# Continuants with equal values, a combinatorial approach

G Ramharter, L.Q. Zamboni

###### Abstract

A regular continuant is the denominator $K$ of a terminating regular continued
fraction, interpreted as a function of the partial quotients. We regard $K$ as
a function defined on the set of all finite words on the alphabet
$1<2<3<\dots$ with values in the positive integers. Given a word
$w=w_{1}\cdots w_{n}$ with $w_{i}\in\mathbb{N}$ we define its multiplicity
$\mu(w)$ as the number of times the value $K(w)$ is assumed in the Abelian
class $\mathcal{X}(w)$ of all permutations of the word $w.$ We prove that
there is an infinity of different lacunary alphabets of the form
$\\{b_{1}<\dots<b_{t}<l+1<l+2<\dots<s\\}$ with $b_{j},t,l,s\in\mathbb{N}$ and
$s$ sufficiently large such that $\mu$ takes arbitrarily large values for
words on these alphabets. The method of proof relies in part on a
combinatorial characterisation of the word $w_{max}$ in the class
$\mathcal{X}(w)$ where $K$ assumes its maximum.

MSC: primary 11J70; secondary 68R15, 68W05, 05A20.

Keywords: Values of continuants; Regular continued fractions; Combinatorial
word problems.

Introduction. Given a sequence $w=(w_{1},\ldots,w_{n}),$ of positive $w_{i},$
let $K(w)$ be the continuant of $w,$ i.e., the denominator of the finite
regular continued fraction
$\frac{1}{w_{1}+}\frac{1}{w_{2}+}\dots\frac{1}{w_{n-1}+}\frac{1}{w_{n}}$. We
shall regard $w$ as a word of length $n$ over the alphabet $\\{1<2<3<\dots\\}$
and write $w=w_{1}\cdots w_{n}.$ Since $K(w)=K(\bar{w}),$ where
$\bar{w}=w_{n}\cdots w_{1}$ denotes the reversal of $w,$ we shall henceforth
identify each word $w$ with its reverse $\bar{w}$. Let $\mathcal{X}(w)$ denote
the Abelian class of $w$ consisting of all permutations of $w.$ The following
problem has attracted much attention and led to a number of applications (see
e.g. [1, 4, 5, 7, 8]): Let $A=\\{a_{1}<\dots<a_{s}\\}$ be a finite ordered
alphabet with $a_{j}\in\mathbb{N}$. Given a word $w=w_{1}w_{2}\cdots w_{n}$
with $w_{i}\in A$, find the arrangements $w_{\max},w_{\min}\in\mathcal{X}(w)$
maximizing resp. minimizing the function $K(\cdot)$ on $\mathcal{X}(w)$. The
first author [3] gave an explicit description of both extremal arrangements
$w_{\max}$ and $w_{\min}$ and showed that in each case the arrangement is
unique (up to reversal) and independent of the actual values of the positive
integers $a_{i}.$ He also investigated the analogous problem for the semi-
regular continuant $K^{\prime}$ defined as the denominator of the semi-regular
continued fraction
$\frac{J^{\prime}}{K^{\prime}}=\frac{1}{w_{1}-}\frac{1}{w_{2}-}\dots\frac{1}{w_{n-1}-}\frac{1}{w_{n}}$
with entries $w_{i}\in\\{2,3,...\\}.$ He gave a fully combinatorial
description of the minimizing arrangement $w^{\prime}_{\min}$ for
$K^{\prime}(\cdot)$ on $\mathcal{X}(w)$ and showed that the arrangement is
unique (up to reversal) and independent of the actual values of the positive
integers $a_{i}.$ However, the determination of the maximizing arrangement
$w^{\prime}_{\max}$ for the semi-regular continuant turned out to be more
difficult. He showed that in the special case of a $2$-digit alphabet
$\\{(2\leq)~{}a_{1}<a_{2}\\},$ the maximizing arrangement $w^{\prime}_{\max}$
is a Sturmian word and is independent of the values of the $a_{i}.$ Recently
the second author together with M. Edson and A. De Luca [8] developed an
algorithm for constructing $w^{\prime}_{\max}$ over any ternary alphabet
$\\{(2\leq)~{}a_{1}<a_{2}<a_{3}\\}$, and showed that the maximizing
arrangement is independent of the choice of the digits. In contrast, they
exhibited examples of words $w=w_{1}\cdots w_{n}$ over a $4$-digit alphabet
$A=\\{(2\leq)~{}a_{1}<a_{2}<a_{3}<a_{4}\\}$ for which the maximizing
arrangement for $K^{\prime}(\cdot)$ is not unique and depends on the actual
values of the positive integers $a_{1}$ through $a_{4}.$ In the course of
these investigations the following problem came up: given an alphabet $A$ of
positive integers, we say that a word $w$ on $A$ has multiplicity $\mu=\mu(w)$
if the value $K(w)$ occurs precisely $\mu$ times in the multi-set
$\\{K(x):x\in\mathcal{X}(w)\\}$. The multiplicity $\mu^{\prime}(w)$ is defined
analogously for the semi-regular continuant $K^{\prime}(w).$ Thus each Abelian
class $\mathcal{X}(w)$ is split into subclasses of equally valued words.
Question: is it true that $\mu$ can take arbitrarily large values for
infinitely many alphabets and is there a combinatorial proof of this? Our aim
here is to give a positive answer to this question in the case of regular
continued fractions.

Theorem. Fix positive integers $1\leq t\hskip-1.42262pt\leq\hskip
0.0ptl\hskip-1.42262pt<\hskip-1.42262pts,\hskip
2.84526ptb_{1}\hskip-2.84526pt<\hskip-1.99168pt...\hskip-1.99168pt<\hskip-1.70717ptb_{t}\hskip-1.42262pt\leq\hskip-1.42262ptl$
and let $A$ be an ordered alphabet of the form
$\\{b_{1}<...<b_{t}<l+1\hskip-1.42262pt<...\hskip-0.56905pt<\hskip-1.42262pts\\}$.
Then for all $s$ sufficiently large, there exists an infinite sequence of
words $w_{k}$ over $A$ with multiplicities $\mu(w_{k})\to\infty$ as
$k\to\infty$.

It should be noted that for fixed $s$ one obtains the largest possible
alphabet
$A^{\prime}\hskip-2.27621pt=\hskip-2.27621pt\\{1\hskip-1.99168pt<\hskip-1.99168pt2<\hskip-2.84526pt\dots<s\\}$
by choosing $b_{1}=t=l=1\ (<s)$. Our proof makes use of the combinatorial
structure of $w_{\max}$ found by the first author in [3].

