Latency Model and Block Size Optimization

Transaction confirmation latency sec:e2e_latency

The transaction confirmation latency of a blockchain can be derived by
considering the following delays:

  1. 1.

Queuing delay ($\text{T}_{\text}{q}$): Clients submit transactions to the
closest miner following a Poisson process where the time between transactions
is given by an exponential distribution with parameter $\mu$. The submitted
transactions wait in a pool before being included in a candidate block, i.e.,
when the block size $b$ is achieved or when the waiting timer $\tau$ expires.
For the sake of simplicity, we assume that the transactions pool is perfectly
shared. We resort to the queue model of Sect. 1 to characterize the queuing
delay.

  2. 2.

Block generation delay ($\text{T}_{\text}{bg}$): Miners run PoW to find the
candidate block’s nonce. This entails solving a computation-intensive
mathematical puzzle. This process is modeled through an exponential random
variable with parameter $\lambda$, so the expected mining time
$\text{T}_{\text}{bg}$ for $M$ miners is given by $(M\lambda)^{-1}$. Notice
that the block size does not affect the mining time.

  3. 3.

Block propagation delay ($\text{T}_{\text}{bp}$): Mined blocks are propagated
throughout the P2P network. The block propagation delay depends on the size of
the P2P network and the links between miners. Blocks arrive simultaneously to
all the miners to avoid synchronization issues.

Considering the effect of forks on the end-to-end latency, we define the
blockchain transaction confirmation $\text{T}_{\text}{BC}$ latency:

$\text{T}_{\text}{BC}=\frac{\text{T}_{\text}{q}+\text{T}_{\text}{bg}+\text{T}_{\text}{bp}}{1-p_{\text}{fork}},$
(1)

where the fork probability $p_{\text}{fork}$ denotes the cases when two or
more miners succeed in mining a block before the winner’s one is completely
propagated. With forks, uninformed miners may mistakenly add non-winner blocks
to their ledger version. Since the time between blocks is a Poisson inter-
arrival process, the fork probability is given by
$p_{\text}{fork}=1-e^{-\lambda(M-1)\text{T}_{\text}{bp}}$.

##  Queue Model

As done in [wilhelmi2021discrete], we consider a finite-length $M/M^{s}/1/K$
queue (see Fig. 1) – where $s$ denotes the number of transactions to be served
in a batch – to derive the transaction confirmation latency in blockchain. To
that purpose, we model the queue using a Markov chain where states indicate
the number of queued transactions before a departure. We apply the Poisson
arrivals see time averages (PASTA) property to obtain the steady-state queue
distribution.

[width=]img/batch_service_queue.pdf

Figure : Batch-service queue for blockchain.

To calculate the expected queue delay, we focus on the queue’s steady-state
distribution $\pi^{k}$, which indicates the probability of each state $k$:

$\text{T}_{\text}{q}=\frac{\sum_{k=0}^{K}k\pi^{k}}{\mu(1-\pi^{K})}.$ (2)

To get $\pi^{k}$, we first derive the departures distribution
$\boldsymbol{\pi}^{d}$, which is obtained by solving
$\boldsymbol{\pi}^{d}=\boldsymbol{\pi}^{d}\boldsymbol{\text{P}}$, being
$\text{P}$ the transition-probability matrix and
$\boldsymbol{\pi}^{d}\boldsymbol{1}^{T}=1$:

$\resizebox{.92}{!}{\small\centering\text{P}=\bordermatrix{&0&1&\ldots&b&\ldots&K-b&\ldots&K\cr\;\;\;0&p_{0,0}&p_{0,1}&\ldots&p_{0,b}&\ldots&p_{0,K-b}&\ldots&\overline{p}_{0,K}\cr\;\;\;1&p_{1,0}&p_{1,1}&\ldots&p_{1,b}&\ldots&p_{1,K-b}&\ldots&0\cr\;\;\;\vdots&\vdots&\vdots&\ddots&\vdots&\ddots&\vdots&\ddots&\vdots\cr\;\;\;b&p_{b,0}&p_{b,1}&\ldots&{p}_{b,b}&\ldots&\overline{p}_{b,K-b}&\ldots&0\cr\;\;\;\>\vdots&\vdots&\vdots&\ddots&\vdots&\ddots&\vdots&\ddots&\vdots\cr
K-b&0&0&\ldots&p_{K-b,b}&\ldots&\overline{p}_{b+1,b+1}&\ldots&0\cr\;\;\;\>\vdots&\vdots&\vdots&\ddots&\vdots&\ddots&\vdots&\ddots&\vdots\cr\;\;\;K&0&0&\ldots&1&\ldots&0&\ldots&0}\@add@centering}$

Being $s(i)$ the number of served transactions from departure state $i$,
transition probabilities $p_{i,j}$ and $\overline{p}_{i,j}$ are computed as

$\resizebox{0.9}{!}{p_{i,j}=\frac{\lambda}{\lambda+\mu}\Big{(}\frac{\mu}{\lambda+\mu}\Big{)}^{j-\big{(}i-s(i)\big{)}},\overline{p}_{i,j}=1-\sum^{K-s(i)-1}_{l=0}p_{i,l}}$
(3)

Forks are captured within the served transactions at departures. Considering
that $\mathcal{T}_{f}$ represents the set of non-conflicting transactions that
remain valid even in the event of a fork (i.e., transactions included in the
winner’s block and not included in other forked blocks), $s(i)$ is obtained as
follows:

