u^{\varepsilon})^{\frac{1}{p_{1}}}\,|f^{\varepsilon}|\,\eta\,|\nabla\eta|\,dx,\end{split}$

for some $C=C(N,p_{1},p_{N})>0$. Since

$1\leq 1+2\,\left(\frac{1}{p_{1}}-\frac{1}{p_{N}}\right)\qquad\mbox{ and
}\qquad\mathcal{G}_{\varepsilon}(\nabla u^{\varepsilon})\geq 1,$

the third term can be absorbed in the first one, up to increasing $C$ if
necessary:

(6.6) $\begin{split}\sum_{i=1}^{N}&\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,|\nabla
u^{\varepsilon}_{x_{i}}|^{2}\,\eta^{2}\,dx\\\ &\leq
C\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{1+2\,\left(\frac{1}{p_{1}}-\frac{1}{p_{N}}\right)}\,\Big{(}|\nabla\eta|^{2}+\eta\,|D^{2}\eta|\Big{)}\,dx+C\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{2-p_{1}}{p_{1}}}\,|f^{\varepsilon}|^{2}\,\eta^{2}\,dx\\\
\ &+C\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{1}{p_{1}}}\,|f^{\varepsilon}|\,\eta\,|\nabla\eta|\,dx.\end{split}$

In the last term, we write

$\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{1}{p_{1}}}\,|f^{\varepsilon}|\,\eta\,|\nabla\eta|=\left(\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{1}{2}}|\nabla\eta|\right)\left(\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{1}{p_{1}}-\frac{1}{2}}\,|f^{\varepsilon}|\,\eta\right),$

and use Young’s inequality:

$\displaystyle\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{1}{p_{1}}}$
$\displaystyle\,|f^{\varepsilon}|\,\eta\,|\nabla\eta|\,dx\leq\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\,|\nabla\eta|^{2}\,dx+\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{2}{p_{1}}-1}\,|f^{\varepsilon}|^{2}\,\eta^{2}\,dx$
$\displaystyle\leq\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{1+2\,\left(\frac{1}{p_{1}}-\frac{1}{p_{N}}\right)}\,\Big{(}|\nabla\eta|^{2}+\eta\,|D^{2}\eta|\Big{)}\,dx+\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{2}{p_{1}}-1}\,|f^{\varepsilon}|^{2}\,\eta^{2}\,dx,$

where in the last line, we have used again that $\mathcal{G}_{\varepsilon}\geq
1$. Inserting this estimate in (6.6), we obtain

$\begin{split}\sum_{i=1}^{N}&\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,|\nabla
u^{\varepsilon}_{x_{i}}|^{2}\,\eta^{2}\,dx\\\ &\leq
C\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{1+2\,\left(\frac{1}{p_{1}}-\frac{1}{p_{N}}\right)}\,\Big{(}|\nabla\eta|^{2}+\eta\,|D^{2}\eta|\Big{)}\,dx+C\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{2-p_{1}}{p_{1}}}\,|f^{\varepsilon}|^{2}\,\eta^{2}\,dx.\end{split}$

On the left-hand side, we use the definition (6.1) of
$\mathcal{V}_{i,\varepsilon}$ which gives that

$|\nabla\mathcal{V}_{i,\varepsilon}|^{2}=g^{\prime\prime}_{i,\varepsilon}(u^{\varepsilon}_{x_{i}})\,|\nabla
u_{x_{i}}^{\varepsilon}|^{2}.$

This yields the the desired conclusion when the exponent $\gamma$ in the
statement of Proposition is equal to $2$. When $\gamma>2$, we only need to
apply the Hölder inequality to the last term of the right-hand side with the
exponents $\gamma/(\gamma-2)$ and $\gamma/2$. The proof is complete. ∎

##  7\. Proofs of the main results

We finally establish the three results presented in the Introduction by
relying on the relevant a priori estimates that we have obtained in the
previous sections. We thus fix a ball $B_{4R}(x_{0})\Subset\Omega$ as in the
statements of Proposition 1.1, Theorem L and Theorem S: we are going to use
the results of the previous sections, with the choice $B=B_{2\,R}(x_{0})$.

We will use the functions $\mathcal{G}_{0}$ and $\mathcal{G}_{\varepsilon}$,
defined by (1.1) and (3.1). Moreover, we will omit to indicate the centers of
the balls, which will always be $x_{0}$.

###  7.1. Proof of Proposition 1.1

In this section, we assume that $1<p_{1}\leq\dots\leq p_{N}<\infty$ and $f\in
L^{\gamma}_{\rm loc}(\Omega)$ for some $\gamma\geq 2$. We consider the ball
$B_{R}\Subset B$. Then for every $0<\varepsilon<\varepsilon_{0}$,

$\|f^{\varepsilon}\|_{L^{\gamma}(B_{R})}\leq\|f\|_{L^{\gamma}(2\,B)}.$

Using also that $\|u^{\varepsilon}\|_{L^{\infty}(B)}\leq M+1$, Proposition 4.1
and Remark 4.2 imply that for every $0<\varepsilon<\varepsilon_{0}$ we have

(7.1) $\int_{B_{\frac{R}{2}}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\gamma}\,dx\leq\Gamma_{1}+\Gamma_{2}\,\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\,dx,$

for two constants $\Gamma_{1},\Gamma_{2}>0$ which do not depend on
$\varepsilon$, but only on

$N,\,p_{N},\,p_{1},\,\gamma,\,R,\,\|f\|_{L^{\gamma}(2\,B)}\mbox{ and
}M=\|U\|_{L^{\infty}(2\,B)}.$

In particular, by using that555The upper bound simply follows from (2.16),
with standard algebraic manipulations.

