_{0})$,
$P^{ab}_{\mu\nu}$ is the $ab$-block of
$(\partial_{\mu}\partial_{\nu}P)(x_{0})$, etc. Notice that in terms of our
previous notation we have

$P^{ab}_{1}=P^{ab}_{\mu}\,\xi^{\mu},\qquad
P^{ab}_{2}=\frac{1}{2}P^{ab}_{\mu\nu}\,\xi^{\mu}\xi^{\nu},$ (116)

etc. Differentiating (115) we obtain

$\displaystyle P_{\mu}(x)=\left(\begin{array}[]{cc}0&P^{ab}_{\mu}\\\
P^{ba}_{\mu}&0\end{array}\right)$ (119) $\displaystyle+$
$\displaystyle\left(\begin{array}[]{cc}-P^{ab}_{\mu}P^{ba}_{\nu}-P^{ab}_{\nu}P^{ba}_{\mu}&P^{ab}_{\mu\nu}\\\
P^{ba}_{\mu\nu}&P^{ba}_{\mu}P^{ab}_{\nu}+P^{ba}_{\nu}P^{ab}_{\mu}\end{array}\right)\xi^{\nu}+\ldots$
(122)

Now we use the expansion (78) of $T_{p}$, which we only need to first order
and which we write as

$T_{p}(x)=\left(\begin{array}[]{cc}I^{aa}&0\\\
0&I^{bb}\end{array}\right)+\left(\begin{array}[]{cc}0&-P^{ab}_{\nu}\\\
P^{ba}_{\nu}&0\end{array}\right)\xi^{\nu}+\ldots,$ (123)

to carry out the change of basis (106). This gives

${\bar{P}}_{\mu}(x)=\left(\begin{array}[]{cc}0&P^{ab}_{\mu}\\\
P^{ba}_{\mu}&0\end{array}\right)+\left(\begin{array}[]{cc}0&P^{ab}_{\mu\nu}\\\
P^{ba}_{\mu\nu}&0\end{array}\right)\xi^{\nu}+\ldots$ (124)

which is off-diagonal, as predicted.

Now we can compute the commutator and thence the curvature. We find

$G^{aa}_{\mu\nu}(x)=(P^{ab}_{\mu}P^{ba}_{\nu}-{\rm
h.c.})+(P^{ab}_{\mu}P^{ba}_{\nu\sigma}+P^{ab}_{\mu\sigma}P^{ba}_{\nu}-{\rm
h.c.})\xi^{\sigma}+\ldots,$ (125)

where subtracting the Hermitian conjugate of the indicated matrices is
equivalent to antisymmetrizing in $\mu,\nu$. The expression for $G^{bb}(x)$ is
obtained from this by swapping $a\leftrightarrow b$, and the off-diagonal
blocks vanish, $G^{ab}=G^{ba}=0$. Finally, replacing $\xi$ by $\lambda\xi$ and
carrying out the integral (98), we obtain the diagonal blocks of the Mead-
Truhlar-Berry connection or derivative couplings in the parallel-transported
basis. We find

$\displaystyle F^{aa}_{\mu}(x)=\frac{1}{2}(P^{ab}_{\nu}P^{ba}_{\mu}-{\rm
h.c.})\xi^{\nu}$
$\displaystyle+\frac{1}{6}(P^{ab}_{\nu}P^{ba}_{\mu\sigma}+P^{ab}_{\sigma}P^{ba}_{\mu\nu}+2P^{ab}_{\nu\sigma}P^{ba}_{\mu}-{\rm
h.c.})\xi^{\nu}\xi^{\sigma}+\ldots,$ (126)

and similarly for the $bb$-block.