Preliminaries. We introduce some notation. Let $w=w_{1}\cdots w_{n}$ be a word
of length $n\geq 2$ with $w_{j}\in\mathbb{N}$ ($j=1,\dots,n)$. The regular
continuant of $w$ has a matrix representation

$K(w_{1})=w_{1}\ \ \mbox{and}\ \
K(w)=\det\left(\begin{matrix}w_{1}&-1&0&\cdots&0\\\
1&w_{2}&-1&\ddots&\vdots\\\ 0&1&\ddots&\ddots&0\\\
\vdots&\ddots&\ddots&w_{n-1}&-1\\\ 0&\cdots&0&1&w_{n}\end{matrix}\right),\
n\geq 2$

It can also be defined recursively by $K(\\{\hskip 1.42271pt\\})=1$
($\\{\hskip 1.42271pt\\}$ = empty word), $K(w_{1})=w_{1}$ and $K(w_{1}\cdots
w_{j})=w_{j}K(w_{1}\cdots w_{j-1})+K(w_{1}\cdots w_{j-2})$ for $j\geq 2$. For
each $1\leq k\leq m\leq n$ we set $w_{k,m}:=w_{k}\cdots w_{m}$ and
$W:=W_{1,n},~{}W_{k,m}:=K(w_{k,m}).$ The following fundamental formula goes
back to the late 19th century and can be found in Perron [2], p.11, (4):
$(W\hskip-2.27621pt=\hskip
1.42262pt)~{}W_{1,n}=W_{1,j}W_{j+1,n}+W_{1,j-1}W_{j+2,n}$
$(j\in\\{1,\dots,n-1\\}).$ From this we infer the simple but useful inequality

$W_{1,n}<2W_{1,j}W_{j+1,n}.$ (1)

Let $A=\\{a_{1}<\dots<a_{s}\\}\subset\mathbb{N}.$ We consider a word
$w=w_{1}\cdots w_{n}:=a_{1}^{p_{1}}\hskip-1.13809pt\cdots a_{s}^{p_{s}}$ of
length $n$ with Parikh vector ${\bf p}=(p_{1},...,p_{s})$ with
$p_{1}+\cdots+p_{s}=n$ where

$a^{r}=\underbrace{aa\cdots a}_{\mbox{$r$-times}}$

denotes a sequence of $r$ equal elements $a$. Let
$\mathcal{X}=\mathcal{X}(A,{\bf p})$ denote the set of all permutations of $w$
where we identify each word $v$ with its reverse $\bar{v}.$ Let $N(A,{\bf p})$
denote the cardinality of $\mathcal{X}.$ Then, $N(A,{\bf
p})\geq\frac{n!}{2p_{1}!\dots p_{s}!}.$ We put $W_{\max}=W_{\max}(A,{\bf
p}):=\max\\{K(v):v\in\mathcal{X}\\}$. It was shown in [3] (see (3), p. 190)
that $W_{\max}$ is uniquely attained (up to reversal) by the arrangement

$a_{s}L_{s-1}a_{s-2}L_{s-3}\cdots a_{1}^{p_{1}}\cdots
a_{s-3}L_{s-2}a_{s-1}L_{s}$ (2)

where $L_{i}=a_{i}^{p_{i}-1}.$ Let $P=P(A,{\bf
p})=\\#\\{K(v):v\in\mathcal{X}\\}.$

###### Proof of the Theorem.

Our first goal is to describe how to specify the last digit $s\ (\geq 2)$ in
an alphabet $A:\\{b_{1}<\dots<b_{t}<l+1<\cdots<s\\}$. We consider
’equipartitioned’ words

$w=w_{1}\cdots w_{n}:=b_{1}^{m}\cdots b_{t}^{m}(l+1)^{m}\cdots s^{m}.$

corresponding to the Parikh vector ${\bf p}=(m,m,\ldots,m)$ in which each
digit of $A$ occurs precisely $m$-times in $w.$ We will give a lower bound for
$s$ (see (7) below). To this end, we introduce the quantities
$Q_{r,m-1}:=K(r^{m-1})~{}(r\in{1,2,...})\hskip 1.42262pt$. They are the
elements of the $r$-th generalised Fibonacci sequence which is determined by
the recursion $Q_{r,0}:=1$, $Q_{r,1}:=K(r)=r$, $Q_{r,j+1}:=rQ_{r,j}+Q_{r,j-1}\
(j=1,2,\dots)$.

Claim: $Q_{r,j-1}<(r+1)^{j}$ for each fixed $r\geq 1$ and all $j\geq 1.$

To prove the claim, we proceed by induction on $j$ : This is obviously true
for $j=1$ and $j=2$. Then by the induction hypothesis

$\displaystyle Q_{r,j-1}$
$\displaystyle=rQ_{r,j-2}+Q_{r,j-3}<r(r+1)^{j-1}+(r+1)^{j-2}$
$\displaystyle=(r+1)^{j-2}(r(r+1)+1)<(r+1)^{j-2}(r+1)^{2}$
$\displaystyle=(r+1)^{j}.$

In order to obtain an upper bound for the number $P(A,{\bf p}),$ it suffices
to consider words over the largest allowed $s$-digit alphabet
$A^{\prime}:\\{1<\dots<s\\}$, $b_{1}=t=l=1\ (<s)$, with Parikh vector ${\bf
p}^{\prime}=(\underbrace{m,m,\ldots,m}_{\mbox{$s$-times}}).$ Clearly

$P(A,{\bf p})\leq W_{\max}(A,{\bf p})\leq W_{\max}(A^{\prime},{\bf
p^{\prime}})$

and by (2)

$w_{\max}(A^{\prime},{\bf p}^{\prime})=s\cdot(s-1)^{m-1}\cdot(s-2)\cdots
1\cdot 1^{m-1}\cdots(s-2)^{m-1}\cdot(s-1)\cdot s^{m-1}.$ (3)

By iteration of (1) applied to the decomposition in (3) we obtain the
inequalities

$\displaystyle W_{\max}(A^{\prime},{\bf p}^{\prime})$
$\displaystyle~{}=K(w_{\max}(A^{\prime},{\bf p}^{\prime}))$
$\displaystyle~{}<2^{2s}~{}s\cdot(s-1)\cdots 3\cdot
2~{}\prod_{j=1}^{s}K(j^{m-1})$
$\displaystyle~{}=2^{2s}~{}s!~{}\prod_{j=1}^{s}Q_{j,m-1}$
$\displaystyle~{}<2^{2s}~{}s!~{}\prod_{j=1}^{s}(j+1)^{m}$
$\displaystyle~{}=2^{2s}~{}s!((s+1)!)^{m}$

and hence

$P(A,{\bf p})<2^{2s}\ s!\ ((s+1)!)^{m}.$ (4)

For each $s\geq 2$ we define $m_{0}=m_{0}(s)$ to be the smallest positive
integer such that

$2^{2s}\ s!\leq\left(\frac{100}{99}\right)^{m_{0}}.$

Then

$P(A,{\bf p})<\left(\frac{100}{99}~{}((s+1)!)\right)^{m}\ \text{\ for all\
~{}}m\geq m_{0}(s).$ (5)