(7.2) $\mathcal{G}_{0}(z)\leq\mathcal{G}_{\varepsilon}(z)\leq
C\,\left(\varepsilon^{\frac{p_{1}}{2}}+\mathcal{G}_{0}(z)\right),\qquad\mbox{
for every }z\in\mathbb{R}^{N},$

for some $C=C(N,p_{N},p_{1})>0$, we can infer

$\int_{B_{\frac{R}{2}}}\mathcal{G}_{0}(\nabla
u^{\varepsilon})^{\gamma}\,dx\leq\Gamma_{1}+C\,\Gamma_{2}\,\int_{B_{R}}\left(\varepsilon^{\frac{p_{1}}{2}}+\mathcal{G}_{0}(\nabla
u^{\varepsilon})\right)\,dx$

In view of Lemma 2.6, there exists an infinitesimal sequence
$\\{\varepsilon_{k}\\}_{k\in\mathbb{N}}$ such that

$(u^{\varepsilon_{k}},\nabla u^{\varepsilon_{k}})\ \textrm{ converges to }\
(U,\nabla U)\qquad\textrm{ a.\,e. in }B.$

We then take the limit on both sides of the estimate above and use Fatou’s
lemma on the left. We get

(7.3) $\int_{B_{\frac{R}{2}}}\mathcal{G}_{0}(\nabla
U)^{\gamma}\,dx\leq\Gamma_{1}+C\,\Gamma_{2}\,\liminf_{k\to\infty}\int_{B_{R}}\mathcal{G}_{0}(\nabla
u^{\varepsilon_{k}})\,dx.$

By Lemma 2.6, the functions $u_{x_{i}}^{\varepsilon}$ converge to $U_{x_{i}}$
in $L^{p_{i}}(B)$. Hence, the continuity of the map $v\in
L^{p_{i}}(B)\mapsto|v|^{p_{i}}\in L^{1}(B)$ implies that

(7.4) $\lim_{\varepsilon\to 0}\left\|\mathcal{G}_{0}(\nabla
u^{\varepsilon})-\mathcal{G}_{0}(\nabla U)\right\|_{L^{1}(B)}=0.$

By using this result in (7.3), we obtain

$\int_{B_{\frac{R}{2}}}\mathcal{G}_{0}(\nabla
U)^{\gamma}\,dx\leq\Gamma_{1}+C\,\Gamma_{2}\,\int_{B_{R}}\mathcal{G}_{0}(\nabla
U)\,dx.$

This concludes the proof, up to rename the constant $\Gamma_{2}$.

###  7.2. Proof of Theorem L

In this section, we assume that $1<p_{1}\leq\dots\leq p_{N}\leq 2$ and $f\in
L^{\gamma}_{\rm loc}(\Omega)$ for some $\gamma>N$. In particular, $\gamma\geq
2$ and thus we can rely on Proposition 4.1.

We introduce the ball $B_{R}\Subset B$ as before. By Proposition 5.1 applied
with $B_{R/4}$ and $B_{R/2}$, for every $\varepsilon\in(0,\varepsilon_{0})$,
we have

$\begin{split}\|\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\|_{L^{\infty}(B_{\frac{R}{4}})}&\leq
C\,\left[\left(\frac{4}{R}\right)^{\frac{N\,\gamma}{\gamma-N}}\,\left(\int_{B_{\frac{R}{2}}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\gamma}\,dx\right)^{\frac{N}{\gamma-N}}+\|f^{\varepsilon}\|_{L^{\gamma}(B_{R})}^{\frac{N\,\gamma}{\gamma-N}}\right]\,\left\|\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\right\|_{L^{1}(B_{R})},\end{split}$

for some $C=C(N,p_{N},p_{1},\gamma)>0$. On the right-hand side, we can apply
(7.1), in order to estimate the term containing
$\mathcal{G}_{\varepsilon}^{\gamma}$. This yields

$\begin{split}\|\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\|_{L^{\infty}(B_{\frac{R}{4}})}&\leq
C\left[\left(\frac{4}{R}\right)^{\frac{N\,\gamma}{\gamma-N}}\\!\\!\left(\Gamma_{1}+\Gamma_{2}\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\,dx\right)^{\frac{N}{\gamma-N}}\\!\\!+\|f^{\varepsilon}\|_{L^{\gamma}(B_{R})}^{\frac{N\,\gamma}{\gamma-N}}\right]\left\|\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\right\|_{L^{1}(B_{R})}.\end{split}$

We now take the same infinitesimal sequence
$\\{\varepsilon_{k}\\}_{k\in\mathbb{N}}$ as in the proof of Proposition 1.1.
By using again (7.2), the lower semicontinuity of the $L^{\infty}$ norm with
respect to almost everywhere convergence, equation (7.4) and the fact that
$f^{\varepsilon_{k}}$ is defined from $f$ by convolution with a smooth kernel,
the limit as $k$ goes to $\infty$ gives

$\begin{split}\left\|\mathcal{G}_{0}(\nabla
U)\right\|_{L^{\infty}(B_{\frac{R}{4}})}&\leq
C\,\left[\left(\frac{4}{R}\right)^{\frac{N\,\gamma}{\gamma-N}}\,\left(\Gamma_{1}+\Gamma_{2}\,\int_{B_{R}}\mathcal{G}_{0}(\nabla
U)\,dx\right)^{\frac{N}{\gamma-N}}+\|f\|_{L^{\gamma}(B_{R})}^{\frac{N\,\gamma}{\gamma-N}}\right]\,\left\|\mathcal{G}_{0}(\nabla
U)\right\|_{L^{1}(B_{R})},\end{split}$

possibly for a different $C=C(N,p_{N},p_{1},\gamma)>0$. This completes the
proof.