###  IV.8 Coordinates and Rotations

So far we have been using the notation $x$ to stand for a point of the nuclear
configuration space in the center of mass frame and $x^{\mu}$ for the
collection of the $3N-3$ components of the $N-1$ Jacobi vectors. We will call
these “Jacobi coordinates.” Thus derivatives $\partial_{\mu}$ are understood
to be taken with respect to Jacobi coordinates. However, there is nothing in
our analysis so far that has required that the coordinates be these;
everything goes through with an arbitrary coordinate system on nuclear
configuration space, that is, a set of $3N-3$ arbitrary, possibly nonlinear
functions $x^{\prime\,\mu}$ of $x^{\mu}$. We only require that the Jacobian
matrix $\partial x^{\prime\,\mu}/\partial x^{\nu}$ be nonsingular in the
neighborhood in which we are working. Therefore all our formulas so far are
valid with $x^{\mu}$ reinterpreted as an arbitrary coordinate system in this
sense.

Our parallel-transported basis has been defined by integrating the (60) along
radial lines emanating from a reference point $x_{0}$, which have the
coordinate representation (62). This means that the parallel-transported basis
at a point $x$ depends on the coordinates, since a line that is straight in
one coordinate system is not straight in another.

Figure 1: Fibers $F_{0}$, $F_{1}$ are 3-dimensional surfaces swept out by
applying rotations to nuclear configurations $x_{0}$, $x_{1}$. Section $S$ is
a $(3N-6)$-dimensional surface transverse to the fibers. To establish a frame
of electronic wave functions over a region of nuclear configuration space, we
may use parallel transport along a section $S$, such as along the path
$x_{0}\to x_{1}$, and then rotation operators to transport the frame along the
fibers, such as along the path $x_{1}\to x_{2}$.

Another issue concerns the direction of integration. Let us assume that the
configuration $x_{0}$ is noncollinear, the typical situation in polyatomic
systems. Then there are three directions in which one can move away from
$x_{0}$ that are purely rotational, and we do not want to use parallel
transport to create a diabatic basis in those directions. In those directions
the nuclear configuration transforms by a rigid rotation, that is, the shape
of the molecule does not change, only its orientation. Those directions are
tangent to the the 3-dimensional surfaces which are generated by applying
rotations to a given configuration, which are the orbits or fibers of the
action of rotations on the nuclear configuration space, as explained by
Littlejohn and Reinsch (1997). When the nuclear configuration changes by a
rigid rotation, the electronic basis functions should change by a rotation
operator, not by parallel transport (or any other method for constructing a
diabatic basis, such as the singular value method). Attention to those
directions that are purely rotational versus those in which the shape changes
is important in the construction of kinetic energy operators on the internal
space (Wang and Tucker Carrington (2000)).

The situation is illustrated in Fig. 1, which is a schematic illustration of a
region of the nuclear configuration space. Configuration $x_{0}$ is some
configuration, and $F_{0}$ is the set of other configurations related to
$x_{0}$ by rigid rotations, that is, configurations of the same shape as
$x_{0}$ but different orientations. Surface $F_{0}$ is called the “fiber
through $x_{0}$”, as explained by Littlejohn and Reinsch (1997). Similarly
$F_{1}$ is the fiber through $x_{1}$. Configurations $x_{1}$ and $x_{2}$ have
the same shape but different orientations.

Surface $S$ is a $(3N-6)$-dimensional surface that cuts transversally through
the fibers in the given region, called a “section” of the fiber bundle
(Littlejohn and Reinsch (1997)). Sections are needed to define orientational
coordinates, that is, the section is the surface upon which the Euler angles
are those of the identity rotation. The section also defines a body frame; it
is the surface upon which the body frame coincides with the space frame.
Sections are also implicitly used in electronic structure calculations, when
it is attempted to fill in some region of nuclear configuration space. This is
because one need only calculate the energy eigenvalues and eigenfunctions for
a single orientation of a given shape; for other orientations the energy
eigenvalues do not change and the energy eigenfunctions change simply by a
rotation operator. Thus, electronic structure calculations take place on a
section.

Now assume that $x_{0}$ is a reference configuration and we have an electronic
basis at $x_{0}$ that block-diagonalizes the Hamiltonian, and suppose we wish
to find a diabatic basis at neighboring points such as $x_{2}$ in the figure.
Then the procedure is to use some rule (singular value, parallel transport,
etc.) to create a diabatic basis at $x_{1}$ on the section, and then to use
rotation operators to create a diabatic basis at $x_{2}$.