$s(i)=(1-p_{\text}{fork})\cdot\min\\{i,b\\}+p_{\text}{fork}\cdot|\mathcal{T}_{f}|.$
(4)

Finally, the steady-state queue distribution $\boldsymbol{\pi}^{k}$ for $k<K$
is given by

$\split\pi^{k}&=\frac{1}{\mu\text{T}_{\text}{d}}\sum_{i=0}^{k}\pi_{i}^{d}\bigg{[}\overline{\varsigma}^{\tau}_{i}\Big{(}\sum_{j=k-s(i)+1}^{K-s(i)}p_{i,j}\Big{)}\\\
+\varsigma^{\tau}_{i}\Big{(}\sum_{j=i}^{b-1}\big{(}\text{Pr}(j-i|\tau)\sum_{l=k-s(j-i)+1}^{K-s(j-i)}p_{j,l}\big{)}\Big{)}\bigg{]},$
(5)

where $\text{T}_{\text}{d}$ is the expected inter-departure time,
$\varsigma^{\tau}_{i}$ is the timer expiration probability from departure
state $i$, and $\text{Pr}(n|\tau)$ is the probability that $n$ packets arrive
to the queue before the timer expires. The probability of finding the queue
full is calculated as $\pi^{K}=1-\sum^{K-1}_{k=0}\pi^{k}$.

##  Optimal block size

We now derive the optimal block size $b^{*}$ that minimizes the transaction
confirmation time in a blockchain system. To that purpose, we use the latency
model provided in Sect. LABEL:sec:e2e_latency together with the following
assumption. The waiting timeout is set to an arbitrarily large number so that
the number of mined transactions is always set to the block size, $b$.
Assumption 1 is necessary to find a convex function that is a close fit to
data obtained from the model. The fact is that the complex interactions at the
queue prevent finding a convex function that allows optimizing the queuing
delay. Furthermore, we approximate $\text{T}_{\text}{q}$ using the Lagrange
interpolation method [abramowitz1970handbook] so that the queuing delay is
approximated by an $n$-order polynomial from a set of $N+1$ data points
obtained from the queuing model. Assuming fixed $\mu$ and $\lambda$ parameters
(provided by the scenario), we approximate $T_{\text}{q}$ as:

$\text{T}_{\text}{q}(b)\approx\hat{\text{T}}_{\text}{q}(b)=\sum_{j=0}^{N}y_{j}\mathcal{L}_{N,j}(b),$
(6)

where $y_{j}$ corresponds to the queuing delays obtained from the model, and
$\mathcal{L}_{N,j}(b)$ are the Lagrange basis polynomials, defined as
$\prod_{k=0,k\neq j}^{N}\frac{b-b_{k}}{b_{j}-b_{k}}$. Being
$C\in\mathbb{R}^{+}$ a positive constant capturing the P2P end-to-end
capacity, the overall transaction confirmation delay, $\text{T}_{\text}{bp}$,
is approximated as follows:

$\hat{\text{T}}_{\text}{BC}=\frac{\hat{\text{T}}_{\text}{q}(b)+1/M\lambda+b/C}{e^{-\lambda(M-1)b/C}}$
(7)

Following Eq. eq:tbc, which is convex for $b\geq 0$, the optimal block size
$b^{*}$ can be derived directly.


# Supporting Information: Ab intio molecular dynamics of temporary anions
using complex absorbing potentials

Jerryman A. Gyamfi<EMAIL_ADDRESS>Thomas-C. Jagau
<EMAIL_ADDRESS>[

###### keywords:

complex absorption potentials, molecular dynamics, temporary anions, transient
anions, electronic resonance

KUL] Department of Chemistry, KU Leuven, Celestijnenlaan 200F, B-3001 Leuven,
Belgium CAP, AIMD, CAP-AIMD

##  1 Lost fraction formula derivation

Following the article, let $P_{n+1}$ be the survival probability of the
temporary anion between the ${n}-$th and $(n+1)$-th steps of the CAP-HF AIMD
simulation. Then,

$\begin{cases}P_{0}&=1\\\ P_{n+1}&=e^{-\Gamma_{n}\cdot\Delta t_{n+1}}\ ,\qquad
n\in\\{0,1,2\ldots\\}\ ,\end{cases}$ (1)

where $\Delta t_{n+1}\equiv t_{n+1}-t_{n}$, and
$\Gamma_{n}\equiv-2\cdot\imaginary E_{n}$. $E_{n}$ is the complex SCF energy
at the $n-$th AIMD step. $P_{0}$ defines the survival probability _at_ the
initial instance $t_{0}$, and is set to be unity. The probability that we
witness autodetachment between the $n-$th and $(n+1)-$th steps, after the
temporary anion had survived down to the $n-$th step, is:

$\begin{split}A_{n+1}&=\left[\prod^{n}_{i=0}P_{i}\right]\cdot(1-P_{n+1})\
\qquad n\in\\{0,1,2\ldots\\}\ .\end{split}$ (2)

We set $A_{0}=0$ since there is no autodetachment at the initial instant
$t_{0}$.

