Density Functional Perturbation Theory Research Articles

Significant improvement in the piezoelectric properties and electromechanical coupling factors of wurtzite AlN compound under high pressures

This work describes a theoretical study of the pressure effect in structural, elastic, piezoelectric and dielectric properties as well as electromechanical coupling factors of wurtzite AlN, obtained by ab-initio calculations using pseudo-potential plane waves (PP-PW) that combine the density functional theory (DFT) and density functional perturbation theory (DFPT). The results of the calculation indicate that the parameters of AlN crystal cells and the volume of AlN crystalline crystal cells decrease notably with increasing pressures from 0 to 40 GPa. Due to an increase in the value of the direct piezoelectric constant (\({e}_{33})\) and a decrease in the value of the elastic constant (\({C}_{33}\)), there is a significant improvement in the value of the converse piezoelectric constant (\({d}_{33}\)). The improvement in the piezoelectric value leads to a higher value in electromechanical coupling coefficient. Our results agree well with previous theoretical and experimental research. We hope that our results will provide guidelines for the realistic application as well as further research of high-performance compounds appropriate for applications in a multitude of fields of study, such as biomedical engineering.

Raman Spectra of Bulk and Few-Layer GeSe From First-Principles Calculations

Raman spectra play a significant role in the study of polar materials. Herein, we report the influence of strain and interlayer shift on vibration responses in bulk and few-layer ferrovalley material GeSe in different polarization states (ferroelectric/FE and antiferroelectric/AFE) based on density functional theory and density functional perturbation theory calculations. We find Ag1 mode shifts by about 10 cm−1 from monolayer to bilayer and trilayer due to the interlayer coupling. The Ag3 mode on behalf of FE mode is observed that is consistent with the experiments in bulk and few-layer GeSe. Meanwhile, in our calculations, with the transition between AFE and FE state in the bilayer and trilayer, the Raman frequency of Ag2 and Ag3 mode decrease obviously whereas that of Ag1 mode increases. Interestingly, the Raman peaks shifted a lot due to the strain effect. We expect these variations in the Raman spectroscopy can be employed to identify the status of GeSe films, e.g., the AFE or FE state, and the number of layers in experiments.

Halogen Interactions in Halogenated Oxindoles: Crystallographic and Computational Investigations of Intermolecular Interactions.

X-ray structural determinations and computational studies were used to investigate halogen interactions in two halogenated oxindoles. Comparative analyses of the interaction energy and the interaction properties were carried out for Br···Br, C-H···Br, C-H···O and N-H···O interactions. Employing Møller–Plesset second-order perturbation theory (MP2) and density functional theory (DFT), the basis set superposition error (BSSE) corrected interaction energy (Eint(BSSE)) was determined using a supramolecular approach. The Eint(BSSE) results were compared with interaction energies obtained by Quantum Theory of Atoms in Molecules (QTAIM)-based methods. Reduced Density Gradient (RDG), QTAIM and Natural bond orbital (NBO) calculations provided insight into possible pathways for the intermolecular interactions examined. Comparative analysis employing the electron density at the bond critical points (BCP) and molecular electrostatic potential (MEP) showed that the interaction energies and the relative orientations of the monomers in the dimers may in part be understood in light of charge redistribution in these two compounds.

Impact of applied biaxial stress on the piezoelectric, elastic, and dielectric properties of scandium aluminum nitride alloys determined by density functional perturbation theory

The effects of biaxial in-plane stress on the elastic, dielectric, and piezoelectric (PE) properties of c-axis textured thin film wurtzite phase scandium aluminum nitride (w-ScxAl1−xN) alloys have been calculated with density functional perturbation theory. The in-plane stress σR was kept below 1 GPa covering compressive and tensile values and applied to alloy supercells represented with special quasi-random structures. An increasingly tensile biaxial stress (σR > 0) produces higher displacement-response internal-strain coefficients for the constituent atoms of the alloy and the related PE properties are more sensitive to σR when the fraction x increases. A significant rise of the relative dielectric permittivity ϵr,33η and softening of the stiffness coefficient c33E are also reported with σR > 0. The effective thin film PE strain coefficient d33,f and coupling coefficient k33,f2 show a relative increase of 22% and 26%, respectively, at σR = 1 GPa and x = 0.438. Both tensile σR and x tend to decrease the c/a cell parameter ratio of the wurtzite structure with a significant impact on the PE coefficients. Based on the decomposition of the stiffness, dielectric, and PE coefficients as well as the structural data, it is suggested that tensile biaxial stress enhances the hexagonal character of w-ScxAl1−xN in a qualitatively similar manner as the scandium nitride fraction x does. The manufacture and PE characterization of a beneficially stressed thin film of w-ScxAl1−xN on a substrate of w-InyAl1−yN with adjusted x, y values are suggested to confirm the calculated values of d33,f.