Thus we see that when we parallel transport a basis from a configuration
$x_{0}$, we do not need to go out in all $3N-3$ directions in nuclear
configuration space, but rather only along the $3N-6$ directions of a section
passing through $x_{0}$. So far we have been interpreting the coordinates
$x^{\mu}$ as $3N-3$ coordinates on nuclear configuration space, but all our
results hold if we reinterpret them as $3N-6$ coordinates on a section, and we
will henceforth do this. These coordinates can be taken to be shape
coordinates, that is, quantities such as bond lengths and angles that are
invariant under overall rotations of the molecule. Similar considerations
apply to all other methods (singular value, etc.) of constructing a diabatic
basis in the neighborhood of $x_{0}$; the purely rotational directions are
handled by rotation operators.

##  V Degeneracy Manifolds

In the following it is assumed that we are working on a section with
coordinates $x^{\mu}$, $\mu=1,\ldots,3N-6$, or some subset of it which for
simplicity we will call $S$. We assume the set ${A}$ (see (1)) contains at
least two levels, so that internal degeneracies are possible. We define the
degeneracy manifold as the subset of $S$ upon which there is an internal
degeneracy. Degeneracy manifolds are usually called “seams” but we prefer the
alternative terminology as it is more descriptive. Degeneracy manifolds are
usually surfaces of conical intersections, but it should be noted that while
all intersections of potential energy surfaces are degeneracies they are not
always conical. We now address the construction of a parallel-transported
basis in a neighborhood of the degeneracy manifold.

###  V.1 Models of the Electronic Hamiltonian

Up to this point we have been assuming that the electronic Hamiltonian was
taken in the electrostatic model, but now we will address the more general
case in which fine structure effects are included. We shall assume that no
external fields are acting so that the molecular Hamiltonian has time-reversal
symmetry. This has an important effect in the case of an odd number of
electrons, in which the electronic eigenstates become Kramers doublets. Let us
agree to call a “level” a single Kramers doublet in the case of an odd number
of electrons, which means the degeneracy is twice the number of levels. In
other cases (electrostatic model or fine structure with an even number of
electrons) the degeneracy is equal to the number of levels.

With this understanding, we now review the simplest case of two levels. If
there is no symmetry in addition to time-reversal, the codimension of the
degeneracy manifold is 2 in the electrostatic model or when fine structure is
included and the number of electrons is even (von Neumann and Wigner (1929)),
and it is 5 if fine structure is included and the number of electrons is odd
(Mead (1980a)). If there are symmetries then the picture is more complicated
since in general the symmetry only holds on submanifolds of configuration
space, but an important exception is the $C_{s}$ symmetry of triatomic
molecules (reflection in the plane of the molecule), which is global. In the
case of triatomics with fine structure and an odd number of electrons, the
codimension of the degeneracy manifold is 3 (Mead (1980a)). We should also add
that in the case of the electrostatic model we are assuming that one is
working within a given subspace of spin states, that is, energy eigenstates of
fixed $S^{2}$ and $S_{z}$, since degeneracies between states of different spin
have codimension 1, whereas when fine structure is included we must enlarge
the electronic Hilbert space to include all spin states.

Let us denote the degeneracy manifold by $D$; its codimension is counted
inside the section $S$. For example, in triatomics $S$ is 3-dimensional. Then
in the electrostatic model or with fine structure and an even number of
electrons, $D$ has codimension 2 or dimension 1, that is, it is a curve in
$S$; while if fine structure is included and number of electrons is odd, then
the codimension of $D$ is 3 and its dimension is 0, that is, $D$ consists of
isolated points inside $S$. These codimension counts apply at generic points
of configuration space, where the Jacobian of the map from configuration space
to Hamiltonian matrix space is of maximal rank; where this is not the case
other phenomena arise, such as the bifurcation of degeneracy manifolds
(Yarkony (2000)).

