The newest members of a two-dimensional material family, involving transition metal carbides and nitrides (called MXenes), have garnered increasing attention due to their tunable electronic and thermal properties depending on the chemical composition and functionalization. This flexibility can be exploited to fabricate efficient electrochemical energy storage (batteries) and energy conversion (thermoelectric) devices. In this study, we calculated the Seebeck coefficients and lattice thermal conductivity values of oxygen terminated M₂CO₂ (where M = Ti, Zr, Hf, Sc) monolayer MXene crystals in two different functionalization configurations (model-II (MD-II) and model-III (MD-III)), using density functional theory and Boltzmann transport theory. We estimated the thermoelectric figure-of-merit, zT, of these materials by two different approaches, as well. First of all, we found that the structural model (i.e. adsorption site of oxygen atom on the surface of MXene) has a paramount impact on the electronic and thermoelectric properties of MXene crystals, which can be exploited to engineer the thermoelectric properties of these materials. The lattice thermal conductivity κ_l, Seebeck coefficient and zT values may vary by 40% depending on the structural model. The MD-III configuration always has the larger band gap, Seebeck coefficient and zT, and smaller κ_l as compared to the MD-II structure due to a larger band gap, highly flat valence band and reduced crystal symmetry in the former. The MD-III configuration of Ti₂CO₂ and Zr₂CO₂ has the lowest κ_l as compared to the same configuration of Hf₂CO₂ and Sc₂CO₂. Among all the considered structures, the MD-II configuration of Hf₂CO₂ has the highest κ_l, and Ti₂CO₂ and Zr₂CO₂ in the MD-III configuration have the lowest κ_l. For instance, while the band gap of the MD-II configuration of Ti₂CO₂ is 0.26 eV, it becomes 0.69 eV in MD-III. The zT_max value may reach up to 1.1 depending on the structural model of MXene.

A procedure to determine optimal vibrational coordinates is developed on the basis of an earlier idea of Thompson and Truhlar [J. Chem. Phys. 77, 3031 (1982)]. For a given molecule, these coordinates are defined as the unitary transform of the normal coordinates that minimizes the energy of the vibrational self-consistent-field (VSCF) method for the ground state. They are justified by the fact that VSCF in these coordinates becomes exact in two limiting cases: harmonic oscillators, where the optimized coordinates are normal, and noninteracting anharmonic oscillators, in which the optimized coordinates are localized on individual oscillators. A robust and general optimization algorithm is developed, which decomposes the transformation matrix into a product of Jacobi matrices, determines the rotation angle of each Jacobi matrix that minimizes the energy, and iterates the process until a minimum in the whole high dimension is reached. It is shown that the optimized coordinates are neither entirely localized nor entirely delocalized (or normal) in any of the molecules (the water, water dimer, and ethylene molecules) examined (apart from the aforementioned limiting cases). Rather, high-frequency stretching modes tend to be localized, whereas low-frequency skeletal vibrations remain normal. On the basis of these coordinates, we introduce two new vibrational structure methods: optimized-coordinate VSCF (oc-VSCF) and optimized-coordinate vibrational configuration interaction (oc-VCI). For the modes that become localized, oc-VSCF is found to outperform VSCF, whereas, for both classes of modes, oc-VCI exhibits much more rapid convergence than VCI with respect to the rank of excitations. We propose a rational configuration selection for oc-VCI when the optimized coordinates are localized. The use of the optimized coordinates in VCI with this configuration selection scheme reduces the mean absolute errors in the frequencies of the fundamentals and the first overtones/combination tones from 104.7 (VCI) to 10.7 (oc-VCI) and from 132.4 (VCI) to 8.2 (oc-VCI) cm⁻¹ for the water molecule and the water dimer, respectively. It is also shown that the degree of coupling in the potential for ethylene is reduced effectively from four modes to three modes by the transformation from the normal to optimized coordinates, which enhances the accuracy of oc-VCI with low-rank excitations.

The symmetric-stretching fundamental (ν₁) and the bending first overtone (2ν₂) of CO₂, which are accidentally degenerate with the same symmetry, undergo a Fermi resonance and give rise to two Raman bands with a frequency difference of 107 cm⁻¹ and an intensity ratio of 2.1. Both the frequency difference and intensity ratio can be varied by pressure applied to CO₂ in condensed phases, which has been utilized as a spectroscopic geobarometer for minerals with CO₂ inclusion. This study calculates the pressure dependence of the Fermi dyad frequency difference and intensity ratio by combining the embedded-fragment second-order Møller–Plesset perturbation calculations of harmonic frequencies of solid CO₂ under pressure and the coupled-cluster singles and doubles with noniterative triples and vibrational configuration-interaction calculations of anharmonic frequencies of molecular CO₂. It reproduces frequency difference quantitatively and intensity ratio qualitatively up to 10 GPa. The analysis of the results is shown to render strong support for one particular order of unperturbed frequencies, ν₁ > 2ν₂, in both the gas and solid phases, which has been a matter of controversy for decades.

The vibrational self-consistent field (VSCF) method is a mean-field approach to solve the vibrational Schrödinger equation and serves as a basis of vibrational perturbation and coupled-cluster methods. Together they account for anharmonic effects on vibrational transition frequencies and vibrationally averaged properties. This article reports the definition, programmable equations, and corresponding initial implementation of a diagrammatically size-extensive modification of VSCF, from which numerous terms with nonphysical size dependence in the original VSCF equations have been eliminated. When combined with a quartic force field (QFF), this compact and strictly size-extensive VSCF (XVSCF) method requires only quartic force constants of the ${\partial }^{4}V/\partial Q_i^2\partial Q_j^2$ type, where V is the electronic energy and Q_i is the ith normal coordinate. Consequently, the cost of a XVSCF calculation with a QFF increases only quadratically with the number of modes, while that of a VSCF calculation grows quartically. The effective (mean-field) potential of XVSCF felt by each mode is shown to be harmonic, making the XVSCF equations subject to a self-consistent analytical solution without matrix diagonalization or a basis-set expansion, which are necessary in VSCF. Even when the same set of force constants is used, XVSCF is nearly three orders of magnitude faster than VSCF implemented similarly. Yet, the results of XVSCF and VSCF are shown to approach each other as the molecular size is increased, implicating the inclusion of unnecessary, nonphysical terms in VSCF. The diagrams of the XVSCF energy expression and their evaluation rules are also proposed, underscoring their connected structures.