High-TCferromagnetic inverse Heusler alloys: A comparative study ofFe2RhSiandFe2RhGe

We report the results of experimental investigations on structural, magnetic, resistivity, caloric properties of Fe$_2$RhZ (Z=Si,Ge) along with \textit{ab-initio} band structure calculations using first principle simulations. Both these alloys are found to crystallize in inverse Heusler structure but with disorder in tetrahedral sites between Fe and Rh. Fe$_2$RhSi has saturation moment of 5.00 $\mu_B$ and while its counterpart has 5.19 $\mu_B$. Resistivity measurement reveals metallic nature in both of them. Theoretical simulations using generalized gradient approximation(GGA) predict inverse Heusler structure with ferromagnetic ordering as ground state for both the alloys. However it underestimates the experimentally observed moments. GGA+$U$ approach, with Hubbard $U$ values estimated from density functional perturbation theory helps to improve the comparison of the experimental results. Fe$_2$RhSi is found to be half metallic ferromagnet while Fe$_2$RhGe is not. Varying $U$ values on Fe and Rh sites does not change the net moment much in Fe$_2$RhSi, unlike in Fe$_2$RhGe. Relatively small exchange splitting of orbitals in Fe$_2$RhGe compared to that of Fe$_2$RhSi is the reason for not opening the band gap in the minority spin channel in the former. High ordering temperature and moment make Fe$_2$RhSi useful for spintronics applications.

First-principles study of the Kohn anomaly in TaTe4

The tetrachalcogenide TaTe4 is known as an excellent example of a charge-density wave (CDW) system that has a commensurately modulated structure at room temperature. Using density function perturbation theory, we find that the unmodulated phase of TaTe4 has a giant Kohn anomaly at room temperature, which manifests itself as softened phonon modes at the CDW vector (1/2a*,1/2b*,1/3c*). Interestingly, after the application of 8 GPa hydrostatic pressure, this CDW instability can be effectively suppressed and disappears at room temperature. By studying the topology of the Fermi surface and the phonon linewidth, we show that the Kohn anomaly in TaTe4 is driven by a large electron–phonon coupling coefficient at the CDW vector and not by Fermi surface nesting.

Spectroscopic Identification of Hydrogen Bond Vibrations and Quasi-Isostructural Polymorphism in N-Salicylideneaniline.

The ortho-hydroxy aryl Schiff base 2-[(E)-(phenylimino)methyl]phenol and its deutero-derivative have been studied by the inelastic incoherent neutron scattering (IINS), infrared (IR) and Raman experimental methods, as well as by Density Functional Theory (DFT) and Density-Functional Perturbation Theory (DFPT) simulations. The assignments of vibrational modes within the 3500–50 cm−1 spectral region made it possible to state that the strong hydrogen bond in the studied compound can be classified as the so-called quasi-aromatic bond. The isotopic substitution supplemented by the results of DFT calculations allowed us to identify vibrational bands associated with all five major hydrogen bond vibrations. Quasi-isostructural polymorphism of 2-[(E)-(phenylimino)methyl]phenol (SA) and 2-[(E)-(phenyl-D5-imino)methyl]phenol (SA-C6D5) has been studied by powder X-ray diffraction in the 20–320 K temperature range.

Split representation of adaptively compressed polarizability operator

The polarizability operator plays a central role in density functional perturbation theory and other perturbative treatment of first principle electronic structure theories. The cost of computing the polarizability operator generally scales as $${\mathcal {O}}(N_{e}^4)$$ where $$N_e$$ is the number of electrons in the system. The recently developed adaptively compressed polarizability operator (ACP) formulation [L. Lin, Z. Xu and L. Ying, Multiscale Model. Simul. 2017] reduces such complexity to $${\mathcal {O}}(N_{e}^3)$$ in the context of phonon calculations with a large basis set for the first time, and demonstrates its effectiveness for model problems. In this paper, we improve the performance of the ACP formulation by splitting the polarizability into a near singular component that is statically compressed, and a smooth component that is adaptively compressed. The new split representation maintains the $${\mathcal {O}}(N_e^3)$$ complexity, and accelerates nearly all components of the ACP formulation, including Chebyshev interpolation of energy levels, iterative solution of Sternheimer equations, and convergence of the Dyson equations. For simulation of real materials, we discuss how to incorporate nonlocal pseudopotentials and finite temperature effects. We demonstrate the effectiveness of our method using one-dimensional model problem in insulating and metallic regimes, as well as its accuracy for real molecules and solids.