###  V.2 Reference point $x_{0}$ on Degeneracy Manifold

Henceforth we will assume that the reference point $x_{0}$ for the
construction of a parallel-transported basis lies on $D$. We will also assume
that at $x_{0}$ the electronic Hamiltonian is completely degenerate in the
coupled subspace, so that $H^{aa}_{0,kl}=\epsilon_{0}\,\delta_{kl}$, where
$\epsilon_{0}$ is the degenerate eigenvalue. In that case we note that
$P^{ab}_{n}$ can be expressed purely in terms of matrix products,

$\displaystyle P^{ab}_{1}$ $\displaystyle=$ $\displaystyle
H^{ab}_{1}R(\epsilon_{0}),$ (127a) $\displaystyle P^{ab}_{2}$ $\displaystyle=$
$\displaystyle[H^{ab}_{2}+H^{ab}_{1}R(\epsilon_{0})H^{bb}_{1}-H^{aa}_{1}H^{ab}_{1}R(\epsilon_{0})]R(\epsilon_{0}),$
(127b)

etc., with $P^{ba}_{n}=(P^{ab}_{n})^{\dagger}$. These are simplified versions
of (36) and (40).

Let us also write the expansion of the Hamiltonian at $x=x_{0}+\xi$ as a power
series in $\xi$, just as we did for the projection operator in (116). We
define, for example, $H^{ab}_{\mu}=(\partial_{\mu}H^{ab})(x_{0})$,
$H^{ab}_{\mu\nu}=(\partial_{\mu}\partial_{\nu}H^{ab})(x_{0})$, etc., so that

$H^{ab}_{1}=H^{ab}_{\mu}\,\xi^{\mu},\qquad
H^{ab}_{2}=\frac{1}{2}H^{ab}_{\mu\nu}\,\xi^{\mu}\xi^{\nu},$ (128)

etc., and similarly for the other blocks. Then with the aid of (127) the
derivatives of $P$ can be expressed in terms of the derivatives of $H$,

$\displaystyle P^{ab}_{\mu}$ $\displaystyle=$ $\displaystyle
H^{ab}_{\mu}R(\epsilon_{0}),$ (129a) $\displaystyle P^{ab}_{\mu\nu}$
$\displaystyle=$
$\displaystyle[H^{ab}_{\mu\nu}+H^{ab}_{\mu}R(\epsilon_{0})H^{bb}_{\nu}+H^{ab}_{\nu}R(\epsilon_{0})H^{bb}_{\mu}$
(129b)
$\displaystyle\quad-(H^{aa}_{\mu}H^{ab}_{\nu}+H^{aa}_{\nu}H^{ab}_{\mu})R(\epsilon_{0})]R(\epsilon_{0}),$

etc.

These can then be used to express the curvature and connection in terms of the
derivatives of the Hamiltonian. For example, (92) gives us the curvature,
which becomes

$G^{aa}_{\mu\nu}=[H^{ab}_{\mu}R(\epsilon_{0})^{2}H^{ba}_{\nu}-{\rm
h.c.}]+\ldots$ (130)

Here we omit the first order correction as it is rather lengthy. As for the
connection, in the parallel-transported basis $|px;k\rangle$ it is given in
terms of the curvature by (103) or (104); thus, to first order we have

$F^{aa}_{\mu}(x)=\frac{1}{2}[H^{ab}_{\nu}R(\epsilon_{0})^{2}H^{ba}_{\mu}-{\rm
h.c.}]\xi^{\nu}+\ldots.$ (131)

The result (130) agrees with Eq. (56) of Mead and Truhlar (1982), in spite of
the fact that those authors were not using the parallel-transported basis. The
reason is that the curvature is a tensor, and has the same form in all frames.
The same cannot be said for the connection. We believe the first order
correction to the curvature seen in (92) is new.

###  V.3 Diabatic Basis in a Neighborhood of the Degeneracy Manifold

We now construct a version of the parallel-transported basis in a neighborhood
of the degeneracy manifold. So far we have constructed such a basis in the
neighborhood of an arbitrary point $x_{0}$; we are free to choose $x_{0}$ to
lie on $D$, but we still have just a small neighborhood of a single point,
where Taylor series expansions are valid. We desire a construction that is
valid over all of $D$, or at least the parts of $D$ that form a smooth
manifold. (Places where $D$ bifurcates require a separate analysis, for which
see Yarkony (2000)). For example, in triatomic molecules where $D$ is a line
inside the 3-dimensional section $S$, we desire a parallel-transported basis
that is valid in a tubular neighborhood of $D$, where the two potential energy
surfaces are strongly coupled.