On the other hand, we have the following lower bound for the number of
different words in $\mathcal{X}(w)$:

$N(A,{\bf p})\geq\frac{((s-l+t)\hskip 2.56073ptm)!}{2(m!)^{s-l+t}}$ (6)

Based on the condition (7) below, we will later make a choice of
$s=s^{\prime}(t,l)$ depending on the parameters $t,l$. We apply the estimates
provided by Sterling’s formula to the factorial terms occurring in relations
(5) and (6) to obtain

$(P(A,{\bf
p}))^{1/m}<\frac{100}{99}(s+1)!<\frac{100}{99}\frac{12}{11}~{}{\textrm{e}}^{-(s+1)}(s+1)^{s+1}\sqrt{2\pi(s+1)}.$
$(((s-l+t)\hskip 1.28036ptm)!)^{1/m}>{\textrm{e}}^{-(s-l+t)}((s-l+t)\hskip
1.42262ptm)^{s-l+t}{\sqrt{2\pi(s-l+t)\hskip 1.42262ptm}}^{\hskip
1.42262pt1/m}.$ $\left(2\hskip
2.56073pt(m!)^{s-l+t}\right)^{1/m}<{\textrm{e}}^{-(s-l+t)}m^{s-l+t}\left(2\
\frac{12}{11}\ \left(\sqrt{2\pi
m}\right)^{s-l+t}\right)^{\hskip-1.13809pt1/m}.$

When we put the right hand sides of the last two inequalities together, the
terms ${\textrm{e}}^{s-l+t}$ and $m^{s-l+t}$ cancel out, and if we keep the
parameters $t,l$ fixed for the moment, the terms of the form $\sqrt{\ \cdot\
}^{~{}1/m}$ tend to 1 as $m\rightarrow\infty$. Letting $m\rightarrow\infty$ we
get

$\displaystyle\lim_{m\to\infty}\left(\frac{N(A,{\bf p})}{P(A,{\bf
p})}\right)^{1/m}$
$\displaystyle\geq\frac{99}{100}\frac{11}{12}\frac{{\textrm{e}}^{s+1}(s-l+t)^{s-l+t}}{\sqrt{2\pi(s+1)}(s+1)^{s+1}}=$
$\displaystyle=\frac{363}{400}\frac{{\textrm{e}}^{s+1}(s+1-l+t-1)^{~{}s+1-l+t-1}}{\sqrt{2\pi(s+1)}(s+1)^{s+1}}$
$\displaystyle=\frac{363}{400}\frac{{\textrm{e}}^{s+1}}{\sqrt{2\pi(s+1)}~{}(s-l+t)^{l-t+1}}\left(1-\frac{l-t+1}{s+1}\right)^{s+1}.$

For fixed $t,l$ ($l-t\geq 1$) the function
$f(t,l,s)=\left(1-\frac{l-t+1}{s+1}\right)^{s+1}$ in the variable $s$ is
strictly increasing on the interval $[l-t+1,\infty)$ with
$f(t,l,s)\nearrow{\textrm{e}}^{-(l-t)-1}$ as $s\rightarrow\infty$. We define
$s_{0}$ to be the lowest integer such that $f(t,l,s_{0})\geq\frac{1}{2}\
{\textrm{e}}^{-(l-t)-1}$. Then

$\displaystyle\lim_{m\to\infty}\left(\frac{N(A,{\bf p})}{P(A,{\bf
p})}\right)^{1/m}\hskip-2.84544pt\geq\frac{363}{400}\frac{{\textrm{e}}^{s+1}}{\sqrt{2\pi(s+1)}~{}(s-l+t)^{l-t+1}}\
\frac{1}{2}\ {\textrm{e}}^{-(l-t)-1}\ =:H(t,l,s)$

for all $s\geq s_{0}$. Obviously there exists some sufficiently large
$s^{\prime}=s^{\prime}(t,l)\geq s_{0}$ such that

$H(t,l,s^{\prime})>1.$ (7)

Therefore the right hand side of

$\left(\frac{N(A,{\bf p})}{P(A,{\bf p})}\right)>(H(t,l,s^{\prime}))^{m}$ (8)

can be made arbitrarily large by letting $m\rightarrow\infty$. We call an
$(s^{\prime}-l+t)$-digit alphabet
$A=\\{(1\leq)~{}b_{1}<\dots<b_{t}<\dots<s^{\prime}\\}$ admissible if
$s^{\prime}=s^{\prime}(t,l)$ fulfills condition (7) We consider the word
$u(A,{\bf
p}_{1})=(b_{1})^{m_{1}}\cdots(b_{t})^{m_{1}}(l+1)^{m_{1}}\cdots(s^{\prime})^{m_{1}}$
of length $n=(s^{\prime}-l+t)\hskip 1.42262ptm_{1}$ with Parikh vector ${\bf
p}_{1}=((m_{1})^{s^{\prime}-l+t})$ where we choose $m_{1}\geq m_{0}$ such that
$\left(\frac{N(A,{\bf p}_{1})}{P(A,{\bf
p}_{1})}\right)>(H(t,l,s^{\prime}))^{m_{1}}$. The multi-set
$\mathcal{X}_{1}=\mathcal{X}(A,{\bf p}_{1})$ is made up of the $N(A,{\bf
p}_{1})=\\#\mathcal{X}_{1}$ permuted arrangements of $u$. There exists at
least one word $w_{1}\in\mathcal{X}_{1}$ with multiplicity $\mu\geq 2$ because
otherwise we would have $N(A,{\bf p}_{1})=P(A,{\bf p}_{1})$ which contradicts
(8) with $m=m_{1}$. Let $\mu_{1}\ (\geq 2)$ be the maximal multiplicity
attained by words $w\in\mathcal{X}_{1}$. Next choose $m_{2}>m_{1}(s^{\prime})$
such that $H(t,l,s^{\prime})^{m_{2}}>\mu_{1}$. We claim that at least one word
$w_{2}$ from $\mathcal{X}_{2}=\mathcal{X}(A,{\bf p}_{2}),\ {\bf
p}_{2}=((m_{2})^{s^{\prime}-l+t})$ has multiplicity $\mu>\mu_{1}$. Otherwise
we would have $N(A,{\bf p}_{2})\leq\mu_{1}P(A,{\bf p}_{2})$ which contradicts
(8) with $m=m_{2}$. Next let $\mu_{2}\ (\geq\mu_{1})$ be the maximal
multiplicity attained by words $w\in\mathcal{X}_{2}$. Proceeding with this
construction step by step we end up with a sequence of words $w_{k}$ on $A$
with multiplicities $\mu_{k}\to\infty$ as $k\to\infty$. The construction can
be carried out for infinitely many different admissible alphabets. This
completes the proof of the Theorem. ∎

The question remains largely unsolved in the case of semi-regular continuants
though it seems certain that the behavior is quite similar to the regular
case.

