# [S2E2] Under The Surface

Andy: I think I have to move out of my dad's place.Maya: I don't know what Miller said, but yes, I approve. Andy: It's time.Maya: Way past.Andy: I mean he's feeling better, he can take care of himself, and I need to do this, right away, before I chicken out and keep living there another 30 years. Maya: Have you told your dad yet? Andy: No, God, how do I tell him? What do I tell him?Maya: You say, "Dad, I've never lived out from under your roof. I need to be an adult now, so I'm moving in with Maya, and I feel confident that you can start folding your own boxers all by yourself."

## [S2E2] Under the Surface

Travis is still on medical leave, and Grant wants him to ask his new captain for more time off to heal. Looking beneath the surface, Grant is scared about the next time Travis runs into a burning building. Who can blame him?

Sullivan and Andy tell Evan that there's still no sign of Max. Andy makes a plan of where to go next, but Sullivan says planning is his part. Andy asks for permission and he grants it. Andy then calls the station and asks for an update on the blueprints. Pruitt asks Andy what the plan is. Andy tries to keep it positive, so Pruitt switches to Spanish so Sullivan won't understand and they talk about Sullivan. There's a handprint on the side of the storm drain, so they change direction to go where Max is heading.

Copper-based catalysts play a pivotal role in many industrial processes and hold a great promise for electrocatalytic CO2 reduction reaction into valuable chemicals and fuels. Towards the rational design of catalysts, the growing demand on theoretical study is seriously at odds with the low accuracy of the most widely used functionals of generalized gradient approximation. Here, we present results using a hybrid scheme that combines the doubly hybrid XYG3 functional and the periodic generalized gradient approximation, whose accuracy is validated against an experimental set on copper surfaces. A near chemical accuracy is established for this set, which, in turn, leads to a substantial improvement for the calculated equilibrium and onset potentials as against the experimental values for CO2 reduction to CO on Cu(111) and Cu(100) electrodes. We anticipate that the easy use of the hybrid scheme will boost the predictive power for accurate descriptions of molecule-surface interactions in heterogeneous catalysis.

Density functional theory (DFT) has been the method of choice for quantitative understanding and developing of complex systems in either quantum chemistry or computational materials science. Often, hybrid functionals are widely used for molecules and solids with localized electrons, while generalized-gradient approximations (GGAs) usually suffice for bulk and surface metals with delocalized electrons13,14,15. However, the choice is not trivial when dealing with systems as in heterogeneous catalysis where molecules meet metal surfaces, both of which ought to be simulated accurately.

Here, we apply a hybrid scheme, XYG3:GGA, that combines the XYG3 functional19,32 and the periodic GGA, to describe some key steps in the copper-based heterogeneous catalysis. The accuracy of XYG3:GGA is validated by a benchmark set, where accurate experimental results are available, which includes (1) the preferred CO adsorption sites on Cu(111) and Cu(100) surfaces, (2) the adsorption energies of CO, H, and O on the Cu(111) surface, and that of NH3 on the Cu(100) surface, (3) the H2 dissociation barrier and the 2 *H desorption barriers on the Cu(111) surface. The benchmark results show that the XYG3:GGA scheme provides a prediction close to chemical accuracy for all these well-established cases. Finally, we utilize the XYG3:GGA scheme to study the electrocatalytic CO2RR to CO on Cu(111) and Cu(100) surfaces. A substantial improvement on the calculated equilibrium and onset potentials is achieved. Taken together, we conclude that the very high accuracy of the XYG3:GGA scheme, as well as its easy use, will enhance the predictive power of the computational catalysis for the copper-based catalysts, which shall offer new mechanistic insights and help catalysts rational design in a quantitative way.

where the basis sets are specified. The PBC@GGA calculation is carried out by the projector augmented-wave (PAW) basis with a high kinetic energy cutoff (see Supplementary Methods for details), which is known to well represent the LB45. Here, the energy difference between XYG3 and GGA for cluster model calculations are effectively carried out by using the SB of def2-SVP, which enables to efficiently simulate metal clusters with sufficiently large size as Cu31 (Fig. 2b) and Cu31 (Fig. 2c) for Cu(111) and Cu(100) surfaces, respectively. It has been demonstrated before that the energy difference between H and L converge well with cluster size of appropriate shapes37. Here, we refer to Supplementary Fig. 2 for illustrative testing on the cluster size effects for CO adsorption on Cu(111) and Cu(100) surfaces. It is worthy of note that the convergence of the cluster size effect for different metals is not necessarily the same (see Supplementary Fig. 3 for CO adsorption on Au(111) as a comparison). Inspection of the cluster size effect is important for achieving reliable results with the hybrid scheme.