Figure 2: The degeneracy manifold $D$ as a subset of the section $S$ and a
neighborhood thereof are described by coordinates $x^{\mu}=(y^{I},z^{i})$,
where $y^{I}$ are coordinates along $D$ and $z^{i}$ are coordinates transverse
to $D$. Coordinates $z^{i}=0$ on $D$ itself.

We let $x^{\mu}$, $\mu=1,\ldots,3N-6$ be coordinates on a section $S$, which
we break up as $x^{\mu}=(y^{I},z^{i})$, where $I=1,\ldots,\dim D$ and
$i=1,\ldots,{\mathop{\textrm{codim}}}D$, making $y^{I}$ coordinates along the
degeneracy manifold, and $z^{i}$ coordinates transverse to it. We let
$z^{i}=0$ on $D$, and we use the summation convention on coordinate indices
$I$ and $i$. See Fig. 2. This is merely a coordinate description of a
neighborhood of $D$. The whole construction takes place in $S$, a subset of
the nuclear configuration space which in general is curved; and $D$ as a
subset of $S$ is generally a curved manifold.

These coordinates draw attention to the manifolds $y=y_{0}={\rm const.}$
illustrated in the figure which are transverse to $D$ and upon which $z^{i}$
are coordinates. These manifolds correspond roughly to what has been called
the “branching space” by Yarkony (2004); there is one such manifold for each
point on $D$. We propose to establish a diabatic basis in a neighborhood of
$D$ by parallel transporting along radial lines in the transverse spaces away
from $D$. We will denote this basis by $|Dx;k\rangle$; it is like the basis
$|px;k\rangle$ except that the reference point for the parallel transport,
instead of being fixed, moves along $D$ as $x$ moves. That is, if $x=(y,z)$,
then for the basis $|Dx;k\rangle$ the reference point for the parallel
transport is $x_{0}=(y,0)$.

This construction requires that a diabatic basis be established on $D$, before
being propagated along the transverse spaces. This can be done by any
convenient means, for example, the singular-value method or by parallel
transport. Once this is done, not only do we have frames for points
$x_{0}=(y,0)\in D$, but the $y^{I}$-components of the connection, that is,

$F^{aa}_{I;kl}(x_{0})=\langle Dx_{0};k|\partial_{I}|Dx_{0};l\rangle,$ (132)

where $\partial_{I}=\partial/\partial y^{I}$, become known at points $x_{0}\in
D$.

Once the basis $|Dx_{0};k\rangle$ is established for $x_{0}=(y,0)\in D$, we
use parallel transport along radial lines in the transverse spaces to create
$|Dx;k\rangle$ in a neighborhood of $D$, that is, for $x=(y,z)$. The radial
line is $x(\lambda)=(y,\lambda z)$, which interpolates between $(y,0)$ and
$(y,z)$ as $\lambda$ goes from 0 to 1. The radial component of the connection
in this basis vanishes, for example, for $k,l\in{A}$ we have

$z^{i}\langle Dx;k|\partial_{i}|Dx;l\rangle=z^{i}F^{aa}_{i;kl}(y,z)=0,$ (133)

which is like (96) and proved in the same way.

###  V.4 Connection and Curvature in a Neighborhood of $D$

To find the connection and curvature in the basis $|Dx;k\rangle$ we begin with
the curvature. Since the curvature is a tensor, the numerical values of its
components at a point $x$, say, $G^{aa}_{\mu\nu;kl}(x)$, depend on the the
coordinates and the basis at $x$. We wish to find these components in the
basis $|Dx;k\rangle$ where $x=(y,z)$. But the basis $|Dx;k\rangle$ is the same
as the parallel-transported basis $|px;k\rangle$ if the reference point for
the latter is taken to be $x_{0}=(y,0)$. So with that understanding about the
reference point, our expressions for the curvature in the basis $|px;k\rangle$
are valid also in the basis $|Dx;k\rangle$. We will, however, want to break
the coordinate indices up into their $y$\- and $z$-parts, by replacing indices
$\mu$, etc., by $I$ or $i$, etc. For example, we can rewrite (125) as