###  7.3. Proof of Theorem S

We assume that $1<p_{1}\leq\dots\leq p_{N}\leq 2$ and $f$ verifying (1.3). We
set for notational simplicity

$\gamma=1+\frac{2}{p_{1}}.$

Consider the ball $B_{R}\Subset B$ and let $\eta\in C^{\infty}_{0}(B_{R/2})$
be such that

$\eta\equiv 1\mbox{ on }B_{\frac{R}{4}},\qquad 0\leq\eta\leq 1\qquad\mbox{ and
}\qquad|\nabla\eta|^{2}+|D^{2}\eta|\leq\frac{C_{0}}{R^{2}},$

for some $C_{0}$ which depends only on $N$. The choice of $\gamma$ entails the
following estimates

$1+2\,\left(\frac{1}{p_{1}}-\frac{1}{p_{N}}\right)<\gamma\qquad\mbox{ and
}\qquad\frac{2-p_{1}}{p_{1}}\frac{\gamma}{\gamma-2}\leq\gamma.$

Then, as a consequence of (6.2), we have

$\begin{split}\sum_{i=1}^{N}\int_{B_{\frac{R}{4}}}\Big{|}\nabla\mathcal{V}_{i,\varepsilon}\Big{|}^{2}\,dx&\leq\frac{C}{R^{2}}\,\int_{B_{\frac{R}{2}}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\gamma}\,dx+C\left(\int_{B_{\frac{R}{2}}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\gamma}\,dx\right)^{\frac{\gamma-2}{\gamma}}\,\|f^{\varepsilon}\|_{L^{\gamma}(B_{R})}^{2},\end{split}$

where we also used that $\mathcal{G}_{\varepsilon}\geq 1$, by definition. We
now rely again on (7.1), to estimate the terms containing
$\mathcal{G}_{\varepsilon}^{\gamma}$. This estimate and Young’s inequality
with exponents $\gamma/2$ and $\gamma/(\gamma-2)$ give

(7.5)
$\sum_{i=1}^{N}\int_{B_{\frac{R}{4}}}\Big{|}\nabla\mathcal{V}_{i,\varepsilon}\Big{|}^{2}\,dx\leq\frac{C}{R^{2}}\,\left(\Gamma_{1}+\Gamma_{2}\,\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\,dx\right)+C\,R^{\gamma-2}\,\|f^{\varepsilon}\|_{L^{\gamma}(B_{R})}^{\gamma},$

possibly for a different constant $C=C(N,p_{N},p_{1})>0$. From this estimate,
we deduce that the family $\nabla\mathcal{V}_{i,\varepsilon}$ is uniformly
bounded in $L^{2}(B_{R/4})$. Moreover, by (2.2) we have

(7.6)
$\sqrt{g_{i,\varepsilon}^{\prime\prime}(t)}\leq(\varepsilon+t^{2})^{\frac{p_{i}-2}{4}}\leq|t|^{\frac{p_{i}-2}{2}},\qquad\mbox{
for }t\not=0.$

Thus, by recalling the definition of $\mathcal{V}_{i,\varepsilon}$, we get

$\begin{split}\int_{B_{\frac{R}{4}}}|\mathcal{V}_{i,\varepsilon}|^{2}\,dx=\int_{B_{\frac{R}{4}}}|V_{i,\varepsilon}(u^{\varepsilon}_{x_{i}})|^{2}\,dx&\leq\left(\frac{2}{p_{i}}\right)^{2}\,\int_{B_{\frac{R}{4}}}|u^{\varepsilon}_{x_{i}}|^{p_{i}}\,dx,\end{split}$

and the latter is uniformly bounded, thanks to Lemma 2.5.

Thus, by taking the same infinitesimal sequence $\\{\varepsilon_{k}\\}_{k\geq
1}$ as in the proof of Proposition 1.1, we have obtained that
$\\{\mathcal{V}_{i,\varepsilon_{k}}\\}_{k\in\mathbb{N}}$ is a bounded sequence
in $W^{1,2}(B_{R/4})$. By appealing to the Rellich-Kondrašov Theorem, we can
infer its convergence to a function $\mathcal{V}_{i}\in W^{1,2}(B_{R/4})$,
weakly in $W^{1,2}(B_{R/4})$ and strongly in $L^{2}(B_{R/4})$ (up to a
subsequence). By the weak lower semicontinuity of the $L^{2}$ norm, (7.2) and
(7.4), we obtain from (7.5) that

$\sum_{i=1}^{N}\int_{B_{\frac{R}{4}}}\Big{|}\nabla\mathcal{V}_{i}\Big{|}^{2}\,dx\leq\frac{C}{R^{2}}\,\left(\Gamma_{1}+\Gamma_{2}\,\int_{B_{R}}\mathcal{G}_{0}(\nabla
U)\,dx\right)+C\,R^{\gamma-2}\,\|f\|_{L^{\gamma}(B_{R})}^{\gamma},$

possibly for a different $C=C(N,p_{N},p_{1})>0$. We claim that for every
$1\leq i\leq N$ and for almost every $x\in B_{R/4}$, we have

(7.7)
$\mathcal{V}_{i}(x)=\frac{2}{p_{i}}\,\sqrt{p_{i}-1}\,|U_{x_{i}}(x)|^{(p_{i}-2)/2}\,U_{x_{i}}(x).$

Indeed, let us take $x\in B_{R/4}$ such that $u_{x_{i}}^{\varepsilon_{k}}$
converges to $U_{x_{i}}$ and such that $|U_{x_{i}}(x)|<+\infty$. Observe that
the collection of these points has full measure in $B_{R/4}$. We then set