Fluorine-substituted cyclobutenes in the solid state: Crystal structures, vibrational spectra and mechanical and thermodynamic properties

The crystal structures, infrared and Raman spectra, and mechanical and thermodynamic properties of four important fluorine-substituted cyclobutene derivatives in the solid state are investigated using first-principles solid-state methods based on density functional theory. These compounds are hexafluorocyclobutene (HFCB, C4F6), 1,3,3,4,4-pentafluoro-2-methoxycyclobut-1-ene (PFMCB, C4F5OCH3), 3,3,4,4-tetrafluoro-1,2-dimethoxycyclobut-1-ene [TFDMCB, C4F4(OCH3)2] and 1,2-dichloro-3,3,4,4-tetrafluorocyclobut-1-ene (DCTFCB, C4Cl2F4). Although some of the properties of the corresponding molecules in the gas phase have been studied, and the structures of the corresponding molecular crystals have been determined by refinement from X-ray diffraction data, their vibrational spectra and properties have not yet been reported. The computed crystal structures and X-ray diffraction patterns are in excellent agreement with their experimental counterparts. The infrared and Raman spectra are calculated from the computed crystal structures using density functional perturbation theory. The results are highly consistent with the corresponding spectra measured experimentally in the gas or liquid phases and, therefore, appropriate normal coordinate analyses of the theoretical results are employed to rigorously assign all the vibrational bands. The elasticity matrices of these materials are computed using the finite deformation technique and a complete set of relevant mechanical properties is determined. Their equations of state are also obtained. These compounds are shown to be weak, highly anisotropic materials displaying the negative Poisson's ratio phenomenon. The TFDMCB also exhibits the negative linear compressibility (NLC) phenomenon for external isotropic pressures in the range of 0.64–1.76 GPa. The computed minimum compressibility, found at P = 0.73 GPa, is substantial (k=−192.9 TPa−1). The NLC effect in TFDMCB is due to a collective rotation of the molecules within the crystal under increasing pressure. Finally, the thermodynamic properties of these materials are determined as a function of temperature using phonon calculations. The computed specific heats of HFCB, PFMCB, TFDMCB and DCMCB at T = 250 K are 127.5, 150.4, 169.9 and 134.8 J·K−1·mol−1, respectively, and corresponding entropies are 152.8, 173.3, 195.4 and 186.6 J·K−1·mol−1.

Electron- and hole-doping on ScH2 and YH2: effects on superconductivity without applied pressure

We present the evolution of the structural, electronic, and lattice dynamical properties, as well as the electron–phonon (el–ph) coupling and superconducting critical temperature (T c) of ScH2 and YH2 metal hydrides solid solutions, as a function of the electron- and hole-doping content. The study was performed within the density functional perturbation theory, taking into account the effect of zero-point energy through the quasi-harmonic approximation, and the solid solutions Sc1−x M x H2 (M = Ca, Ti) and Y1−x M x H2 (M = Sr, Zr) were modeled by the virtual crystal approximation. We have found that, under hole-doping (M = Ca, Sr), the ScH2 and YH2 hydrides do not improve their el–ph coupling properties, sensed by λ(x). Instead, by electron-doping (M = Ti, Zr), the systems reach a critical content x ≈ 0.5 where the latent coupling is triggered, increasing λ as high as 70%, in comparison with its λ(x = 0) value. Our results show that T c quickly decreases as a function of x on the hole-doping region, from x = 0.2 to x = 0.9, collapsing at the end. Alternatively, for electron-doping, T c first decreases steadily until x = 0.5, reaching its minimum, but for x > 0.5 it increases rapidly, reaching its maximum value of the entire range at the Sc0.05Ti0.95H2 and Y0.2Zr0.8H2 solid solutions, demonstrating that electron-doping can improve the superconducting properties of pristine metal hydrides, in the absence of applied pressure.