$G^{aa}_{ij}(y,z)=(P^{ab}_{i}P^{ba}_{j}-{\rm
h.c.})+(P^{ab}_{i}P^{ba}_{jk}+P^{ab}_{ik}P^{ba}_{j}-{\rm h.c.})z^{k}+\ldots,$
(134)

and similarly for the $(iI)$, $(Ii)$ and $(IJ)$ components of $G$. Here we
have replaced $\xi^{\mu}$ by $(0,z^{i})$, since the parallel transport is
taking place only in the transverse manifold (at constant $y$). Also, all
derivatives of the projection operator in (134) are understood to be evaluated
on the degeneracy manifold at point $(y,0)$. As for (130), it can be expressed
in the basis $|Dx;k\rangle$, whereupon it becomes

$G^{aa}_{ij}(y,z)=[H^{ab}_{i}R(\epsilon_{0})^{2}H^{ba}_{j}-{\rm
h.c.}]+\ldots,$ (135)

and similarly for the other components $(iI)$, $(Ii)$ and $(IJ)$, where now
the derivatives of $H$ are evaluated at $(y,0)$.

To obtain the connection in a neighborhood of $D$ we begin with the components
$F_{i}$. These are related to the components $G_{ij}$ of the curvature by an
integral formula like (98), that is,

$F^{aa}_{i}(y,z)=\int_{0}^{1}d\lambda\,\lambda z^{j}\,G_{ji}(y,\lambda z).$
(136)

This is really a special case of (98), which applies to any parameter space
that the Hamiltonian depends on, upon which the coordinates are $x^{\mu}$; by
interpreting that parameter space as the transverse space $y={\rm const.}$ and
by replacing $x^{\mu}$ with $z^{i}$, we obtain (136). One can also repeat the
derivation of (98) with a changed notation. This then implies a version of
(104),

$\displaystyle F^{aa}_{i}$ $\displaystyle=$ $\displaystyle 0,$ (137a)
$\displaystyle\partial_{j}F^{aa}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}G^{aa}_{ji},$ (137b)
$\displaystyle\partial_{j}\partial_{k}F^{aa}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{3}[\partial_{k}G^{aa}_{ji}+\partial_{j}G^{aa}_{ki}],$
(137c)

etc., where both sides are understood to be evaluated at $(y,0)$. Now
expressing the curvature in terms of the Hamiltonian, we obtain a Taylor
series for $F_{i}$ in terms of the coordinates $z^{i}$,

$F^{aa}_{i}(y,z)=\frac{1}{2}[H^{ab}_{j}R(\epsilon_{0})^{2}H^{ba}_{i}-{\rm
h.c.}]z^{j}+\ldots,$ (138)

a version of (131). Here we omit the second order term (which is available) as
it is rather lengthy, and the derivatives on the right hand side are
understood to be evaluated at $(y,0)$.

As for the components $F_{I}$, we use the integral formula,

$F^{aa}_{I}(y,z)=F^{aa}_{I}(y,0)+\int_{0}^{1}d\lambda\,z^{j}\,G^{aa}_{jI}(y,\lambda
z),$ (139)

which may be compared with (98). In this formula it is understood that
components of the connection and curvature are taken with respect to the basis
$|Dx;k\rangle$ (while the basis $|px;k\rangle$ was used in (98)). To prove
(139) we express $G$ in terms of $F$ so that the integral becomes

$\displaystyle\int_{0}^{1}d\lambda\,z^{j}\\{(\partial_{j}F^{aa}_{I})(y,\lambda
z)-(\partial_{I}F^{aa}_{j})(y,\lambda z)$ (140) $\displaystyle+$
$\displaystyle[F^{aa}_{j}(y,\lambda z),F^{aa}_{I}(y,\lambda z)]\\},$

and we use (133), which causes the commutator and the second term to vanish.
As for the first term, it is

$\int_{0}^{1}d\lambda\,\frac{d}{d\lambda}F^{aa}_{I}(y,\lambda
z)=F^{aa}_{I}(y,z)-F^{aa}_{I}(y,0),$ (141)

which proves (139).