There is some evidence supporting the following

Conjecture. Given any ordered alphabet
$A=\\{a_{1}<\dots<a_{s}\\}~{}(a_{j}\in\mathbb{N},\ s\geq 2)$, let $\mu\geq 2$
be a positive integer. Then there exist infinitely many words on $A$ whose
multiplicity is precisely $\mu$. The problem appears to require a difficult
investigation into the values of continuants. Most likely our theorem and the
conjecture also hold for continuants of semi-regular continued fractions.
Unfortunately no higher-dimensional analogue of the theorem is available at
present for $s\geq 4$ due to the fact that very little is known about the
maximizing arrangements $w^{\prime}_{\max}$ for $s\geq 4$ (see [8]).

## References

  * [1] C. Baxa, Extremal values of continuants and transcendence of certain continued fractions, Adv. Appl. Math. 32 (2004), 754–790.
  * [2] O. Perron, Die Lehre von den Kettenbrüchen, Bd. 1, Vieweg&Teubner, 1977.
  * [3] G. Ramharter, Extremal values of continuants, Proc. Amer. Math. Soc. 89 (2) (1983), 189–201.
  * [4] G. Ramharter, Some metrical properties of continued fractions, Mathematica 30 (2) (1983), 117–132.
  * [5] G. Ramharter, Maximal continuants and the Fine-Wilf theorem, J. Comb. Th. A 111 (2005), 59–77.
  * [6] G. Ramharter, Maximal continuants and periodicity, Integers (2006-01), (2005), 10 p.
  * [7] J. Shallit, J. Sorensen, Analysis of a left shift binary GCD algorithm, J. Symb. Comput. 17 (6) (1994), 473–486.
  * [8] A. De Luca, M. Edson, L.Q Zamboni, Extremal values of semi-regular continuants and codings of interval exchange transformations, preprint 2021, arXiv:2105.00496.

Gerhard Ramharter
Dept. of Convex and Discrete Geometry
Technische Universität Wien
Wiedner Hauptstrasse 8-10
A-1040 Vienna, Austria
e-mail: Gerhard Ramharter<EMAIL_ADDRESS>

Luca Q. Zamboni
Institut Camille Jordan
Université Claude Bernard Lyon 1
43 boulevard du 11 novembre 1918
F-69622 Villeurbanne Cedex
e-mail: zamboni<EMAIL_ADDRESS>


# A quest for 5-point condition a la Alexandrov

Anton Petrunin

###### Abstract.

Here I give a description of Alexandrov 4-point comparison via quadratic forms
and then propose a natural 5-point condition which might have some future.
Consider this note as a letter from me — do not take it seriously.

###### Key words and phrases:

Alexandrov spaces, CAT(0), CBB(0)

## Associated form

Let us construct a quadratic form $W_{\bm{x}}$ on $\mathbb{R}^{n-1}$ for given
$n$-point array $\bm{x}=\nobreak(x_{1},\dots,x_{n})$ in a metric space $X$.

Fix a nondegenerate simplex $\triangle$ in $\mathbb{R}^{n-1}$; denote by
$v_{1},\dots,v_{n}$ its vertices; if $e_{i}$ is the standard basis on
$\mathbb{R}^{n-1}$, we may assume that $v_{i}=e_{i}$ for $i<n$ and $v_{n}=0$.

Let us denote by $|a-b|_{X}$ the distance between points $a$ and $b$ in the
metric space $X$. Set

$W_{\bm{x}}(v_{i}-v_{j})=|x_{i}-x_{j}|^{2}_{X}$

for all $i$ and $j$. Note that this identity define $W_{\bm{x}}$ uniquely.

The constructed quadratic form $W_{\bm{x}}$ will be called _associated form_
for the point array $\bm{x}$ with respect to the simplex $\triangle$.

Note that an array $\bm{x}=(x_{1},\dots,x_{n})$ in a metric space $X$ is
isometric to an array in Euclidean space if and only if
$W_{\bm{x}}(v)\geqslant 0$ for any $v\in\mathbb{R}^{n-1}$.

In particular, for $n=3$ the condition $W_{\bm{x}}\geqslant 0$ means that for
the points $x_{1}$, $x_{2}$ and $x_{3}$, all three triangle inequalities hold.

## Alexandrov’s 4-point comparison

Now let us discuss the relation between form $W_{\bm{x}}$ and geometry of the
space for $n=4$. In this case $\triangle$ is a tetrahedron on
$\mathbb{R}^{3}$.

$\text{\rm Quad}(4)$$\text{\rm Quad}(3)$

From the case $n=3$ it follows that $W_{\bm{x}}$ is nonnegative on every plane
parallel to a face of the tetrahedron $\triangle$. In particular, $W_{\bm{x}}$
can have at most one negative eigenvalue.

Assume $W_{\bm{x}}(w)<0$ for some $w\in\mathbb{R}^{3}$. It follows from above
that $w$ is transversal to each of 4 planes parallel to a faces of
$\triangle$. Let us project $\triangle$ along $w$ to a transversal plane. Note
that after the projection the 4 vertices of $\triangle$ lie in general
position; i.e., no three of them lie on one line. Therefore in the projection
we can see one of two combinatorial pictures shown on the diagram. It is easy
to see that the combinatorics of the picture does not depend on the choice of
$w$.

If we see the diagram on the left we say that $\bm{x}$ is of type $\text{\rm
Quad}(4)$ and otherwise we say that it is of type $\text{\rm Quad}(3)$.

The following statements give a connection between the associated forms
$W_{\bm{x}}$ and the curvature bounds in the sense of Alexandrov. Their proofs
are left to the reader.

Assume $X$ is a complete space with intrinsic metric. Then

  * $\diamond$

If $W_{\bm{x}}\geqslant 0$ for any quadrilateral $\bm{x}=(x_{1},\dots,x_{4})$
then $X$ is isometric to a closed convex set in a Hilbert space.

  * $\diamond$

$X$ has no quadruples of type $\text{\rm Quad}(3)$ if and only if $X$ has
nonnegative curvature in the sense of Alexandrov.

  * $\diamond$

$X$ has no quadruples of type $\text{\rm Quad}(4)$ if and only if $X$ is a
$\mathrm{CAT}[0]$ space.

## 5-point conditions

Let us try to do the same for 5-points arrays $\bm{x}=(x^{1},\dots,x^{5})$ in
a metric space. In this case the associated form $W_{\bm{x}}$ is defined on
$\mathbb{R}^{4}$ and it has to be nonnegative on any plane which is parallel
to any of 10 two-dimensional faces of the $4$-simplex
$\triangle\subset\mathbb{R}^{4}$. In particular $W_{\bm{x}}$ has at most two
negative eigenvalues.

