1 An evaluation for g eometries formation enthalpies and dissociation energies of diatomic and triatomic C H N O NO 3 and HNO 3 molecules from the PAW DFT method

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An evaluation for geometries, formation enthalpies, and dissociation energies of

diatomic and triatomic (C, H, N, O), NO3, and HNO3 molecules from the PAW DFT method

with PBE and optB88-vdW functionals (AIP Advances, in press, 2022)

Yong Han

Division of Chemical and Biological Sciences, Ames National Laboratory, Ames, Iowa 50011, USA.

Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA.

*Email: y27h@ameslab.gov

ABSTRACT

Structural geometries, formation enthalpies, and dissociation energies of all diatomic and triatomic molecules consisting

of the four basic elements C, H, N, and/or O are calculated from the projector augmented wave (PAW) density functional

theory (DFT) method with the PBE and optB88-vdW exchange-correlation functionals. The calculations are also extended to

two larger molecules NO3 and HNO3, which consist of 4 and 5 atoms, respectively. In total, 82 molecules or isomers are

considered in the calculations. The geometric parameters including 42 bond lengths and 15 bond angles of these molecules

from the planewave DFT method are highly satisfactory, relative to available experimental data. The error analysis is also

performed for 49 formation enthalpies and 138 dissociation energies (including 51 atomization energies as well as

corresponding bond dissociation energies). The results are also compared with the previous data from various atomic-

orbitals-based methods for molecules and from similar or different planewave DFT methods for various solids and other

molecules. This provides an informative and instructive evaluation, especially for calculating the large-size material systems

containing these small molecules as well as for further developing DFT methods.

I. INTRODUCTION

Atomic-orbitals-based methods (AOBMs) [1, 2, 3] are generally regarded by the computational chemical community as

more accurate numerically than the planewave density functional theory (DFT) methods [4, 5]. The reason is that, in earlier

years, the pseudopotentials in the planewave DFT methods were not carefully generated when used in many planewave codes.

However, since the projector augmented wave (PAW) method was suggested and used by Blöchl [6], a lot of effort has been

made and the reliable PAW pseudopotentials have been generated [7] with constantly updated versions released, as

implemented in the Vienna Ab initio Simulation Package (VASP) code [4]. Due to the high computational costs, AOBMs with

higher accuracies are generally used to calculate relatively small systems like molecules. In contrast, the planewave DFT

methods can be computationally much more efficient [4, 5] by using the supercell technology with the periodic boundary

conditions to simulate any crystalline materials.

Due to the fundamental importance in various scientific areas including chemical physics, biophysics, environmental

science, interstellar medium, etc., research on small molecules is constantly developing both theoretically and experimentally.

Specifically, adsorption of small molecules on materials surfaces (including outer surfaces of materials or the surfaces of pores

in materials, e.g., molecular sieves [8, 9]) is often considered for both theoretical studies and applications. The supercell for

such a system by using a planewave DFT method often contains hundreds to thousands of atoms at least for obtaining reliable

adsorption properties. In addition, ab initio molecular dynamics (AIMD) simulations [10] for such systems are also selectively

implemented for, e.g., visualizing the diffusion paths of molecules on the surfaces, while AIMD simulations are even much

more demanding than normal structural optimizations. Thus, using AOBMs with higher accuracies but high computational

costs is impractical for such computations. Instead, the planewave DFT methods can be computationally practical due to the

high efficiency. However, when a planewave DFT method is applied to a specific system, the reliability of the method must be

first assessed because the accuracy of the DFT results can significantly depend on the electronic exchange-correlation energy

functionals. To this end, we mention an example. It is well-known that Perdew-Burke-Ernzerhof (PBE) functional [11] cannot

predict the interlayer spacing of graphite due to the absence of the dispersion corrections, e.g., the predicted lattice constant

= 8.870 Å along the [0001] direction of graphite from our previous PBE calculations [12] is notoriously much larger than the

experimental value of 6.6720 Å [13]. In contrast, the value of 6.701 Å from our optB88-vdW [14] calculations with dispersion

corrections can reproduce the above experimental  value very well [12]. Thus, one should be particularly careful when

selecting a functional with or without dispersion corrections to calculate the weakly bonded systems like graphite.