Consider now an ensemble $N$ of identical temporary anions who share the same
initial geometry and initial velocities. It is sufficient to run a single CAP-
AIMD simulation for this particular ensemble to obtain the data set
$\\{\Gamma_{n}\\}$, which informs us of the width of each member of the
ensemble along that given trajectory. Without loss of generality, we may take
$A_{n+1}$ to be equivalent to the fraction of the initial number $N$ of
temporary anions we lose between time steps $n$ and $n+1$. Thus, the sum

$f_{n}\equiv A_{0}+A_{1}+A_{2}+\ldots+A_{n}$ (3)

is the total fraction lost between time steps $0$ and $n$ (included). That is,
for $n\geq 1$,

$f_{n}=\sum^{n}_{m=1}A_{m}=\sum^{n}_{m=1}\left[\prod^{m-1}_{i=0}P_{i}\right]\cdot(1-P_{m})\
,$ (4)

which can be simplified to:

$f_{n}=1-\prod^{n}_{i=1}P_{i}=1-e^{-1/\hbar\
\sum^{n}_{i=1}\Gamma_{i-1}\cdot\Delta t_{i}}\ .$ (5)

There are a number of ways to prove this. We give below a simple proof by
induction. Let’s begin by writing explicitly the expressions for
$A_{1},A_{2}$:

$\displaystyle A_{1}$ $\displaystyle=(1-P_{1})$ (6a) $\displaystyle A_{2}$
$\displaystyle=P_{1}(1-P_{2})=P_{1}-P_{1}P_{2}$ (6b)

Note that $P_{1}=1-A_{1}$, and so we may rewrite $A_{2}$ as

$A_{2}=1-A_{1}-P_{1}P_{2}$ (7)

from which we derive that

$A_{1}+A_{2}=1-P_{1}P_{2}\ .$ (8)

If we now consider $A_{3}$, we have

$A_{3}=P_{1}P_{2}(1-P_{3})=P_{1}P_{2}-P_{1}P_{2}P_{3}\ .$ (9)

Since, from Eq. (8), $P_{1}P_{2}=1-A_{1}-A_{2}$, then we may also rewrite Eq.
(9) as,

$A_{3}=1-A_{1}-A_{2}-P_{1}P_{2}P_{3}\ ,$ (10)

from which we also derive that

$A_{1}+A_{2}+A_{3}=1-P_{1}P_{2}P_{3}\ .$ (11)

Following the same line of reasoning, we can see that, in general,

$\sum^{n}_{m=1}A_{m}=1-\prod^{n}_{m=1}P_{i}\ .$ (12)

##  2 Initial molecular structures (Cartesian coordinates in Angstroms)

    N2 R_N-N = 1.067 Angstroms:
        I     Atom           X                Y                Z
     ----------------------------------------------------------------
        1      N       0.0000000000     0.0000000000     0.0000000000
        2      N       0.0000000000     0.0000000000     1.0671420000
     ----------------------------------------------------------------

    N2 R_N-N = 1.400 Angstroms:
        I     Atom           X                Y                Z
     ----------------------------------------------------------------
        1      N       0.0000000000     0.0000000000     0.0000000000
        2      N       0.0000000000     0.0000000000     1.4000000000
     ----------------------------------------------------------------

    N2 R_N-N = 1.150 Angstroms:
        I     Atom           X                Y                Z
     ----------------------------------------------------------------
        1      N       0.0000000000     0.0000000000     0.0000000000
        2      N       0.0000000000     0.0000000000     1.1500000000
     ----------------------------------------------------------------

    Ethylene:
        I     Atom           X                Y                Z
     ----------------------------------------------------------------
        1      C       0.0000000000     0.0000000000     0.0000000000
        2      H       0.0000000000     0.0000000000     1.0742160000
        3      H       0.9595703355     0.0000000000    -0.4828713968
        4      C      -1.1191466626     0.0000000000    -0.6896850753
        5      H      -2.0787169982     0.0000000000    -0.2068136786
        6      H      -1.1191466626     0.0000000000    -1.7639010753
     ----------------------------------------------------------------

    Chloroethylene:
         I     Atom           X                Y                Z
     ----------------------------------------------------------------
        1      Cl      0.0000000000     0.0000000000     0.0000000000
        2      C       0.0000000000     0.0000000000     1.7316720000
        3      H       0.9853392606     0.0000000000     2.1483176034
        4      C      -1.0935317241    -0.0000000000     2.4493744736
        5      H      -2.0698140500    -0.0000000000     2.0064002052
        6      H      -1.0220874542    -0.0000000000     3.5199672631
     ----------------------------------------------------------------

    cis-dichloroethylene:
         I     Atom           X                Y                Z
     ----------------------------------------------------------------
        1      Cl      0.0000000000     0.0000000000     0.0000000000
        2      C       0.0000000000     0.0000000000     1.7158570000
        3      H       0.9754145626     0.0000000000     2.1546605495
        4      C      -1.0642674138    -0.0000000000     2.4775763725
        5      H      -0.9635098987    -0.0000000000     3.5423909325
        6      Cl     -2.6884303018    -0.0000000000     1.9241686045
     ----------------------------------------------------------------

    trans-dichloroethylene:
         I     Atom           X                Y                Z
     ----------------------------------------------------------------
        1      Cl      0.0000000000     0.0000000000     0.0000000000
        2      C       0.0000000000     0.0000000000     1.7251020000
        3      H       0.9725182492     0.0000000000     2.1682755046
        4      C      -1.1108641214    -0.0000000000     2.4124445514
        5      H      -2.0833803441    -0.0000000000     1.9692665997
        6      Cl     -1.1108732143     0.0000000000     4.1375455514
     ----------------------------------------------------------------