Vibrational spectroscopy and molecular dynamics simulation of choline oxyanions salts

The structure of choline salts containing the anions acetate, [Chol][Ac], and dihydrogen phosphate, [Chol][DHP], were investigated by infrared, Raman, and inelastic neutron scattering (INS). The chosen systems allow for the comparison of structural effects related to the bond acceptor characteristic of [Ac] and the simultaneous acceptor and donor characteristics of [DHP] in forming hydrogen bonds (H-bond) in salts of [Chol], which is itself prone to forming H-bonds. Different computational tools were used for the analysis of different spectral ranges. The calculation of the low-frequency range of Raman and INS spectra of the crystalline phases at low-temperatures by solid state DFPT (density functional perturbation theory) unveils the coupling between vibrations of the H-bonds and intramolecular modes. Changes observed in the spectral pattern of lattice and [DHP] modes upon heating crystalline [Chol][DHP] are analogous to the ferroelectric–paraelectric phase transition known in the potassium salt of [DHP]. The fingerprint region of the vibrational spectra provides information concerning the [Chol] conformation in the solid phase (gauche in [Chol][Ac] and anti in [Chol][DHP]) and in aqueous solution. DFT calculations of ionic pairs and ionic clusters unveil the interplay between [Chol] conformation and the [DHP] ability to form H-bonded dimers of anions. The high-frequency spectral range and the structures driven by H-bonds are discussed using classical molecular dynamics (MD) simulations. The MD simulations of aqueous solutions highlight the strong anion-cation H-bond in [Chol][Ac], in contrast to the strong anion–anion H-bond in [Chol][DHP] due to occurrence of dimers and larger clusters of [DHP].

Engineering the Thermal Conductivity of Doped SiGe by Mass Variance: A First-Principles Proof of Concept

Thermal conductivity of bulk Si0.5 Ge0.5 at room temperature has been calculated using density functional perturbation theory and the phonon Boltzmann transport equation. Within the virtual crystal approximation, second- and third-order interatomic force constants have been calculated to obtain anharmonic phonon scattering terms. An additional scattering term is introduced to account for mass disorder in the alloy. In the same way, mass disorder resulting from n- and p-type dopants with different concentrations has been included, considering doping with III-group elements (p-type) such as B, Al, and Ga, and with V-group elements (n-type) such as N, P, and As. Little effect on the thermal conductivity is observed for all dopants with a concentration below 1021 cm−3. At higher concentration, reduction by up to 50% is instead observed with B-doping in agreement with the highest mass variance. Interestingly, the thermal conductivity even increases with respect to the pristine value for dopants Ga and As. This results from a decrease in the mass variance in the doped alloy, which can be considered a ternary system. Results are compared to the analogous effect on the thermal conductivity in doped Si.

A temperature-dependent Raman scattering and X-ray diffraction study of K2Mo2O7·H2O and ab initio calculations of K2Mo2O7

This study reports a temperature-dependent Raman scattering and X-ray diffraction study of K2Mo2O7·H2O. The high-temperature Raman scattering analysis shows that the material remains structurally stable, with triclinic symmetry, in a temperature range from 300 to 413 K and undergoes a structural phase transition between 413 and 418 K. This phase transition is most likely connected with the dehydration process of K2Mo2O7·H2O. The temperature-dependent X-ray diffraction patterns are measured from 30 to 573 K. The results show that the discovered phase transition occurs between 419 and 433 K, in good agreement with the Raman scattering results. According to the Raman data, with increasing temperature, the dehydrated crystal of K2Mo2O7 undergoes a new phase transformation at 603 K and melts at ~843 K. Principal component and hierarchical cluster analyses are performed based on the treatment of the raw spectral data to infer the phase transformations occurring in the material. Assignments of the Raman modes for the K2Mo2O7 system at ambient conditions are studied through first-principles calculations based on density functional perturbation theory. These calculations are applied to understand the electronic properties, including the band structure and the associated projected density of states, of K2Mo2O7 under the local density approximation.

Structural, electronic, dynamic, optic and elastic properties of MgScGa via density functional theory

The structural, electronic and phonon properties of the MgScGa compound were investigated by density functional theory using the generalized gradient approximation. Some basic structural properties of this compound, such as the lattice constants, bulk modulus and pressure derivative of the bulk module have been studied. Electronic properties were investigated by calculating and analyzing the electronic band structure and total density of states graphs for the MgScGa compound. Electronic band structure calculations showed that MgScGa compound has a semiconductor structure. Phonon spectra were calculated using a linear response method within the framework of the total predicted state density, density functional perturbation theory for the MgScGa compound. A factor group analysis has been performed in order to get the decomposition of whole representation of Γ of MgScGa compound into irreducible representation. For investigation of optic properties; real and imaginary components of complex dielectric function, reflectivity (R), refractive index (n), extinction coefficients (k), energy-loss functions for volume (LV) and surface (LS), the effective number of valence electrons per unit cell (Neff) were calculated. Elastic properties are revealed by calculating the elastic stiffness constants, Bulk, Shear and Young modulus, Poisson Ratio, Flexibility Coefficient and Zener anisotropy constant.