In the case if $W_{\bm{x}}$ has exactly two negative eigenvalues, one can
choose a plane $\Pi$ such that the restriction of $W_{\bm{x}}$ to $\Pi$ is
negative. Let us project $\triangle$ along $\Pi$ to a transversal plane. The
same argument as in case $n=4$ shows that after projection the vertices of
$\triangle$ lie in general position. Therefore we may get one of the following
three combinatorial pictures.

$\text{\rm Pent}(5)$$\text{\rm Pent}(4)$$\text{\rm Pent}(3)$

I.e., for any 5-point array $\bm{x}$ either $W_{\bm{x}}$ has at most one
negative eigenvalue or it has exactly two negative eigenvalues and belongs to
one of these three types $\text{\rm Pent}(5)$, $\text{\rm Pent}(4)$ or
$\text{\rm Pent}(3)$.

We may consider the metric spaces which do not have a 5-point arrays of some
of these types. For example, denote by $\text{\rm Pent}(\hat{3},\hat{4})$ the
class of all complete length-metric spaces without 5-point arrays of type
$\text{\rm Pent}(3)$ and $\text{\rm Pent}(4)$.

Let us list some easy observations about these new classes of metric spaces.

  1. (i)

Any $\mathrm{CAT}[0]$ space is a $\text{\rm Pent}(\hat{3},\hat{4},\hat{5})$
space. In other words, the form $W_{\bm{x}}$ for any 5-point array $\bm{x}$ in
any $\mathrm{CAT}[0]$ space has at most one negative eigenvalue.

  2. (ii)

Any space with nonnegative curvature in the sense of Alexandrov has no 5-point
arrays of type $\text{\rm Pent}(3)$ and $\text{\rm Pent}(4)$; i.e., it is a
$\text{\rm Pent}(\hat{3},\hat{4})$ space.

  3. (iii)

If a complete Riemannian manifold $M$ has no 5-point array of type $\text{\rm
Pent}(5)$ then it is a simply connected and it has non-positive sectional
curvature. In other words, if a complete Riemannian manifold $M$ belongs to
$\text{\rm Pent}(\hat{5})$ then it is $\mathrm{CAT}[0]$.

From the observations (i) and (ii) it follows that the $\text{\rm
Pent}(\hat{3},\hat{4})$ includes all $\mathrm{CAT}[0]$ spaces as well as the
spaces with nonnegative curvature in the sense of Alexandrov.

Question. Do $\text{\rm Pent}(\hat{3},\hat{4})$ spaces have meaningful
geometry?

In particular we do not know the answer to the following question.

Question. Let $X$ be a complete $\text{\rm Pent}(\hat{3},\hat{4})$ space with
intrinsic metric. Assume two minimizing geodesics in $X$ share three distinct
points. Does it imply that these geodesics share an arc containing these three
points?

Question. Let $X$ be the coordinate plane with metric induced by norm. Assume
$X$ is a $\text{\rm Pent}(\hat{3},\hat{4})$ space. Is it true that $X$ has to
be isometric to Euclidean plane?

Here is an other question related to the observations (i) and (iii).

Question. Is it true that any complete $\text{\rm
Pent}(\hat{3},\hat{4},\hat{5})$ space with intrinsic metric has to be
$\mathrm{CAT}[0]$?


# Probing Metal-Molecule Contact at the Atomic Scale via Conductance Jump

Biswajit Pabi, Debayan Mondal, Priya Mahadevan*, and Atindra Nath Pal
Department of Condensed Matter Physics and Material Science, S. N. Bose
National Center for Basic Science, Sector III, Block JD, Salt Lake, Kolkata -
700106  Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Understanding the formation of metal-molecule contact at the microscopic level
is the key towards controlling and manipulating atomic scale devices.
Employing two isomers of bipyridine, $4,4^{\prime}$ bipyridine and
$2,2^{\prime}$ bipyridine between gold electrodes, here, we investigate the
formation of metal-molecule bond by studying charge transport through single
molecular junctions using a mechanically controlled break junction technique
at room temperature. While both molecules form molecular junctions during the
breaking process, closing traces show the formation of molecular junctions
unambiguously for $4,4^{\prime}$ bipyridine via a conductance jump from the
tunneling regime, referred as ‘jump to molecular contact’, being absent for
$2,2^{\prime}$ bipyridine. Through statistical analysis of the data, along
with, molecular dynamics and first-principles calculations, we establish that
contact formation is strongly connected with the molecular structure of the
electrodes as well as how the junction is broken during breaking process,
providing important insights for using a single-molecule in an electronic
device.

KEYWORDS: MCBJ. jump to contact. single molecular junction. Bipyridine. Break
junction.

##  I Introduction

The drive towards miniaturization has led to envisaged electronic components
having elements which have dimensions in the nanometre scale. These elements
could be molecules 1, and before placing them in a circuit, it becomes
essential to know how they would behave. While there have been tremendous
advances made 2, 3, 4, 5, probing the contact formation at the microscopic
level has not been straightforward. A route towards exploring atomic scale
transport has involved forming mechanical break junctions 6. Considering
metallic electrodes, examining the metal-metal contact formed in a break
junction at liquid helium temperatures, one finds a regime in which one has
quantum tunneling, followed by the formation of contacts. This transition from
a tunneling transport regime to contact formation occurs, in most cases, via a
sudden jump in the conductance, referred to as a jump to contact 7, 8. Various
metals like Au, Pt, Cu and Ag exhibit a jump to contact, while there are
others like Ir, Ni and W in which a jump to contact can be absent. In a
controlled experiment involving an Au break junction, using a highly ordered
gold electrode, it was shown that the phenomena could be explained by a
generic potential energy model with the elastic constant of the metal being
the only free parameter 9.

Formation of metal-molecule bond is rather complex and highly dependent on the
nature of binding groups and geometry of the electrodes 10, 11, 12, 13, 14,
15, 16, 17. Jump to molecular contact was reported for several flat molecules
through scanning tunneling microscopy (STM) based experiments at cryopgenic
temperatures 18, 19, 20, 21. It is believed that there exist two energetically
close adsorption geometries with one of them bridging the junction via a soft
phonon from the molecular side groups 19. In case of mechanically controllable
break junctions, however, the shape of the electrodes are not expected to
remain in line while approaching each other, especially at room temperature
22. This may lead to the formation of asymmetric junctions, which, in contrast
to the breaking process, may not provide precise conductance of single
molecular junctions. In recent break junction experiments with an
organometallic compound (BdNC), carried out in solution, a jump was observed
in the conductance for closing traces 23. This was attributed to the formation
of one-dimensional coordination polymers in those junctions. The knowledge
about the role of metal-molecule interaction towards bond formation is rather
limited and a general picture is still lacking.