    1,1-dichloroethylene:
         I     Atom           X                Y                Z
     ----------------------------------------------------------------
        1      Cl      0.0000000000     0.0000000000     0.0000000000
        2      C       0.0000000000     0.0000000000     1.7223030000
        3      C       1.0985662562     0.0000000000     2.4310014204
        4      H       1.0546069317     0.0008291316     3.5008653643
        5      H       2.0551413131    -0.0005294646     1.9498419721
        6      Cl     -1.5693772271     0.0020344528     2.4319816909
     ----------------------------------------------------------------

    trichloroethylene:
         I     Atom           X                Y                Z
     ----------------------------------------------------------------
        1      Cl      0.0000000000     0.0000000000     0.0000000000
        2      C       0.0000000000     0.0000000000     1.7092420000
        3      C       1.0751798641     0.0000000000     2.4583761272
        4      H       0.9959360068     0.0004721400     3.5236907956
        5      Cl      2.6789106202    -0.0002059649     1.8573907787
        6      Cl     -1.5586905277     0.0010844173     2.4407910370
     ----------------------------------------------------------------

    tetrachloroethylene:
         I     Atom           X                Y                Z
     ----------------------------------------------------------------
        1      Cl      0.0000000000     0.0000000000     0.0000000000
        2      C       0.0000000000     0.0000000000     1.7227670000
        3      C       1.1127702200     0.0000000000     2.4402796195
        4      Cl      2.6820100527    -0.0002426958     1.7293487076
        5      Cl      1.1129142371     0.0003344598     4.1630555810
        6      Cl     -1.5693071217     0.0005641705     2.4335709971
     ----------------------------------------------------------------

    chloroethane:
         I     Atom           X                Y                Z
     ----------------------------------------------------------------
        1      Cl      0.0000000000     0.0000000000     0.0000000000
        2      C       0.0000000000     0.0000000000     1.7959090000
        3      H       1.0352683828     0.0000000000     2.0949364710
        4      H      -0.4531548755     0.9308229065     2.0949364710
        5      C      -0.7464826193    -1.1936557262     2.3509867800
        6      H      -0.2904814026    -2.1205141302     2.0296230487
        7      H      -1.7794323605    -1.1893612135     2.0296230487
        8      H      -0.7229945558    -1.1560973682     3.4350270623
     ----------------------------------------------------------------

##  3 CAP box parameters

As mentioned in the article, the CAP Hamiltonian, $H^{\eta}$, is given by 1,
2, 3, 4, 5

$H^{\eta}=H-i\eta W\ ,$ (13)

where $H$ is the physical Hamiltonian, $\eta$ is a real scalar parameter and
$W$ is a Hermitian operator. The potential $W$ was chosen to be quadratic of
the following form,

$W=\sum_{\alpha=x,y,z}\ W_{\alpha}\ ,\qquad
W_{\alpha}=\theta(\Delta_{\alpha}-r^{o}_{\alpha})\cdot(\Delta_{\alpha}-r^{o}_{\alpha})^{2}\
,\ \Delta_{\alpha}\equiv\absolutevalue{r_{\alpha}-o_{\alpha}}$ (14)

where $\theta(x)$ is a piecewise function defined as

$\theta(x)\equiv\begin{cases}1\ &\text{if }x>0\\\ 0\ &\text{if }x\leq
0\end{cases}\ .$ (15)

$r_{\alpha}$ is the electronic coordinate along axis $\alpha$. The vector
$(o_{x},o_{y},o_{z})$ is the origin of the CAP, which in our calculations is
chosen to coincide with the center of nuclear charge 6, that is

$o_{\alpha}=\frac{\sum_{k}R_{k,\alpha}Z_{k}}{\sum_{k}Z_{k}}\ ,$ (16)

where $Z_{k}$ and $R_{k}$ are the charge and coordinate of the $k-$th nucleus,
respectively. This choice of $o_{\alpha}$ allows the origin of the CAP to move
with the nuclear frame during the AIMD simulation.

Similar to single point energy CAP calculations, to run a CAP-AIMD simulation,
one needs to specify the values of the set $\\{r^{o}_{\alpha},\eta\\}$, which
parameterize the CAP. It has been advocated 7, 4, 6 to set $r^{o}_{\alpha}$
for temporary radical anions equal to the second moment of the neutral
molecule’s wave function along the axis $\alpha$, i.e.
$r^{o}_{\alpha}=\sqrt{\left<{r_{\alpha}^{2}}\right>}$. To minimize the
perturbation of the resonance wave function by the CAP, one runs an
$\eta-$trajectory to determine an optimal $\eta$, i.e. $\eta_{opt}$, which is
chosen such that 1

$\eta_{opt}=\min_{\eta}\absolutevalue{\eta\frac{dE}{d\eta}}\ .$ (17)

However, in our calculations, we resorted to a more complete approach, whereby
the CAP box dimensions, $r^{o}_{\alpha}$, are also optimized. The details of
the approach will be expounded on in an upcoming work. For now, it suffices to
say that the optimized CAP parameters, $\\{r^{o}_{\alpha,opt},\eta_{opt}\\}$,
were chosen according to criterion

$\\{r^{o}_{\alpha,opt},\eta_{opt}\\}=\min_{\\{r^{o}_{\alpha},\,\eta_{opt}\\}}\absolutevalue\Big{\imaginary\left<-i\eta
W\right>}\ .$ (18)

The optimized CAP parameters used in our CAP-AIMD simulations are summarized
in Tab. 1. The vertical attachment energies (VAE) computed using these
parameters are reported in Tab. 2 and compared with results from static CAP-
EOM-EA-CCSD calculations taken from Ref. 8. It is interesting to note that the
ratio between the CAP-EOM-EA-CCSD VAE’s and those from CAP-HF is always
between 0.6 and 0.7.