Now we expand the integrand in (139) in powers of $\lambda$ and do the
integral to obtain

$F^{aa}_{I}(y,z)=F^{aa}_{I}(y,0)+z^{j}\,G^{aa}_{jI}(y,0)+\frac{1}{2}z^{j}z^{k}\,(\partial_{k}G^{aa}_{jI})(y,0)+\ldots$
(142)

This implies

$\displaystyle\partial_{i}F^{aa}_{I}$ $\displaystyle=$ $\displaystyle
G^{aa}_{iI},$ (143a) $\displaystyle\partial_{i}\partial_{j}F^{aa}_{I}$
$\displaystyle=$
$\displaystyle\frac{1}{2}[\partial_{j}G^{aa}_{iI}+\partial_{i}G^{aa}_{jI}],$
(143b)

where everything is evaluated at $(y,0)$. Finally, we may express the
connection in terms of the Hamiltonian and its derivatives,

$F^{aa}_{I}(y,z)=F^{aa}_{I}(y,0)+[H^{ab}_{i}R(\epsilon_{0})^{2}H^{ba}_{I}-{\rm
h.c.}]z^{i}+\ldots,$ (144)

where again we omit the available second order term due to its length. Taken
with (138) this provides an expansion of the connection in a neighborhood of
$D$. We see that the components along $D$ and those transverse to $D$ can both
be expressed in terms of the curvature, but the expansions are different.
Again, the $bb$-block is obtained by swapping $a\leftrightarrow b$.

We make one final comment. In the case of triatomic molecules in the
electrostatic model the degeneracy manifold $D$ is a one-dimensional curve
inside the section $S$, as noted. Thus if we use parallel transport along $D$
to create frame on $D$, the component of $F$ along $D$ will vanish. That is,
$F^{aa}_{I}(y,0)=0$, where there is only one index $I$ since $D$ is one-
dimensional. But then (138) shows that the transverse components $F^{aa}_{i}$
also vanish on $D$. Thus we have shown that in the 3-body problem with the
electrostatic model for the electronic Hamiltonian, there exists a diabatic
basis such that the derivative couplings vanish on the degeneracy manifold.
This construction only works when the degeneracy manifold is one-dimensional.

##  VI Conclusions

In this article we have given two versions of a parallel-transported diabatic
basis, one valid in a neighborhood of a point (which is allowed to lie on a
degeneracy manifold or seam), and another which is valid in a neighborhood of
a degeneracy manifold of possibly global extent. In both cases we have given
Taylor series expansions of the basis vectors, the derivative couplings and
the curvature. We have demonstrated the close relationship between the
parallel-transported basis and the singular-value basis, showing that they
agree to second order in a Taylor series expansion about a point. Our
expansion of the singular-value basis seems to be new. We have promoted a
method of carrying out these expansions that relies on the projection operator
and that avoids small or vanishing energy denominators or other singularities.
This approach seems to be new in the literature on diabatic bases. We have
also exploited integral relationships that hold between the connection and
curvature in the parallel-transported basis, which are generalizations of
Poincaré gauge to a non-Abelian context and which provide a convenient means
for computing the derivative couplings. Our goal is to give analytic
treatments of connection and curvature in the neighborhood of degeneracy
manifolds that will be useful for future work, including multi-dimensional
Landau-Zener normal forms.

###### Acknowledgements.

One of us (R.L.) would like to thank Alden Mead for his kind hospitality and
many stimulating and memorable conversations during the spring of 1993, as
well as his support of the present work. The authors would also like to
acknowledge many stimulating conversations and much good advice from Enzo
Aquilanti, Tucker Carrington, Eric Heller and David Yarkony. JES was supported
by the National Science Foundation under Grant No. CHE-2102402.

##  Appendix A Some Proofs

Identities involving the projection operators $P(x)$ and $Q(x)$ include
$P+Q=1$, $P^{2}=P$, $Q^{2}=Q$, $QP=PQ=0$, $P^{\prime}=P^{\prime}P+PP^{\prime}$
and $PP^{\prime}P=0$, where prime means differentiation with respect to a
parameter $\lambda$.