In the present study, we report the observation of jump to molecular contact
in a single molecular junction by studying the charge transport using a
mechanically controllable break junction setup at ambient temperature. Through
statistical analysis of conductance-distance traces, along with density
functional theory based calculations and ab-initio molecular dynamics, we
provide the microscopic origin behind the metal-molecular contact formation.
Considering two isomers of bipyridine, $4,4^{\prime}$ bipyridene
($4,4^{\prime}$-BPY) and $2,2^{\prime}$ bipyridene ($2,2^{\prime}$-BPY),
attached to gold electrodes, we find that molecular junctions form in the
breaking traces for both the molecules. The surprising aspect is that
$4,4^{\prime}$-BPY forms molecular junctions isotropically with conductance
jumps in the closing traces also, an aspect that is absent for
$2,2^{\prime}$-BPY.Our DFT and MD simulations reveal two important mechanisms
for jump to molecular contact. Firstly, while the $2,2^{\prime}$-BPY prefers
to lie flat on the Au surface, $4,4^{\prime}$-BPY has two stable minima - one
with the molecule lying flat on the surface and the other with the molecule
standing vertically, despite having similar anchoring group involving
nitrogen. This unusual aspect of the $4,4^{\prime}$-BPY emerges from the fact
that the nitrogen forms a very strong bond in $4,4^{\prime}$-BPY 24 while the
orientation on the Au surface in $2,2^{\prime}$-BPY doesn’t allow a similar
strong bond due to steric effects from the hydrogens attached to the carbon
atoms, with the Au-N bondlength equal to 2.24 $\AA$ here in contrast to 2.12
$\AA$ for the $4,4^{\prime}$-BPY molecule. The possibility of having two
stable energy conformations may lead to a jump to contact for
$4,4^{\prime}$-BPY, similar to Ref[19]. Additionally, we propose another
mechanism, according to which, the observed jump to molecular contact in case
of $4,4^{\prime}$-BPY may arise from the fact that the molecule breaks by
pulling few gold atoms with it due to strong Au-N bond compared to the Au-Au
bond. Consequently, while making the contacts during the closing loops, one
does not have to worry about the directionality of the bonding. Through a
statistical analysis of the experimental data the second mechanism was found
to be the dominant one.

Figure 1:  (a) Schematic of a mechanically controllable break junction (MCBJ)
set up, along with the schematic representation of the molecules used in this
work: $4,4^{\prime}$ bipyridine ($4,4^{\prime}$-BPY) and $2,2^{\prime}$
bipyridine ($2,2^{\prime}$-BPY). Conductance traces for (b) $4,4^{\prime}$-BPY
and (d) $2,2^{\prime}$-BPY. Blue (red) represents trace during pull (push). 1D
logarithmic conductance histogram for (c) $4,4^{\prime}$-BPY and (e)
$2,2^{\prime}$-BPY, constructed from 5000 and 8000 traces respectively,
without any data selection. Blue and red represent the corresponding pull and
push respectively. White dashed line shows the Gaussian fit to the
corresponding molecular peaks of the histogram.

##  II Methods

A single molecular junction was formed in a custom built mechanically
controllable break junction (MCBJ) technique at ambient condition 25. A gold
wire (0.1 mm dia, 99.998$\%$, Alfa aesar) with a notch at its center was fixed
onto a flexible substrate (Phosphor Bronze) with a two component epoxy glue
(Stycast 2850 FT with catalyst 9). For creating molecular junctions, molecules
($4,4^{\prime}$-BPY and $2,2^{\prime}$-BPY, 98$\%$, Alfa aesar) were either
evaporated or drop casted (0.1mM Ethanol solution) on top of the notch and the
wire was controllably broken using a piezo stack. Formation of molecular
junctions were confirmed from the appearance of additional peaks in the
conductance histograms below 1G0 (where G${}_{0}=2e^{2}/h$, is the quantum of
conductance). In the conductance-distance measurements, the typical moving
rate of the piezo was 3.2 $\mu$m/s, which translates into an opening and
respective closing rate for the two leads of about $2-4$ nm/s (attenuation
factor of $\sim 10^{-3}$) 26.

In order to model the experiments, we use density functional theory based ab-
intio electronic structure calculations using the generalised gradient
approximation (Perdew Burke-Ernzerhof)27 for the exchange correlational
functional with dispersion-corrections(DFT-D2) 28, 29 as implemented within
Vienna ab initio simulation package (VASP) 30, 31, 32, 33. We have used
$\Gamma$ point K-space sampling 34 and a plane-wave cut-off of 500 eV . The
structural relaxations were carried out till forces were less than $10^{-3}$
eV/atom and energy convergence in each case was $10^{-5}$ eV. The ab-initio MD
simulations 35, 36, 37 were carried out at room temperature (300K) and the
whole system is kept at the desired temperature in NVT canonical ensemble
using a Nosé-Hoover thermostat 38, 39, 40 for several molecular junctions
formed by $4,4^{\prime}$-BPY and $2,2^{\prime}$-BPY molecules with gold
electrodes. The optimized structure of a gold slab-molecule-gold slab is used
to construct the junction atomic geometry, where 8 layers were used to
construct the slab. The slabs are separated by a vacuum of 15 $\AA$. The
electrodes are pulled apart in steps of 0.1$\AA$. The entire system is relaxed
at every step by taking 200 MD steps of 0.5 fs.

##  III Results and Discussion

Typical opening or pull (blue) and closing or push (red) conductance traces
for both the molecules are shown in Figure 1b and 1d respectively (see SI for
measurement details). During junction breaking process (pull), well-developed
molecular plateaus in the range of $10^{-3}$ to $10^{-4}$ G0 is noticed for
both $4,4^{\prime}$-BPY $\&$ $2,2^{\prime}$-BPY. Closing traces (push) of
$4,4^{\prime}$-BPY, however, exhibit a plateau like feature which is longer
and slanted compared to the pull traces, in line with the previous
observations 41. In case of $2,2^{\prime}$-BPY, no such plateau was observed
in push traces, instead, a continuous tunneling-like behavior observed. A
closer look at the push traces for $4,4^{\prime}$-BPY reveals that atomic
contacts of Au forms via two jumps in the conductance: first jump occurring
from the background noise or tunneling to a conductance value of $\sim$
$10^{-3}$ G0 and the second one from $\sim$ $0.1$ G0 to the few tenths of G0.
First jump is referred as ‘jump to molecular contact’ (abbreviated as J2C) and
the second jump corresponds to the formation of metallic contact. No such jump
is observed in case of $2,2^{\prime}$-BPY. In the following paragraphs we
discuss more details about origin of the J2C phenomena.

Figure 2:  2D Conductance-displacement density plot for (a) $4,4^{\prime}$-BPY
and (b) $2,2^{\prime}$-BPY, for push traces used in Figure 1d and 1e, with 100
bins per decade. Inset: Histogram of the conductance of the molecular contact
formed after the jump, having 3750 traces (see SI for details).