$M_{i}(x)=\sup_{k\in\mathbb{N}}|u_{x_{i}}^{\varepsilon_{k}}(x)|,$

which is finite, by construction. Then, for every $k\geq 0$ we have

$\begin{split}\left|\mathcal{V}_{i,\varepsilon_{k}}(x)-\frac{2}{p_{i}}\,\sqrt{p_{i}-1}\,|U_{x_{i}}(x)|^{\frac{p_{i}-2}{2}}\,U_{x_{i}}(x)\right|&=\left|V_{i,\varepsilon_{k}}(u^{\varepsilon_{k}}_{x_{i}}(x))-\frac{2}{p_{i}}\,\sqrt{p_{i}-1}\,|U_{x_{i}}(x)|^{\frac{p_{i}-2}{2}}\,U_{x_{i}}(x)\right|\\\
&\leq\left|\int_{0}^{u_{x_{i}}^{\varepsilon_{k}}(x)}\left(\sqrt{g_{i,\varepsilon_{k}}^{\prime\prime}(\tau)}-\sqrt{(p_{i}-1)|\tau|^{p_{i}-2}}\right)\,d\tau\right|\\\
&+\frac{2}{p_{i}}\,\sqrt{p_{i}-1}\,\left||u_{x_{i}}^{\varepsilon_{k}}(x)|^{\frac{p_{i}-2}{2}}\,u_{x_{i}}^{\varepsilon_{k}}(x)-|U_{x_{i}}(x)|^{\frac{p_{i}-2}{2}}\,U_{x_{i}}(x)\right|\\\
&\leq\int_{0}^{M_{i}(x)}\left|\sqrt{g_{i,\varepsilon_{k}}^{\prime\prime}(\tau)}-\sqrt{(p_{i}-1)|\tau|^{p_{i}-2}}\right|\,d\tau\\\
&+\frac{2}{p_{i}}\,\sqrt{p_{i}-1}\,\left||u_{x_{i}}^{\varepsilon_{k}}(x)|^{\frac{p_{i}-2}{2}}\,u_{x_{i}}^{\varepsilon_{k}}(x)-|U_{x_{i}}(x)|^{\frac{p_{i}-2}{2}}\,U_{x_{i}}(x)\right|.\end{split}$

Thanks to (7.6) one can apply the dominated convergence to conclude that the
first term in the right-hand side converges to $0$ when $k$ goes to $+\infty$.
By also using that $u^{\varepsilon_{k}}_{x_{i}}(x)$ converges to
$U_{x_{i}}(x)$, we finally get (7.7).

By using the Chain Rule in Sobolev spaces, we also obtain that $U_{x_{i}}\in
W^{1,p_{i}}(B_{R/4})$ and satisfies the estimate claimed in the statement of
Theorem S. The proof is now over.

##  Appendix A A weak maximum principle

Let $G:\mathbb{R}^{N}\to[0,+\infty)$ be a strictly convex function such that
$G(0)=0$. Let $\zeta:\mathbb{R}\to\mathbb{R}$ be a Lipschitz function, with
the following property: there exists $M>0$ such that

$\zeta(t)=\left\\{\begin{array}[]{cc}M,&\mbox{ if }t\geq M,\\\ -M,&\mbox{ if
}t\leq-M.\end{array}\right.$

Given a ball $B\subset\mathbb{R}^{N}$, $f\in L^{1}(B)$ and $U\in
W^{1,1}(B)\cap L^{\infty}(B)$ such that

$\|U\|_{L^{\infty}(B)}\leq M\qquad\mbox{ and }\qquad\int_{B}G(\nabla
U)\,dx<+\infty,$

we consider the functional

$\mathcal{F}(v)=\int_{B}\Big{[}G(\nabla v)+f\,\zeta(v)\Big{]}\,dx,\qquad\mbox{
for every }v\in U+W^{1,1}_{0}(B).$

######  Lemma A.1.

If $u$ is a minimum of $\mathcal{F}$, then $\|u\|_{L^{\infty}(B)}\leq M$.

###### Proof.

We want to test the minimality of $u$ against the truncated function

$v:=\max\\{-M,\min\\{u,M\\}\\}.$

By construction, we still have $v\in U+W^{1,1}_{0}(B)$ and by minimality of
$u$, we get

$\mathcal{F}(u)\leq\mathcal{F}(v).$

By the properties of $\zeta$ and the fact that $G(0)=0$, we have

$\mathcal{F}(v)=\int_{\\{|u|\leq M\\}}\Big{[}G(\nabla
u)+f\,\zeta(u)\Big{]}\,dx+\int_{\\{u>M\\}}f\,\zeta(M)\,dx+\int_{\\{u<-M\\}}f\,\zeta(-M)\,dx.$

Hence, using that $\zeta(u)=\zeta(M)$ when $u\geq M$ and $\zeta(u)=\zeta(-M)$
when $u\leq-M$, by comparing the last two displays we get

$\int_{\\{|u|>M\\}}G(\nabla u)\,dx=0.$

Since $G\geq 0$ and $G(\xi)=0$ if and only if $\xi=0$, we deduce that $\nabla
u=0$ almost everywhere on the set $\\{|u|>M\\}$. It follows that $\nabla
u=\nabla v$ almost everywhere and since $u=v=U$ on $\partial B$, this implies
that $u=v$ almost everywhere in $B$. In particular, $|u|\leq M$ almost
everywhere in $B$. ∎

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# Hydrogen atom/molecule adsorption on 2D metallic porphyrin: A first-
principles study