As a semilocal functional, the generalized gradient approximation (GGA) generally has comparably good accuracies for

calculating ground state properties of neither weakly bonded nor strongly correlated systems. For example, recent extensive

tests on the lattice constants, bulk moduli, and cohesive energies (or atomization energies) of 44 strongly and 17 weakly

bonded solids from various local, semilocal, and nonlocal functionals (so-called “DFT Jacob’s ladder”) have been reported [15,

16]. For the weakly bonded systems, dispersion corrections need to be considered [17], while for strongly correlated systems,

DFT+U corrections are usually needed [18]. In addition, more nonlocal functionals (upper rungs of the DFT Jacob’s ladder) can

have higher accuracies but higher computational costs than more local functionals (lower rungs). Therefore, to appropriately

choose a functional before calculating a specific system, both accuracy and computational cost need to be balanced.

Recently, we have selected and applied the optB88-vdW functional [14], which typically considers dispersion correction

including van der Walls (vdW) interactions, to various vdW materials with guest atoms [12, 19–30] and silica polymorphs with

molecular groups consisting of C, H, and O [31]. These applications have already been proven very successful. The success is

not surprising, given that these systems include weakly bonded interlayer spaces, while for the weakly bonded systems, the

GGA plus dispersion corrections generally have satisfactory accuracies, as described above. In the near future, we will

extensively involve the adsorption and diffusion properties of small molecules on the outer or inner pore surfaces of such

weakly bonded materials. The elements that make up these small molecules will be C, H, N and/or O, which are also 4 most

fundamental elements of organisms. Before extensively calculating these large-size systems, an evaluation even only for these

small molecules in gas phase will be very instructive and necessary.

In this work, we use the PAW DFT method with the optB88-vdW functional to calculate the structural geometries,

formation enthalpies, and dissociation energies of all diatomic and triatomic molecules consisting of C, H, N, and/or O, by

considering 80 linear or triangular molecules or isomers. Then, we extend our calculations to two larger molecules NO3 and

HNO3, containing 4 and 5 atoms, respectively, because these molecules will be the first candidates which will be involved in

our ongoing projects for, e.g., separation of rare earth elements. As comparison, we also obtain the results using the most

popular PBE GGA [11] without dispersion corrections.

The paper is organized as follows. The computational details are described in Sec. II. In Sec. III, we tabulate and discuss the

structural geometries, formation enthalpies, dissociation energies, and corresponding spin states of all 82 molecular molecules

or isomers consisting of C, H, N and/or O from our DFT calculations and previous experiments or AOBM calculations available

in literature. In Sec. IV, we also discuss our results and make error analysis by comparing with other relevant DFT results in

the literature. Sec. V provides a summary of this work. The formulation of formation enthalpies and dissociation energies is

provided in Supplementary Material (SM) Sec. S1 and the original data for error analysis are also provided in SM.

II. COMPUTATIONAL DETAILS

We use the VASP code [4] to perform all DFT calculations in this work, with the PAW pseudopotentials [7] developed by

the VASP group and released in 2015. For the electron-electron exchange correlation part, as described in Sec. I, we use PBE

and optB88-vdW functionals without and with dispersion corrections, respectively. In all DFT calculations, we take the energy

cutoff to be 600 eV with sufficient accuracy (cf. the default energy cutoffs of 400 eV for C, 250 eV for H, 400 eV for N, and 400

eV for O in the PAW pseudopotential data files). Spin polarizations are always considered because the energy of a molecular

configuration depends on the spin state. For 80 diatomic or triatomic molecules or isomers, the supercell is always taken to be

a rectangular box with the size of 23 Å×22 Å×21 Å. For two larger molecules (NO3 and HNO3), the supercell is taken to be

slightly larger with the size of 24.3 Å×24.2 Å×24.1 Å. These supercell sizes are sufficiently large so that the interactions

between replicas can be ignored. The k mesh is always taken to be 1×1×1 which is sufficient because of the large supercell

sizes. During energy minimization, the atoms in the supercell are fully relaxed after the initial configurations are judiciously

selected based on previous experiments or AOBM calculations in the literature. The convergence of total energy is reached

when the force exerted on each atom is less than 0.01 eV/Å.

III. RESULTS AND DISCUSSION

In Table I, we list our PAW PBE and optB88-vdW results for all diatomic and triatomic molecules or isomers consisting of

C, H, N, and/or O. As comparison, the corresponding AOBM and experimental data available in the previous literature are also

listed. The theoretical and experimental data for two larger molecules NO3 and HNO3 are listed in Table II.