Thousands of individual traces were collected, analyzed and plotted as
histograms to get a meaningful estimate of the molecular conductance from
statistically independent junction configurations. Figure 1c and 1e display
the logarithmically binned normalized one-dimensional (1D) conductance
histogram for $4,4^{\prime}$-BPY and $2,2^{\prime}$-BPY respectively,
constructed from 5,000 and 8,000 conductance traces without any data
selection. For both the molecules, the pull histogram (blue) exhibits a clear
peak around 1G0 corresponding to the Au atomic contact, followed by a
prominent molecular conductance feature in the range of $5\times 10^{-2}$ G0
to $5\times 10^{-4}$ G0 ($4,4^{\prime}$-BPY) and $1\times 10^{-3}$ G0 to
$1\times 10^{-4}$ G0 ($2,2^{\prime}$-BPY). For $4,4^{\prime}$-BPY, conductance
peak can be resolved into two separate peaks ($9.33\pm 0.29\times 10^{-4}$ G0
and $4.28\pm 0.07\times 10^{-4}$ G0), obtained by fitting Gaussian to the
characteristic maxima, similar to the previous observation 42, 43 (see SI
Figure S5 for details). The most probable conductance for $2,2^{\prime}$-BPY
was found to be $1.27\pm 0.05\times 10^{-4}$ G0 (Figure 1e). Push histogram
(Red) for $4,4^{\prime}$-BPY (Figure 1c) shows a clear conductance peak in the
region ($5\times 10^{-4}$ G0 to $2\times 10^{-1}$ G0), being higher than the
pull conductance, whereas, no such conductance peak is observed in case of
$2,2^{\prime}$-BPY.

To get a clear picture about the evolution of conductance during the formation
processes and understand the possible origin of observed J2C in case of
$4,4^{\prime}$-BPY, we have constructed conductance-displacement histogram
from the push traces for both the molecules. Figure 2a and 2b display such 2D
histograms for the push data for $4,4^{\prime}$-BPY and $2,2^{\prime}$-BPY,
respectively. The plots were constructed from the same traces used in the 1D
histogram, by aligning them at 2G0 and having 100 bins per decade.
Corresponding pull histograms are shown in SI Figure S2. By analyzing the
length histogram, we see that $4,4^{\prime}$-BPY (Figure 2a) exhibits a longer
and slanted conductance plateau in push with an average length of $9.77\pm
0.05$$\AA$ (SI Figure S3c), compared to the corresponding pull plateaus with
an average plateau length of $3.03\pm 0.02$$\AA$ (SI Figure S3a). Similar
analysis for $2,2^{\prime}$-BPY (pull) shows an average plateau length of
$2.57\pm 0.01$$\AA$ (SI Figure S3b). The most intriguing aspect is the
unambiguous formation of single molecular junction for $4,4^{\prime}$-BPY
during the closing process with a conductance-jump occuring mostly at
$\sim(5.98\pm 0.23)\times 10^{-4}$G0, calculated from the Gaussian fit of the
histogram of conductance values where jump to molecular contact takes place,
shown in the inset of Figure 2a (see SI for details about the jump detection
algorithm). This value bears close resemblance to the conductance of of
Au-$4,4^{\prime}$-BPY-Au junctions in the stretched configuration, reported
earlier 42. Jump to molecular contact is clearly absent for $2,2^{\prime}$-BPY
(Figure 2b), where a continuous tunneling slope is observed from the noise
background while closing the electrodes and no clear formation of molecular
junction was noticed.

Figure 3:  Conditional histogram of $4,4^{\prime}$-BPY for (a) pull and (d)
push traces. Black dashed-dotted line represent the histogram of the selected
pull (push) traces for which plateaus are formed in the corresponding push
(pull) traces, respectively. Blue and red area graph represents the histogram
for all traces, as a reference for (a) and (d), respectively. (b) 2D
correlation histogram, indicating the correlation between the pull (horizontal
axis) and push (vertical axis ). Histogram is constructed from 10000 traces
without any data selection using 50 bins per decade. (c) Bar diagram showing
percentage of traces forming molecular junction in push via jump (black) and
percentage of traces having both molecule in the pull and push (grey) for 8
different data set.

A statistical correlation analysis of the conductances traces, as described in
Ref 44, has been used to distinguish various structural configurations and
understand their origin. Using the same procedure, we have carried out
statistical correlation analysis between the pull and push traces 45 to find
out if there is any possible correlation between them (see SI for details).
Figure 3b shows the two-dimensional cross correlation plot for
$4,4^{\prime}$-BPY, constructed from 10,000 traces with 50 logarithmic bins
per decade. The vertical axis belongs to the push traces and the horizontal to
the corresponding pull traces. A clear positive correlation between the pull
and corresponding push traces is observed, shown by a dark spot near the
region where pull and push peak appear, as depicted by the black dashed
lines.This is also evident from the conditional histogram of the corresponding
pull (push) traces, shown by the black dash-dotted line in Figure 3a (3d).
While selecting the push (pull) traces having conductance plateau in the
selected region between the black dashed lines, we observe a clear enhancement
in the conductance peak of the pull (push) histogram. A positive correlation
may indicate a structural memory effect 46 that any molecular junction in push
is possible, mostly when a molecular junction is formed during breaking
process (pull). Similar positive correlation between pull and push traces was
observed for Au-CO-Au junction at cryogenic temperature 47. It was concluded
that the molecule is bound rigidly to the apex of one electrode and the
similar single molecular configuration was reestablished during closing
process. At room temperature, however, due to the enhanced surface diffusion,
electrodes are flattened and a correlation between the opening and closing
traces is uncommon. Figure 3c compares the correlation observed in push from
eight different data sets from three different samples with a varying
probability of molecular junction formation. A conditional analysis of eight
statistically independent data sets indicate that J2C was observed for more
than $80\%$ cases whenever a molecular junction is formed during pull,
establishing the fact that J2C is primarily associated with the formation of a
molecular junction in pull (see SI for details).

Figure 4:  (a) MD snapshot of molecular junction with $4,4^{\prime}$-BPY for
effective length of electrode (1) 11.10$\AA$ , (2) 11.60$\AA$ , (3) 12.20$\AA$
, and (4) 12.45$\AA$ (b) MD snapshot of molecular junction with
$2,2^{\prime}$-BPY for effective length of electrode (1) 6.65$\AA$ , (2) 6.80
$\AA$, and (3) 6.89$\AA$ . (c) Change in total energy with respect to
stretching of the bond of Molecule-Au (black) and Au-Au atom (red).