In Tab. 2, we also report values for $\absolutevalue{\real\left<-i\eta
W\right>/\,\text{VAE}}$ and $\absolutevalue{2\imaginary\left<-i\eta
W\right>/\,\Gamma}$ from our CAP-HF calculations. These ratios may be
interpreted as relative errors in the computed attachment energies and
resonance widths, respectively. The closer they are to zero, the better.

Table 1: CAP parameters optimized at the equilibrium geometry of the neutral
molecule, used in the CAP-AIMD simulations. Basis set is cc-pVTZ+3p for all
molecules except C2Cl4 where aug-cc-pVDZ+3p is used. The extra diffuse
functions are placed on all atoms except H.

Molecule |  $\eta_{opt}$_a_ |  $r^{o}_{x,opt}$_b_ | $r^{o}_{y,opt}$ | $r^{o}_{z,opt}$
---|---|---|---|---
N2 | 2490 | 3535 | 3535 | 8102
C2H5Cl | 650 | 3994 | 4148 | 4163
C2H4 | 230 | 4360 | 2680 | 4360
C2H3Cl | 460 | 8097 | 4490 | 16823
cis-C2H2Cl2 | 500 | 8300 | 6500 | 8300
trans-C2H2Cl2 | 500 | 6388 | 5355 | 32786
1,1-C2H2Cl2 | 10 | 23800 | 10856 | 25131
C2HCl3 | 500 | 7880 | 6200 | 7880
C2Cl4 | 2340 | 28080 | 7840 | 43774

_a_ $\eta$ values are in $\times 10^{-5}$ a.u.

_b_ $r^{o}$ values are in $\times 10^{-3}$ a.u. (bohr).

Table 2: Vertical Attachment Energies (VAE) and resonance widths ($\Gamma$) in
eV of dinitrogen, chloroethane, ethylene and the mono-, di-, tri- and tetra-
chloro-substituted ethylenes computed at the optimized geometry of the neutral
molecules with CAP-HF. Values computed with CAP-EOM-EA-CCSD and experimental
values are also shown.

Molecule | expt _a_ | CAP-EOM-EA-CCSD _b_ | CAP-HF
---|---|---|---
| VAE | VAE | $\Gamma$ | VAE | $\Gamma$ | $\absolutevalue{\frac{\real\left<-i\eta W\right>}{\text{VAE}}}$ | $\absolutevalue{\frac{2\imaginary\left<-i\eta W\right>}{\Gamma}}$
N2 | 2.3 | – | – | 3.396 | 0.362 | $1.91\cdot 10^{-2}$ | $4.08\cdot 10^{-4}$
C2H5Cl | 2.35 | – | – | 4.033 | 1.488 | $2.67\cdot 10^{-3}$ | $1.83\cdot 10^{-3}$
C2H4 | 1.73 | 2.155 | 0.446 | 2.908 | 0.661 | $2.30\cdot 10^{-2}$ | $1.27\cdot 10^{-3}$
C2H3Cl | 1.28 | 1.730 | 0.266 | 2.368 | 0.341 | $4.91\cdot 10^{-3}$ | $1.11\cdot 10^{-2}$
cis-C2H2Cl2 | 1.11 | 1.573 | 0.056 | 2.158 | 0.015 | $4.41\cdot 10^{-4}$ | $6.32\cdot 10^{-5}$
trans-C2H2Cl2 | 0.80 | 1.335 | 0.109 | 1.942 | 0.059 | $1.68\cdot 10^{-3}$ | $1.26\cdot 10^{-3}$
1,1-C2H2Cl2 | 0.76 | 1.285 | 0.245 | 1.858 | 0.535 | $5.37\cdot 10^{-6}$ | $4.29\cdot 10^{-4}$
C2HCl3 | 0.59 | 1.052 | 0.169 | 1.637 | 0.034 | $1.48\cdot 10^{-3}$ | $1.12\cdot 10^{-3}$
C2Cl4 | 0.3 | 0.982 | 0.026 | 1.286 | 0.007 | $8.51\cdot 10^{-4}$ | $8.23\cdot 10^{-2}$

_a_ From Ref. 9.

_b_ From Ref. 8.

##  4 Results

Below, we show scatter plots related to different specific trajectories in
different grades of gray. Average i) real part of the complex SCF energy, ii)
resonance width, and iii) lost fraction at each time step are shown in i) dark
orange, ii) red, and iii) blue dashed lines, respectively. To make some graphs
more intelligible, we plot the related data only for some few initial steps.
Unless otherwise stated, the initial geometry is always the optimized geometry
of the neutral molecule. All trajectories were run in the microcanonical
ensemble; the reported temperatures define the Maxwell-Boltzmann distribution
from which the initial velocities were sampled.