TABLE I. Theoretical and experimental data for 82 diatomic and triatomic molecules or isomers consisting of C, H, N, and/or O.

Molecular formulas adopt the NIST notations [32] for convenient indexing, where molecules are listed in alphabetic order of

element symbols and the numbers of elemental atoms in the molecule as subscripts 1, 2, 3, …, etc., but the conventional

notations and names for molecules or isomers are also listed. All ball-and-stick geometric structures are from our PAW

optB88-vdW calculations with the gray, white, blue, and red balls representing C, H, N, and O atoms, respectively. “PBE” and

“optB88” are our PAW DFT calculations with PBE and optB88-vdW functionals, respectively. The results from other

functionals in the previous literature (see Sec. IV) are not provided in this list, except for a few specific molecules or isomers.

“AOBM” denotes the data from various atomic-orbitals-based methods in the literature. “Exp.” denotes available experiment

data from the literature.  (in unit of Bohr magneton B) is the spin magnetic moment of the molecule and taken to be the

value with the lowest energy for a given configuration. αβ (in Å) is the bond length (or interatomic distance or atomic spacing)

between atom  and atom . αβγ (in degree °) is the angle between βα and βγ. ∆f (in eV) is the formation enthalpy from Eq.

(S2). p1+p2+p3+⋯ (in eV) is the dissociation energy for products p1, p2, p3, … from Eq. (S4). The dissociation energy in the

final column is also called the atomization energy from Eq. (S6). The data for ∆f and p1+p2+p3+⋯ are obtained always at 0 K.

Under the corresponding values in eV, the available experimental data from Active Thermochemical Tables (ATcT) [33] for

∆f and p1+p2+p3+⋯ in units of kJ/mol with the uncertainties are also listed, as indicated by “±”, corresponding to estimated

95% confidence limits [34, 35]. The species names from ATcT [33] are adopted partly.

Formula: C1H1  ∆f C+H

CH or HC (Methylidyne) CH

PBE 1 1.1369 6.429 3.697

optB88 1 1.1300 6.384 3.895

AOBM [36] 1.1204

Exp. [37] 1.1199

Exp. [38] 1.119786

Exp. [33] 6.14432

592.837±0.097

3.46791

334.602±0.087

Formula: C1H1N1  ∆f CH+N CN+H C+HN C+H+N

Linear HCN or NCH (Hydrogen cyanide) CH CN

PBE 0 1.0749 1.1611 1.240 10.386 5.544 10.220 14.083

optB88 0 1.0699 1.1561 1.336 10.311 5.765 10.181 14.206

AOBM [39] 1.067 1.160

AOBM [40] 1.0651-

1.0826

1.1527-

1.1758

Exp. [41] 1.06549 1.15321

Exp. [33] 1.34402

129.678±0.089

9.6775

933.74±0.12

5.4215

523.09±0.12

9.7473

940.47±0.18

13.14542

1268.340±0.085

Linear HNC or CNH (Hydrogen isocyanide) NH NC

PBE 0 1.0050 1.1774 1.857 9.769 4.927 9.603 13.466

optB88 0 1.0019 1.1728 1.946 9.701 5.155 9.571 13.596

AOBM [39] 0.996 1.175

AOBM [40] 0.9952-

1.0063

1.1686-

1.1895

Exp. [42] 0.9940 1.1689

Exp. [33] 1.9906

192.06±0.32

9.0310

871.36±0.32

4.7749

460.71±0.32

9.1008

878.09±0.35

12.4989

1205.96±0.31

Linear CHN or NHC HC HN

PBE 0 1.1043 1.0155 11.553 0.072 -4.769 -0.094 3.769

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1Anevaluationforgeometries,formationenthalpies,anddissociationenergiesofdiatomicandtriatomic(C,H,N,O),NO3,andHNO3moleculesfromthePAWDFTmethodwithPBEandoptB88-vdWfunctionals(AIPAdvances,inpress,2022)YongHanDivisionofChemicalandBiologicalSciences,AmesNationalLaboratory,Ames,Iowa50011,USA.DepartmentofP...

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1 An evaluation for g eometries formation enthalpies and dissociation energies of diatomic and triatomic C H N O NO 3 and HNO 3 molecules from the PAW DFT method

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