Raphael M. Tromer  [  Isaac M. Felix  [  Levi C. Felix  [  Leonardo D. Machado
[  Cristiano F. Woellner  [  Douglas S. Galvao<EMAIL_ADDRESS>[

###### Abstract

Hydrogen is a promising element for applications in new energy sources like
fuel cells. One key issue for such applications is storing hydrogen. And, to
improve storage capacity, understanding the interaction mechanism between
hydrogen and possible storage materials is critical. This work uses DFT
simulations to comprehensively investigate the adsorption mechanism of H/H2 on
the 2D metallic porphyrins with one transition metal in its center. Our
results suggest that the mechanism for adsorption of H (H2) is chemisorption
(physisorption). The maximum adsorption energy for atomic hydrogen was $-3.7$
eV for 2D porphyrins embedded with vanadium or chromium atoms. Our results
also revealed charge transfer of up $-0.43$ e to chemisorbed H atoms. In
contrast, the maximum adsorption energy calculated for molecular hydrogen was
$-122.5$ meV for 2D porphyrins embedded with scandium atoms. Furthermore,
charge transfer was minimal for physisorption. Finally, we also determined
that uniaxial strain has a minimal effect on the adsorption properties of 2D
metallic porphyrins.

###### keywords:

Hydrogen storage, MOF’S, 2D porphyrin like graphene

State University of Campinas] Applied Physics Department, State University of
Campinas, Campinas, SP, 13083-970, Brazil [State University of Campinas]
Center for Computational Engineering and Sciences, State University of
Campinas, Campinas, SP, 13083-970, Brazil Federal University of Natal]
Departamento de Física Teórica e Experimental, Universidade Federal do Rio
Grande do Norte, Natal, RN, 59072-970, Brazil. State University of Campinas]
Applied Physics Department, State University of Campinas, Campinas, SP,
13083-970, Brazil [State University of Campinas] Center for Computational
Engineering and Sciences, State University of Campinas, Campinas, SP,
13083-970, Brazil Federal University of Natal] Departamento de Física Teórica
e Experimental, Universidade Federal do Rio Grande do Norte, Natal, RN,
59072-970, Brazil. Federal University of Parana] Physics Department, Federal
University of Paraná, UFPR, Curitiba, PR, 81531-980, Brazil State University
of Campinas] Applied Physics Department, State University of Campinas,
Campinas, SP, 13083-970, Brazil

##  1 Introduction

The use of nanostructured systems in applications is predicated on obtaining
new 2D materials and then understanding and manipulating their electrical1,
thermal2, 3, 4, 5, magnetic properties6, among others.

Since Andre Geim and Konstantin Novoselov extracted a graphene layer from
graphite by a simple exfoliation process 7, many methods have been proposed to
allow the synthesis of new 2D 8, 9, 10, 11, 12, 13 nanostructured materials.
Among these materials, there is a preference for systems that are organic and
that do not cause pollution when discarded 14, 15. In this quest for future
materials, one common objective is to find solids that support the use of
alternative energy sources, which aim to replace fossil fuels 16, 17. One such
alternative fuel is hydrogen, and intense research has been carried out to
investigate nanostructured systems that could serve as hosts for its storage.
Many materials have been proposed for hydrogen storage applications, such as
covalent organic frameworks (COFs) 18, 19 and metal-organic frameworks (MOFs)
20, 21. Still, to improve hydrogen storage cells, understanding the
interaction of nanostructured systems with hydrogen is vitally important.

In addition to applications in hydrogen storage 22, 23, 24, 25, MOFs have also
been used as catalysts in hydrogen evolution reactions 26, 27, 28, 29. A type
of system commonly used for the latter purpose are MOFs constructed from
porphyrin molecules 26, 27, 28. To assemble these systems, various
experimental techniques have been used to link the porphyrin molecules through
covalent bonds 26, 27, 28. Other porphyrin systems have been investigated
recently, including two-dimensional (2D) porphyrins that contain metal atoms
30, 31. Still, this type of 2D system has yet to be investigated for possible
uses in hydrogen storage applications.

In this work, we investigate the interaction between 2D porphyrin metallic
systems and hydrogen atoms and molecules. The 2D porphyrin systems were
assembled from a unit cell containing a porphyrin molecule with one transition
metal in the center. We considered 2D porphyrin systems with ten different
transition metals embedded in the center, each corresponding to one of the ten
elements of period 4 of the periodic table. For each 2D metallic porphyrin
system, we calculated the adsorption energy with H/H2, for both relaxed and
strained systems. We verified that the interaction between 2D-porphyrin and
hydrogen atoms (molecules) corresponds to chemisorption (physisorption).

##  2 Methodology

As mentioned above, we investigated the interaction between an H atom and an
H2 molecule with 2D porphyrin systems containing one transition metal atom
from period 4 of the periodic table. The following metals were considered:
scandium (Sc), titanium (Ti), vanadium (V), chromium (Cr), manganese (Mn),
iron (Fe), cobalt (Co), nickel (Ni), copper (Cu), and zinc (Zn). Here, we use
the name 2D-por-M to refer to the investigated structures in general, and we
replace M by an element symbol to refer to a specific structure. For example,
2D-por-Sc refers to a 2D porphyrin system containing a scandium atom.