###  4.1 Dinitrogen, N2

The dinitrogen anion was simulated at three different initial geometries: i)
$R^{o}_{N-N}=1.067$ Å, ii) $R^{o}_{N-N}=1.150$ Å and iii) $R^{o}_{N-N}=1.400$
Å. The time step chosen for the simulation does not matter as long as the
condition $\Gamma_{0}\Delta t<1$ is respected, as explained in the article.

####  4.1.1 Dinitrogen, N2, $R^{o}_{N-N}=1.067$ Å

Time step = 10 a.u. (0.2419 fs); Number of steps = 1500; Temperature = 300 K;
Number of trajectories = 200; $\eta=0.02490$ a.u.

Figure 1: $\real E$ v time for dinitrogen anion. $R^{o}_{N-N}=1.067$ Å. An
offset of –108.8 is used for the y-axis. Figure 2: Resonance width, $\Gamma$,
v time for dinitrogen anion. $R^{o}_{N-N}=1.067$ Å Figure 3: Lost fraction v
time for dinitrogen anion. $R^{o}_{N-N}=1.067$Å. The equilibrium lost fraction
is 1.0 for all trajectories.

####  4.1.2 Dinitrogen, N2, $R^{o}_{N-N}=1.400$ Å

Time step = 2 a.u. (0.0484 fs); Number of steps = 2000; Temperature = 300 K;
Number of trajectories = 40; $\eta=0.00500$ a.u.

Figure 4: $\real E$ v time for dinitrogen anion. $R^{o}_{N-N}=1.400$Å. An
offset of –108.8 is used for the y-axis. Figure 5: Resonance width, $\Gamma$,
v time for dinitrogen anion. $R^{o}_{N-N}=1.400$ Å Figure 6: Lost fraction v
time for dinitrogen anion. $R^{o}_{N-N}=1.400$ Å. The equilibrium lost
fraction is 1.0 for all trajectories.

####  4.1.3 Dinitrogen, N2, $R^{o}_{N-N}=1.150$ Å.

Time step = 2 a.u. (0.0484 fs); Number of steps = 2000; Temperature = 300 K;
Number of trajectories = 51; $\eta=0.00070$ a.u.

We can see in these simulations that the $\Gamma$ time profile (Fig. 8) is not
uniform as in the other initial geometries of the same anion considered above;
in particular, it could increase or decrease in the beginning. This is related
to the high dependence of $\Gamma$ on the initial velocities at this initial
geometry, which is in the neighborhood of the anion’s equilibrium structure.
The plot of the real part of the complex SCF energy, Re$\ E$, against time
(Fig. 7) also shows a small difference between the maxima and minima of the
peaks compared to the other initial geometries considered above. Just like the
$\Gamma$ profile (Fig. 8) the lost fraction time profile also shows less
homogeneity compared to Figs. 3 and 6.

Figure 7: $\real E$ v time for dinitrogen anion. $R^{o}_{N-N}=1.150$ Å. An
offset of –108.8 is used for the y-axis. Figure 8: Resonance width, $\Gamma$,
v time for dinitrogen anion. $R^{o}_{N-N}=1.150$ Å. Figure 9: Lost fraction v
time for dinitrogen anion. $R^{o}_{N-N}=1.150$ Å. The equilibrium lost
fraction is 1.0 for all trajectories.

####  4.1.4 Dinitrogen, variation of $\Gamma$ with N-N bond length

In Fig. 2 of the article, we reported the real part of the CPES of N2-
obtained from CAP-AIMD simulation. We now report in Fig. 10 the imaginary part
of the same CPES.

Starting from the compressed N-N bond length of $R_{N-N}\sim 1.0$Å, we see
that $\imaginary E$ increases as $R_{N-N}$ increases. The upper limit of
$\imaginary E$ is zero, which is associated to bound N2-.

We observe in Fig. 10 a discontinuity in $\imaginary E$ as $\imaginary E\to
0$, which is around $R_{N-N}\approx 1.2$Å. The discontinuity is an indication
of the transition between bound and unbound N2-. It is also a consequence of
the limitations of the CAP-HF method.

Figure 10: The plot shows variation of the imaginary part of the CPES of N2-
with N-N bond length, $R_{N-N}$. The points derive from a CAP-AIMD simulation
performed at a time step of 2 a.u. $\approx$ 0.05 fs, for a total of 2000 time
steps.

###  4.2 Ethylene, C2H4

Time step = 5 a.u. (0.1209 fs); Number of steps = 1000; Temperature = 200 K;
Number of trajectories = 200; $\eta=0.00230$ a.u.

Looking at the actual trajectories (in shades of gray) in Fig. 12, we see that
$\Gamma$ becomes zero intermittently, indicating the extra electron has become
bound; the average $\Gamma$ (in red) on the other hand, never becomes zero.
This is due to the large variance among the trajectories in regards to the
$\Gamma$ profile. However, as we can see from Fig. 13, there is a great
homogeneity among the trajectories in regards to the lost fraction’s time
profile. This is due to the high homogeneity we see in the $\Gamma$ profiles
in the first 10fs.

Figure 11: $\real E$ v time for ethylene anion. Figure 12: Resonance width,
$\Gamma$, v time for ethylene anion.