Even though CO interactions with Cu surfaces are of particular interest, there exists a large gap between the experimental observation and the theoretical prediction. While the experiment observed that CO preferred the top site on the Cu(111) surface, previous theoretical calculations showed various possibilities, depending critically on the methods used23,26,37,38,39,46.

We now pay more attention to the performance of the XYG3:GGA scheme and some other methods in describing the absolute adsorption energies (Supplementary Table 5). It is worth noting that the benchmark values should consider the vibrational zero-point energy (ZPE) contribution contained in the low-temperature experimental surface reaction energy51. This approach has also been employed here to consider the ZPE contributions to all the experimental values (see Supplementary Methods 1.2 and Supplementary Table 2 for details). The performances of some selected DFAs on calculating the adsorption energies of CO, H, and O on the Cu(111) surface, as well as the NH3 adsorption energy on the Cu(100) surface, are tested, while the calculation errors of different functionals are presented in Fig. 3b.

The errors for the predicted barriers with different functionals are presented in Fig. 3d. While PBE predicts a good desorption barrier, it significantly underestimates the dissociative adsorption barrier. Both PBE-D3BJ and M06-L follow the same trend as PBE, further exacerbating the problem. On the contrary, B3LYP overestimates the dissociative adsorption barrier. Such a tendency is eliminated by adding the dispersion correction as in B3LYP-D3BJ. B3LYP also overestimates the 2 *H desorption barrier to some extent, while B3LYP-D3BJ does not help in this context. Encouragingly, XYG3 can correctly predict both barriers for the H2 dissociative adsorption and the 2 *H desorption, which represents an important advance.

All adsorption energy calculations using cluster models were performed by using the Q-Chem 5.0 computational package68. All the structures of cluster models cut from extended systems were fixed. The cluster model calculations were performed with a small basis set of def2-SVP69. For calculating the formation energy of the surface species in the gas phase, the large basis set of def2-QZVP69 was used. More details and discussions can be found in the Supplementary Methods.

All gas-phase molecules were treated as ideal gas, whose thermodynamic quantities contain all the transitional, rotational, and vibrational contributions. All surface species were treated as an immobile model containing the vibrational contribution only. The thermodynamic quantities of the gas-phase molecules can be directly obtained from the Q-Chem calculation results with the vibrational contribution treated by the harmonic oscillator approximation. A brief introduction of partition functions also has been shown in the Supplementary Methods.

Magma is a molten and semi-molten rock mixture found under the surface of Earth. This mixture is usually made up of four parts: a hot liquid base, called the melt; minerals crystallized by the melt; solid rocks incorporated into the melt from the surrounding confines; and dissolved gases.

Much like heat transfer, flux melting also occurs around subduction zones. In this case, water overlying the subducting seafloor would lower the melting temperature of the mantle, generating magma that rises to the surface.

Magma can intrude into a low-density area of another geologic formation, such as a sedimentary rock structure. When it cools to solid rock, this intrusion is often called a pluton. A pluton is an intrusion of magma that wells up from below the surface.

Let $x_0^2+x_1^2+x_2^2=0$ be a conic in $\mathbbP^2$. Its image in $\mathbbP^5$ is the intersection of $y_0+y_3+y_5=0$ and the surface $S$. Making a change of variable so $y_5=0$ and plugging $-y_0-y_3$ into $y_5$ of the three defining equations of $S$, I got$$y_1^2=-(y_3^2+y_4^2)\\y_1^2=-(y_0^2-y_2^2)\\y_1^2=y_0y_3$$ 041b061a72