The first step of our calculations consisted in optimizing all 2D-por-M
structures. Our calculations were based on the density functional theory (DFT)
formalism, as implemented in the Quantum Espresso (QE) software 32. In the
approach used, the wavefunctions were expanded in the plane wave basis set and
pseudopotentials were used to represent the core electrons 33, 34. To choose
the calculation parameters, we carried out convergence tests for the total
energy against the number of k-points and the cut-off energy. After these
tests, we set the cut-off energy to $75$ Ry and used a $10\times 10\times 1$
k-point mesh. The van der Waals vdw-df is a functional which was used to
describe the exchange-correlation term 35, 36, 37. During optimization, ions
and lattice vectors were varied simultaneously, and we assumed that
convergence had been achieved when the force on each atom was less than $0.05$
eV/Å. After all 2D-por-M structures were optimized, we investigated their
interaction with H atoms and H2 molecules. For these calculations, the initial
position for H/H$2$ was always located above the transition metal. Preliminary
tests indicated that electrostatic interactions were stronger when a hydrogen
atom/molecule was placed in this region.

For the calculations with hydrogen atoms, in the initial step, we placed a H
atom $2.0$ Å above one of the considered metals. Then we optimized the
2D-por-M structure and the hydrogen position, with the constraint that H was
only allowed to move in the direction perpendicular to the 2D plane
($z$-direction). Note that we also tested an initial distance of $1.0$ Å, but
found that this change did not affect the final optimized distance. For the
calculations with hydrogen molecules, in the initial step we placed a H2
molecule $3.0$ Å above one of the considered metals, with a vertical
orientation (the H-H bond was perpendicular to the surface). During the
optimization process, the hydrogen atom that was initially closer to the metal
was constrained to only move along the $z$-direction, whereas the other
hydrogen was allowed to move in all directions.

After the 2D-por-M structure with a hydrogen atom or molecule is optimized,
the adsorption energy is obtained using this expression 38, 1:

$E_{ad}=E_{\mathrm{2D-por-M+H/H_{2}}}-E_{\mathrm{2D-por-M}}-E_{\mathrm{H/H_{2}}},$
(1)

where $E_{\mathrm{2D-por-M+H/H_{2}}}$ is the total energy of a system where
2D-por-M and H/H2 are interacting, $E_{\mathrm{2D-por-M}}$ is the energy of an
isolated 2D-por-M system, and $E_{\mathrm{H/H_{2}}}$ is the energy of an
isolated H atom/H2 molecule.

We also calculate the formation energy per atom for the various 2D-por-M
structures using the following expression:

$E_{f}=(E_{\mathrm{2D-por-M}}-N_{\mathrm{C}}E_{\mathrm{C}}-N_{\mathrm{N}}E_{\mathrm{N}}-E_{\mathrm{M}})/N_{t},$
(2)

where $E_{\mathrm{2D-por-M}}$ is the energy of an isolated 2D-por-M system,
$N_{\mathrm{C}}$/$N_{\mathrm{N}}$ is the number of carbon/nitrogen atoms in
the unit cell, $E_{\mathrm{C/N/M}}$ is the energy of an isolated
carbon/nitrogen/metal atom, and $N_{t}$ is the total number of the atoms in
the unit cell.

##  3 Results and discussion

Figure 1 presents the 2D metallic porphyrin (2D-por-M) structure for the
transition metals M present in period 4 of the periodic table. For the ten
different transition metals considered here, the optimized lattice was a
square ($L_{x}=L_{y}=L$) with $L$ values ranging from $8.37$ and $8.52$ Å.
Hence, the difference between lattice vectors is minimal. We also observed
that the transition metal remained at the 2D plane ($xy$) for all cases. As a
result, the electrostatic potential is the same above and below the 2D plane.
Here, we do not consider isomeric effects on the magnetic properties, as Singh
et al. did for metallic 2d-porphyrin-vanadium 30.

Figure 1: Structure of the 2D metallic porphyrin, with the square unit cell
highlighted. The central metallic atom varied in our calculations, and we
considered ten transition metals from period 4.

Table 1 presents the formation energy per atom obtained using expression 2 for
the optimized 2D-por-M structures. Note that the calculated values are very
close, with a slight difference of about $0.3$ eV/atom. Consequently, the
energy necessary to obtain all the structures investigated here is quite
similar, although experimental procedures could vary for different metallic
atoms 39, 40, 41, 42, 12.

2D-por-M |  $E_{f}$(eV/atom)
---|---
Sc | -8.2
Ti | -8.2
V | -8.2
Cr | -8.2
Mn | -8.1
Fe | -8.1
Co | -8.1
Ni | -8.1
Cu | -8.0
Zn | -7.9
Table 1: In the first column, we have the metallic element attached to the
porphyrin structure. In the second column, we have the corresponding formation
energy per atom.

Figure 2 displays the spin-polarized density of states for the optimized
2D-por-M structures. The $\mu$ value in each graph indicates the corresponding
total magnetic moment. It can be observed that all structures are metallic,
i.e., without a bandgap. Furthermore, notice that structures containing metals
with intermediate atomic numbers present high magnetic moment, whereas those
with smaller or higher atomic numbers have either null or insignificant
magnetic moment. Finally, for systems with high $\mu$ value, an apparent
asymmetry in the DOS between spin up and down states occurs, which is due to
unpaired electrons.

Figure 2: Spin-polarized Density of states for 2D metallic porphyrin.

Let us now discuss the interaction between the optimized 2D-por-M structures
and H atoms/H2 molecules. As mentioned in the methods section, we initially
placed the H atom (or H2 molecule) above the metal. Then, for the H atom, we
constrained the hydrogen to relax only in the $z$-direction, that is, the
direction perpendicular to the plane of 2D-por-M. For the H2 molecule, we
constrained the atom closer to the plane and allowed the other atoms to move
freely. Figure 7 in the supplementary material displays the optimized
structures. After optimization, we calculated the adsorption energy using
expression 1, and we present the results for H and H2 in column 2 from tables
2 and 3, respectively. The adsorption energy is negative for both H and H2,
due to the attractive electrostatic interaction between 2D-por-M and hydrogen,
with larger negative energy values indicating stronger mutual attraction.