Figure 13: Lost fraction v time for ethylene anion. The equilibrium lost
fraction is $1.0$ across all trajectories.

###  4.3 Chloroethylene, C2H3Cl

Time step = 10 a.u. (0.2419 fs); Number of steps = 1000; Temperature = 300 K;
Number of trajectories = 200; $\eta=0.00460$ a.u.

Statistics on equilibrium lost fraction: minimum= $0.7480$; maximum=$0.9973$;
mean =$0.9220$; median=$0.9305$; standard deviation = $0.0416$.

In all trajectories, we observe an initial stage where $\Gamma$ decreases as
the nuclei relax. In some trajectories, that stage is followed by a stage
where $\Gamma$ increases again due to the nuclear configurations reached,
before eventually becoming zero; this gives rise to the curve centered around
25 fs in Fig. 15.

Figure 14: $\real E$ v time for chloroethylene anion. Figure 15: Resonance
width, $\Gamma$, v time for chloroethylene anion. Figure 16: Lost fraction v
time for chloroethylene anion.

####  4.3.1 What happens when the wrong CAP parameters are used?

To illustrate that it is important to run CAP-AIMD simulations with the right
CAP parameters, we show below results for chloroethylene anion using the same
initial geometry and CAP box parameters used for the simulations above (and
listed in Tab. 1), but this time with $\eta=0.010$ a.u. We chose this value
for $\eta$ because the $\eta-$trajectory (with the same box dimensions as in
Tab. 1) indicates that the electronic state of the anion at this $\eta$ value
has pseudocontinuum character and is not a resonance.

Fig. 17 and 18 show disagreement between $\real E$, $\Gamma$ and their
respective deperturbed counterparts. In Fig. 20, we show as a counterexample,
the $\real E$ and $\real\widetilde{E}$ values for the trajectories discussed
in Fig. 3 of the article; in Fig. 21 we also show the $\real E$ and
$\real\widetilde{E}$ values for the trajectories discussed in Fig. 1 of the
article for N2-. The neat agreement between $\real E$, $\Gamma$ and their
respective deperturbed counterparts in Fig. 20 and 21 is in sharp contrast to
what we see in Fig. 17 and 18.

Figure 17: $\real E$ and $\real\widetilde{E}$ v time for chloroethylene anion
using $\eta=0.010$ a.u. An offset of –536.9 is used for the y-axis. Figure 18:
$\Gamma$ and $\widetilde{\Gamma}$ v time for chloroethylene anion using
$\eta=0.010$ a.u. Figure 19: Lost fraction $f$ and deperturbed $\widetilde{f}$
v time for chloroethylene anion using $\eta=0.010$ a.u. Figure 20: Lost
fraction, $f$, $\real E$ and resonance width, $\Gamma$, profiles and their
deperturbed counterparts, $\widetilde{f}$, $\real\widetilde{E}$ and
$\widetilde{\Gamma}$ for randomly chosen CAP-AIMD trajectories of A)
chloroethylene, B) cis-, C) trans- and D) 1,1-dichloroethylene anions,
simulated with the appropriate CAP parameters from Tab. 1. Figure 21: Lost
fraction, $f$, $\real E$ and resonance width, $\Gamma$, profiles (and their
deperturbed counterparts, $\widetilde{f}$, $\real\widetilde{E}$ and
$\widetilde{\Gamma}$) for randomly chosen CAP-AIMD trajectories of N2-,
simulated with the appropriate CAP parameters from Tab. 1.

###  4.4 cis-dichloroethylene, C2H2Cl2

Time step = 10 a.u. (0.2419 fs); Number of steps = 1000; Temperature = 350 K;
Number of trajectories = 199; $\eta=0.00500$ a.u.

Statistics on equilibrium lost fraction: minimum= $0.0606$; maximum=$0.1736$;
mean =$0.0995$; median=$0.0981$; standard deviation = $0.0168$.

Figure 22: $\real E$ v time for cis-dichloroethylene anion. Figure 23:
Resonance width, $\Gamma$, v time for cis-dichloroethylene anion.  Figure 24:
Lost fraction v time for cis-dichloroethylene anion.

###  4.5 trans-dichloroethylene, C2H2Cl2

Time step = 10 a.u. (0.2419 fs); Number of steps = 1000; Temperature = 350 K;
Number of trajectories = 200; $\eta=0.00500$ a.u.

Statistics on equilibrium lost fraction: minimum= $0.2089$; maximum=$0.4473$;
mean =$0.3264$; median=$0.3294$; standard deviation = $0.0503$.

Figure 25: $\real E$ v time for trans-dichloroethylene anion. Figure 26:
Resonance width, $\Gamma$, v time for trans-dichloroethylene anion. Figure 27:
Lost fraction v time for trans-dichloroethylene anion.

###  4.6 1,1-dichloroethylene, C2H2Cl2

Time step = 10 a.u. (0.2419 fs); Number of steps = 1000; Temperature = 350 K;
Number of trajectories = 194; $\eta=0.00010$ a.u.

Statistics on equilibrium lost fraction: minimum= $0.7911$; maximum=$0.9990$;
mean =$0.9338$; median=$0.9383$; standard deviation = $0.0482$.

Figure 28: $\real E$ v time for cis-dichloroethylene anion. Figure 29:
Resonance width, $\Gamma$, v time for cis-dichloroethylene anion. Figure 30:
Lost fraction v time for cis-dichloroethylene anion.