To simulate the experiment, we take two pyramidal gold electrodes and place
the molecule $4,4^{\prime}$-BPY symmetrically between the electrodes. The
optimised Au-molecule bond length was found to be $2.12\AA$ in the static
calculation. Placing the molecule at this distance from the electrode, led to
an effective length (distance between two electrode tips) of 11.10$\AA$. The
Au-Au bondlengths in the electrodes ware set equal to the bond length of 2.65
$\AA$ found in the experimental structure. Molecular dynamics simulations were
then carried out at a temperature of 300 K. After 200 iterations, we find that
while the positions of the Au atoms in the electrode do not change, two atoms
from the electrode come closer to the molecule making shorter bonds. We have
increased the effective length to several values as shown in Figure 4a and
there are two aspects that emerge. An Au atom gets removed from the electrode
and moves towards the molecule. Additionally the molecule is no longer
symmetrically placed with respect to electrodes, and the distance of the
molecule and one end of the electrode becomes shorter (on one side the length
is 2.45 $\AA$ and on the other it is 2.55 $\AA$). The converged effective
length is now 12.20 $\AA$ (Figure 4a (3)). If we increase the effective length
further (13.00 $\AA$), we still find that the Au atom moves towards the
molecule (Figure 4a (4)). Similar calculation was done for $2,2^{\prime}$-BPY.
The N-Au bondlength is found to be 2.24$\AA$ where we optimised the structure
at 0K. When we increase the separation, the Au atom from the electrode is not
detached as found for $4,4^{\prime}$-BPY ( Figure 4b).

In order to understand this further, we have examined the energy variations as
we stretch the Au-Au bond of a metal-metal contact. The energy is found to
vary continuously till a bond length of 3.20 $\AA$ (a stretch of 8.8% in the
bondlength) (Figure 4c). At this point there is a discontinuous change in the
bondlength and a jump in the energy also. The energy variations now lie on a
new curve and are again continuous. As the hopping interaction strength
between any two atoms varies inversely with their distance according to a
power law, this jump in bondlength would translate into a jump in the hopping
interaction strength. This then leads to a jump in the conductance, and is
consistent with experiments carried out with Au electrodes in which a jump to
contact is observed 7, 9, 8. When we went on to examine the variations in the
energy of the molecule connected to a Au atom, on subjecting it to a similar
elongation we found a continuous variation (Figure 4c). This suggests that one
should see only a monotonic variation in the conductivity in this case. This
is, however, contrary to experiments where one has a jump to contact.

Figure 5: Complete optimised structure for vertically aligned and horizontally
aligned orientations of the molecules shown in panels (a) and (b) for
$4,4^{\prime}$-BPY and in panel (c) for $2,2^{\prime}$-BPY. Charge density of
a state formed as a result of interactions between Au and the molecule (see SI
Figure S7) is shown in the inset of panel (a).

To shed more light on this, we have placed both molecules on gold surface and
carried out static calculations for different orientations of both molecules
at a fixed distance. These results suggest that there are two minima present
for $4,4^{\prime}$-BPY on gold, while there is only one minima for the
$2,2^{\prime}$-BPY. Allowing for a complete structural optimization in each
case leads to the orientations of the molecules shown in panels (a) and (b) of
Figure 5 for $4,4^{\prime}$-BPY and what is shown in panel (c) for
$2,2^{\prime}$-BPY. While in panel (a) we find that the molecule prefers an
orientation almost vertical to the surface, in (b) and (c) we find that the
molecule likes to lie horizontal to the surface. The reason for this is easy
to understand. In the horizontal position the molecule interacts through the
carbon backbone with the Au surface. Hence for both molecules, this
corresponds to a stable orientation. In the vertical orientation,
$4,4^{\prime}$-BPY interacts strongly through the nitrogen with Au surface.
This is evident from the charge density shown in the inset of Figure 5a, where
one finds significant weight on several Au atoms beyond it’s nearest
neighbour. In contrast the Au-N bondlength is much longer for
$2,2^{\prime}$-BPY and found to be 2.24 $\AA$ as the steric effects from the
neighbouring atoms don’t allow the molecule to come closer to the Au surface.
This leads to only one stable minima for $2,2^{\prime}$-BPY. The strong
bonding interaction of the nitrogen in $4,4^{\prime}$-BPY that we find is
extremely directional. This would therefore imply that in the push cycle, we
would not have have the desired reproducibility as only certain orientations
would form strong bonds. However, as we saw in the MD simulations and also
supported by an analysis of the charge density of the bonding state between
$4,4^{\prime}$-BPY and the Au surface, we find the interaction between them to
be strong enough to pull one or more Au atoms out. This then leads to a
contact between two Au atoms in the push cycle, needless to say the bonding
between Au atoms involves s orbitals and is isotropic (it does not need to
attach back to its initial position, can attach with any other gold atom) 7,
48, 49, and hence we have the reproducibility. From Figure 3c we observe that
a jump to contact is observed in $\sim 20\%$ cases despite not forming any
molecular junction during pull. As we found that $4,4^{\prime}$-BPY has two
stable orientation on the Au surface, a jump may occur between them via a soft
phonon mode of the molecule during push19.

##  IV conclusion

In conclusion, we have studied the formation and the post rupture evolution of
molecular junctions at room temperature of two isomers of bipyridene
($4,4^{\prime}$-BPY and $2,2^{\prime}$-BPY) having the same anchoring groups
using MCBJ techniques. Transport measurements indicate formation of a stable
molecular junction during opening cycle for both of these molecules. During
the closing cycle, however, we see that $4,4^{\prime}$-BPY shows clear
formation of a molecular junction via a jump in the conductance from the
tunneling regime, while a continuous quantum tunneling like behavior was
observed for $2,2^{\prime}$-BPY. As such formations of molecular junctions via
a jump in the conductance at room temperature are rare, we use DFT
calculations as well as ab-initio MD simulations to explain them. Two possible
mechanisms have come out from our study. Firstly, $4,4^{\prime}$-BPY was found
to have two possible conformations on the Au surface due to the formation of
strong Au-N bond, whereas, $2,2^{\prime}$-BPY prefers to lay flat on the Au
surface. These two stable conformations of $4,4^{\prime}$-BPY may provide jump
to contact while approaching the two electrodes. Secondly, we show that while
breaking the Au-$4,4^{\prime}$-BPY junctions, primarily the Au-Au bond that is
ruptured instead of the molecule-Au bond. As the Au atom attached to the
molecule can bind isotropically with the other Au atoms of the electrodes, one
has single molecular junctions formed during the closing process via
conductance jumps similar to what one has seen for Au-Au contacts. Statistical
analysis of our experimental data indicate that the second mechanism is more
dominant one in our case. We provide an important insight, describing the role
of metal-molecule interaction on the formation of chemical bond, important for
the development of molecular-scale electronics.

##  V Acknowledgement

A. N. Pal acknowledge the funding from Department of Science and Technology
(grant no. CRG/2020/004208), B. Pabi acknowledges support from DST-Inspire
fellowship, D. Mondal acknowledges support from CSIR-HRDG fellowship and P.
Mahadevan acknowledges support from DST - Nanomission for the research through
the project DST/NM/NS/2018/18.

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