For an H atom interacting with 2D-por-M, the adsorption energy varies between
-1.5 eV (for Cu) and -3.7 eV (for V and Cr). Column 3 of table 2 presents the
corresponding equilibrium distance for H atoms, which varies between 1.44 Å
(for Co) and 1.84 Å (for Sc). Together, these results denote that H atoms
chemisorb on 2D-por-M. We also analyzed the charge transfer between H atoms
and 2D-por-M, and the results are presented in column 4 of table 2. Note that
negative values indicate charge transfer from 2D-por-M to a hydrogen, and the
opposite is true for positive values. Overall, we observe high charge transfer
for calculations with H atoms, confirming the occurrence of chemisorption.
Additionally, we observe a tendency for higher charge transfer in systems
containing metals with higher electropositivity 43. For instance, the least
electronegative metal investigated here (Sc) produced the largest charge
transfer (-0.43 $e$).

For H2 molecules, adsorption energy values range from -33.4 meV (for Co) to
-122.5 meV (for Sc). All other structures present adsorption energies around
-50/-60 meV, as we see in column 2 of table 3. Concerning the equilibrium
distance between metal atoms and H2 molecules, values are presented in column
3 of table 3. When comparing these results with those previously discussed for
H atoms, we observe considerably larger equilibrium distances for H2
molecules, varying between 2.64 Å (for Ti) and 3.21 Å (for Mn). Together,
these results indicate that H2 molecules are physisorbed on 2D-por-M. In this
case, interactions are mainly due to Van der Waals forces. Charge transfer
results for H2 molecules are presented in column 4 of table 3. Transferred
charge values are much smaller in this instance, supporting our argument that
physisorption occurs for H2 molecules. Note that our results regarding the
H/H2 charge transfer process are in agreement with the literature 38, 44.

The last columns of tables 2 and 3 shows the total magnetic moment of the
system after H or H2 adsorbed on 2D-por-M. Comparing these results with those
presented in figure 2, we observe that the adsorbed H atom affects the
magnetic moment value considerably. In contrast, the total magnetic moment
remains practically unaffected with H2 adsorption. For chemisorption, the
magnetic moment value changes because the charge transfer process changes the
electronic distribution of the monolayer.

M | [$E_{ad}$-H](eV) |  $R_{H-M}$ (Å) |  $q_{H}$(e) |  $\mu_{H}$ ($\mu_{B}$)
---|---|---|---|---
Sc | -2.9 | 1.84 | -0.43 | 0.0
Ti | -3.6 | 1.70 | -0.26 | 0.0
V | -3.7 | 1.62 | -0.17 | 1.6
Cr | -3.7 | 1.57 | -0.24 | 1.7
Mn | -3.6 | 1.53 | -0.09 | 2.0
Fe | -3.4 | 1.49 | -0.07 | 1.0
Co | -3.0 | 1.44 | -0.05 | 0.0
Ni | -1.9 | 1.46 | 0.06 | 0.3
Cu | -1.5 | 1.53 | -0.24 | 0.0
Zn | -1.7 | 1.59 | -0.21 | 1.0
Table 2: In the first column, we have the transition metal element considered in the calculation (M). Columns 2 and 3 present adsorption energies and equilibrium distance between H and transition metal, while columns 4 and 5 show the charge transferred to H (negative values) or from H (positive values) and, the total magnetic moment in the structure after adsorption of H on 2D-por-M. M | [$E_{ad}$-H2](meV) |  $R_{H_{2}-M}$ (Å) |  $q_{H_{2}}$ ($e$) |  $\mu_{H_{2}}$ ($\mu_{B}$)
---|---|---|---|---
Sc | -122.5 | 3.10 | 0.009 | 0.0
Ti | -65.7 | 2.64 | 0.020 | 0.0
V | -52.8 | 3.07 | -0.003 | 2.6
Cr | -54.4 | 3.14 | 0.00 | 2.7
Mn | -56.2 | 3.21 | 0.00 | 3.1
Fe | -59.1 | 3.20 | -0.004 | 0.2
Co | -33.4 | 3.16 | -0.004 | 1.0
Ni | -62.6 | 3.11 | 0.00 | 0.0
Cu | -60.7 | 3.08 | 0.00 | 0.1
Zn | -56.2 | 3.13 | 0.008 | 0.2
Table 3: In the first column, we have the transition metal element considered
in the calculation (M). Columns 2 and 3 present adsorption energies and
equilibrium distance between H2 and transition metal, while columns 4 and 5
show the charge transferred to H2 (negative values) or from H2 (positive
values) and, the total magnetic moment in the structure after adsorption of H
on 2D-por-M.

In order to gain insight into the electronic density distribution after
adsorption, we calculated the charge density difference of (i) an H atom on
2D-por-Cr and (ii) an H2 molecule on 2D-por-Sc. We show the electron density
of 2D-por-Cr and 2D-por-Sc because the former presents the highest interaction
energy with H and the latter with H2. Moreover, the results from Fig. 3
illustrate well typical charge distributions obtained for the other structures

Figure 3-a)/b) presents the results for the H atom/H2 molecule. In Figs. 3-a)
and 3-b) we used isosurface values of $0.008$e/V3 and $0.0008$e/V3,
respectively. In addition, the blue/red regions represent electron
depletion/accumulation after adsorption.