###  4.7 On the out-of-plane motion of the dichloroethylene anion isomers

The top row graphs in Fig. 31 report the dihedral angle between the plane
containing the two C atoms and: i) the dissociating Cl, and ii) the other
non-C atom geminal to the dissociating Cl atom. The normal vectors of these
two planes are defined such that they are anti-parallel to each other at
$t=0$. We observe from the dihedral plots that right from the onset of their
time evolutions these ethylene derivative TAs go through an inexorable
decrease in the dihedral angle $\phi$ in question, accompanied by the R-Cl
bond elongation, where Cl refers to the dissociating chlorine. In the vast
majority of our simulations, we see the R-Cl bond contracts just a little bit
in the beginning before elongating as $\phi$ decreases. With the exception of
1,1-dichloroethylene (column D of Fig. 31), we also see a wiggly but steep
drop of $\phi$ towards zero once $90^{\circ}\lesssim\phi\lesssim 120^{\circ}$.
When $90^{\circ}\lesssim\phi\lesssim 120^{\circ}$, the R structure is often
almost perpendicular to the R-Cl bond; this is usually when we begin to see a
sharp increase in the R-Cl bond length.

Overall, as shown in the same plots, the sharp decrease in $\phi$ in the first
$90-120$ fs follows more or less the same trend in time as $\real E$. The
Mulliken charge on the dissociating Cl also decreases rapidly (see second row
in Fig. 31) in the same interval of time that $\phi$ drops sharply.
Interestingly, the time step at which this charge begins to stabilize is the
same point at which we see a stabilization in $\real E$.

Figure 31: Dihedral angle, Mulliken charge (on dissociating Cl) and R-Cl bond
length profiles for the trajectories shown in Fig. 3 of article (and Fig. 20
above).

###  4.8 Trichloroethylene, C2HCl3

We ran 24 trajectories, the dissociative electron attachment (DEA) process led
to the formation of trans-dichloroethylene radical + Cl- in 21 trajectories,
1,1-dichloroethylene \+ Cl- in 2 trajectories, and cis-dichloroethylene + Cl-
in 1 trajectory. We show below results for the first two channels. Results for
the cis-dichloroethylene + Cl- channel not shown below are very similar to the
first two.

####  4.8.1 trans-dichloroethylene channel

Time step = 20 a.u. (0.4838 fs); Number of steps = 1000; Temperature = 370 K;
Number of trajectories = 21; $\eta=0.00500$ a.u.

Statistics on equilibrium lost fraction: minimum= $0.0861$; maximum=$0.2229$;
mean =$0.1148$; median=$0.1037$; standard deviation = $0.0389$.

Figure 32: $\real E$ v time for trichloroethylene anion. An offset of –1454.0
is used for the y-axis. Figure 33: Resonance width, $\Gamma$, v time for
trichloroethylene anion. Figure 34: Lost fraction v time for trichloroethylene
anion.

####  4.8.2 1,1-dichloroethylene channel

Time step = 20 a.u. (0.4838 fs); Number of steps = 1000; Temperature = 370 K;
Number of trajectories = 2; $\eta=0.00500$ a.u.

Figure 35: $\real E$ v time for trichloroethylene. An offset of –1454.0 is
used for the y-axis. Figure 36: Resonance width, $\Gamma$, v time for
trichloroethylene anion. Figure 37: Lost fraction v time for trichloroethylene
anion.

###  4.9 Tetrachloroethylene, C2Cl4

Time step = 20 a.u. (0.4838 fs); Number of steps = 1000; Temperature = 410 K;
Number of trajectories = 21; $\eta=0.02340$ a.u.

Statistics on equilibrium lost fraction: minimum= $0.0052$; maximum=$0.0053$;
mean =$0.0053$; median=$0.0053$; standard deviation = $3.71\cdot 10^{-5}$.

The $\Gamma$ profile here, Fig. 39, shows a very small width for the anion.
Also, the width becomes zero almost immediately, i.e., after 1 fs. This is the
reason why this particular DEA can be simulated without CAP. This observation
is also in agreement with the very low lost fraction that we get from our
simulations, see Fig. 40.

Figure 38: $\real E$ v time for tetrachloroethylene anion. An offset of
–1913.0 is used for the y-axis. Figure 39: Resonance width, $\Gamma$, v time
for tetrachloroethylene anion. Figure 40: Lost fraction v time for
tetrachloroethylene anion.

###  4.10 Chloroethane, C2H5Cl

Time step = 5 a.u. (0.1209 fs); Number of steps = 1200; Temperature = 260 K;
Number of trajectories = 67.

The resonance width for chloroethane anion is higher compared to the
chlorinated ethylene anions discussed above because we are dealing with a
$\sigma^{*}$ resonance here. We observe a very sharp rise of the lost fraction
in the beginning. The average equilibrium lost fraction,
$\left<f_{\infty}\right>$, is practically unity, indicating that the DEA
process for chloroethane anion is very inefficient. This is supported by
experimental results 10.

Figure 41: $\real E$ v time for chloroethane anion. Figure 42: Resonance
width, $\Gamma$, v time for chloroethane anion. Figure 43: Lost fraction v
time for chloroethane anion. The equilibrium lost fraction is $\sim 1$ for all
trajectories.

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