To prove (58) we assume (57) is true so that the left hand side of (58) can be
written

$\displaystyle\left(\frac{d}{d\lambda}+P^{\prime}\right)[(1-P)\psi]=-P^{\prime}\psi+(1-P)\psi^{\prime}+P^{\prime}(1-P)\psi$
$\displaystyle\quad=[-P^{\prime}+(1-P)P^{\prime}+P^{\prime}(1-P)]\psi$
$\displaystyle\quad=(P^{\prime}-P^{\prime}P-PP^{\prime})\psi=0.$ (145)

A similar proof is to show that if $\psi$ satisfies (60) then $Q\psi$
satisfies (58). We have

$\displaystyle\left(\frac{d}{d\lambda}+P^{\prime}\right)[(1-P)\psi]$
$\displaystyle=[-P^{\prime}+(1-P)(P^{\prime}P-PP^{\prime})+P^{\prime}(1-P)]\psi=0.$
(146)

Next we show that if $\psi$ satisfies (60) and $Q\psi=0$ then
$\psi^{\prime}=P^{\prime}\psi$. First, $Q\psi=0$ implies $\psi=P\psi$. Next,
we are given $\psi^{\prime}=P^{\prime}P\psi-PP^{\prime}\psi$, of which the
first term is $P^{\prime}\psi$ and the second is $-PP^{\prime}P\psi=0$. Thus,
$\psi^{\prime}=P^{\prime}\psi$.

Now we prove (87). We work in any basis for which the derivatives are defined,
which we denote simply by $|x;k\rangle$. We are carrying out parallel
transport along a curve $x(\lambda)$ according to (60), where the prime means
$d/d\lambda$.

First we define

$F_{\lambda,kl}=\langle
x;k|\frac{d}{d\lambda}|x;l\rangle=F_{\mu,kl}\,\frac{dx^{\mu}}{d\lambda}=-\left(\frac{d}{d\lambda}\langle
x;k|\right)|x;l\rangle.$ (147)

Next we note that $P|x;k\rangle=|x;k\rangle$ if $k\in A$ and $0$ otherwise.
This implies $\langle x;k|P|x;l\rangle=\delta_{kl}$ if $k,l\in A$, and $0$
otherwise. Then by differentiating this we obtain

$\langle x;k|P^{\prime}|x;l\rangle=\begin{cases}-F_{\lambda,kl}&\text{if $k\in
A$ and $l\notin A$,}\\\ +F_{\lambda,kl}&\text{if $k\notin A$ and $l\in A$,}\\\
0&\text{otherwise.}\end{cases}$ (148)

This then implies

$\langle x;k|[P^{\prime},P]|x;l\rangle=\begin{cases}F_{\lambda,kl}&\text{if
$k\in A$ and $l\notin A$ or $k\notin A$ and $l\in A$,}\\\
0&\text{otherwise.}\end{cases}$ (149)

Now we differentiate (86) and substitute into (60), obtaining

$\frac{d|\psi\rangle}{d\lambda}=\sum_{{\rm
all}\,k}\left[\left(\frac{d}{d\lambda}|x;k\rangle\right)\psi_{k}+|x;k\rangle\frac{d\psi_{k}}{d\lambda}\right]=\sum_{{\rm
all}\,k}[P^{\prime},P]|x;k\rangle\psi_{k},$ (150)

or,

$\frac{d\psi_{k}}{d\lambda}=\sum_{{\rm all}\,l}\left(\langle
x;k|[P^{\prime},P]|x;l\rangle-F_{\lambda,kl}\right)\psi_{l}.$ (151)

Taking now the cases $k\in A$ and $k\notin A$, we find

$\frac{d\psi_{k}}{d\lambda}=\begin{cases}-\sum_{l\in
A}F_{\lambda,kl}\,\psi_{l}&\text{if $k\in A$,}\\\ -\sum_{l\notin
A}F_{\lambda,kl}\,\psi_{l}&\text{if $k\notin A$.}\end{cases}$ (152)

Comparing this with (85) we obtain (87).

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