In Fig. 3-a), we note electron depletion at the Cr atom and accumulation at
the hydrogen. This result agrees with that presented in table 2, which
indicated a charge transfer of $-0.24$ e from 2D-por-Cr to the H atom. We
typically observed charge accumulation in the hydrogen when chemisorption
occurred. In Fig. 3-b), we first note the charge transfer is tiny for
physisorption. Looking at the H2 molecule, we observe the formation of a
dipole, with charge accumulation (red) at the H atom near the metal and
depletion (blue) at the other one. Notice that the blue region is larger than
the red one, as the total charge in the molecule is positive (0.009$e$
according to table 3). We also present the total density of states after H/H2
adsorption in Figures 8 and 9 of the Supplementary Material. These results
reveal that all investigated systems remain metallic after hydrogen
adsorption.

Figure 3: Charge density difference map (a) for a hydrogen atom adsorbed on
2D-por-Cr and (b) for a hydrogen molecule adsorbed on 2D-por-Sc. The red and
blue colors represent electron accumulation and depletion, respectively.

##  4 Adsorption on strained 2D-por-M

When chemisorption occurs, we observed the that the monolayer: (i) transfers
charge to H and (ii) it had its total magnetic moment reduced. In contrast, H2
physisorbed on 2D-por-M, had little charge transfer and total magnetic moment
changes. In this section, we investigate how an uniaxial strain applied along
the $x$-direction affects adsorption. We did not apply strain to the
$y$-direction because the system is isotropic in the $xy$ plane.

Figure 4: Adsorption energies for an H atom placed on a strained 2D-por-M
monolayer. We considered different metallic elements and strain values.

Figure 4 presents the adsorption energy of an H atom as a function of the
applied strain along the $x$-direction. We considered strain values of 3, 6,
and 9$\%$. We note that the strain altered the adsorption energy slightly in
all cases. The change is more perceptible for Co, Ni, Cu, and Zn, where the
adsorption energy decreased with the strain. Still, the maximum variation was
only $0.25$ eV (for Zn). Figure 5-a) shows the hydrogen-metal distance as a
function of the applied uniaxial strain. In all cases, we observed that this
distance remained nearly unaffected. Figure 5-b) presents the change transfer
between an H atom and a metal against the strain. In this case, we observed a
slight decrease in the transferred charge.

Overall, for 2D-por-M structures with an H atom, we observed that adsorption
energies and transferred charges decreased slightly with the strain, whereas
the hydrogen-metal distance remained almost constant. Finally, note that the
applied strain did not significantly change the total magnetic moment of the
investigated structures.

Figure 5: a) Equilibrium distance $R$ between an H atom and 2D-por-M and b)
charge in an H atom ($q_{H}$) as a function of the uniaxial strain.

Figure 6 displays adsorption energies for a H2 molecule adsorbed on a strained
2D-por-M monolayer. We again considered monolayers under 3%, 6%, and 9% strain
in the $x$-direction. The results reveal that the strain had little effect on
the adsorption energies for all investigated structures. We also found that
the applied strain did not modify charge transfer, H2-metal distance, and
magnetic moment values for the structures where physisorption occurred. In
summary, for structures with H2 molecules adsorbed on 2D-por-M, we found that
the strain had no appreciable effect on all studied quantities.

Figure 6: Adsorption energies for an H2 molecule placed on a strained 2D-por-M
monolayer. We considered different metallic elements and strain values.

##  5 Conclusions

In summary, we used density functional theory calculations to study the
structural and electronic properties of an H atom/H2 molecule adsorbed on the
2D metallic porphyrin with a transition metal in its center (2D-por-M). We
considered all transition metals of row four of the periodic table. Our
results revealed chemisorption of atomic hydrogens on the monolayer, with
adsorption energies ranging from $-1.5$ eV (for Cu) to $-3.7$ eV (for V and
Cr). In contrast, we found physisorption of molecular hydrogens on 2D-por-M,
with adsorption energies ranging from $-33.4$ meV (for Co) to $-122.5$ eV (for
Sc). We also analyzed the charge transferred between the monolayer and H/H2
and found an appreciable charge transfer in chemisorption (up to $-0.43$ e for
Sc) but a negligible one in physisorption. Negative values indicate electron
accumulation in the hydrogen. Moreover, we observed that chemisorption changed
the total magnetic moment moderately, as the charge transfer process changed
the electronic distribution of 2D-por-M, particularly in the cases of Fe and
Zn. Finally, we observed that strain slightly changes the properties of
monolayers with hydrogen chemisorbed. However, the strain had practically no
effect on the properties of monolayers where physisorption occurred.

In general, we conclude that 2D-por-M can be useful in applications involving
hydrogen atoms or molecules. The sizeable mutual interaction between the
monolayer and hydrogen is crucial for applications in hydrogen storage.
Moreover, it is possible to adjust the charge transferred to the adsorbed
hydrogen by changing the metal in the monolayer, an important feature for
catalysis applications. Finally, we found that the considered monolayers have
varied magnetic moments and that these can be changed through hydrogen
chemisorption. This characteristic could be useful in spintronic applications.

## Acknowledgements

This work was financed in part by the Coordenacão de Aperfeiçoamento de
Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001, CNPq, and
FAPESP. The authors thank the Center for Computational Engineering & Sciences
(CCES) at Unicamp for financial support through the FAPESP/CEPID Grant
2013/08293-7. LDM would also like to thank the support of the High Performance
Computing Center at UFRN (NPAD/UFRN).

##  6 Supplementary Material

###  6.1 H/H2 adsorbed on the 2D-porphyrin-M

Figure 7: Equilibrium distance for H/H2 atom/molecule adsorbed on the
2D-porphyrin-M for all transition metal investigated in this manuscript.

###  6.2 Density of states of porphyrin metalic

Figure 8: Total density of states for H atom adsorbed on the 2D metallic
porphyrin.. Figure 9: Total density of states for H2 atom adsorbed on the 2D
metallic porphyrin..

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