Enhanced piezoelectric effect in MoS2 and surface-engineered GaN bilayer

The piezoelectric effect of MoS2 and surface-engineered GaN bilayers is systematically investigated using the density functional perturbation theory based on first-principles calculations. The results show that the piezoelectric coefficients increased in the stacking with same polarization orientation between MoS2 and of surface-engineered GaN monolayers. We further find that the increment extent is also affected by different stacking configurations. Furthermore, the uniaxial and normal strains can effectively modulate the piezoelectric coefficients in a reasonable range. Such variations can be revealed by analyzing the charge distribution at different strain states. Our results show that MoS2 and surface-engineered GaN bilayers have potential applications in nanosensors and piezotronics.

Density functional study of the variants of inter-Coulombic decay resonances in the photoionization of Cl@C60

Inter-Coulombic decay (ICD) resonances in the photoionization of Cl@C60 endofullerene molecule are calculated using a perturbative density functional theory (DFT) method. This is the first ICD study of an open shell atom in a fullerene cage. Three classes of resonances are probed: (i) Cl inner vacancies decaying through C60 outer continua, (ii) C60 inner vacancies decaying through Cl outer continua, and (iii) inner vacancies of either system decaying through the continua of Cl-C60 hybrid levels, the hybrid Auger-ICD resonances. Comparisons with Ar@C60 results reveal that the properties of hybrid Auger-ICD resonances are affected by the extent of level hybridization.

Temperature-dependent elastic properties of high entropy ceramic (ZrTaNbTi)C from self-consistent quasi-harmonic approximation

High-entropy ceramics have recently attracted considerable attentions because of excellent combination of exceptional properties. In this work, the temperature dependent elastic properties of (ZrTaNbTi)C have been systematically studied from the density functional perturbation theory combined with self-consistent quasi-harmonic approximation. High entropy ceramic (ZrTaNbTi)C is thermodynamically stable due to the negative formation enthalpy, and is also mechanically stable from the obtained elastic properties. The phonon dispersion relation computed at equilibrium volume contains no imaginary-frequency, implying the dynamical stability. At 0 K, (ZrTaNbTi)C possesses evidently high elastic moduli and hardness. The calculated electronic density of state and Bader charge at 0 K and 2000 K show that (ZrTaNbTi)C have covalent characteristics accompanied by ionic feature while the covalency decreases and the ionicity increases as temperature increases. The temperature-dependent elastic properties show that (ZrTaNbTi)C is mechanically stable in temperature range studied, and remains high strength and hardness at high temperature due to the slight softening of strong covalent bonding. Poisson's ratio, Pugh's ratio and Cauchy pressure suggest the higher brittle-ductile transformation trend as the temperature increases. Moreover, (ZrTaNbTi)C shows less anisotropy at higher temperature from the Zener index AZ and three-dimensional projections, being beneficial to reduce cracking and improve durability. The present study provides more insight into the high temperature behavior of mechanical properties, would be valuable for understanding and design of high-temperature properties of high entropy carbide ceramics.

Relative stabilities of Si polytypes under biaxial stress: A first-principles study

We explore the possibility of the production of Si polytypes via the total energies and free energies obtained in the density-functional theory and the density-functional perturbation theory. We first calculate the total energies of three polytypes of the tetrahedrally coordinated $s{p}^{3}$ Si atoms having the two-layer, three-layer, and four-layer stacking periods in the hexagonal unit cell (``$2H$'', ``$3H$'', and ``$4H$'') as a function of the in-plane lattice constant $a$ using the local-density approximation (LDA) and generalized gradient approximation (GGA) comparatively. In the LDA calculation, the $4H$ phase is energetically the most favorable among three phases in the range of 3.55 \AA{} $\ensuremath{\le}$ $a$ $\ensuremath{\le}$ 3.73 \AA{}. While the $4H$ phase is found to be more stable than the $2H$ phase at a relatively wide range of the $a$ value we studied, the $2H$ phase is found to be more stable than the $3H$ ($3C$) phase in the range of 3.57 \AA{} $\ensuremath{\le}a\ensuremath{\le}$ 3.61 \AA{}. In the GGA calculation, on the other hand, the $4H$ phase becomes the most favorable phase for almost the same range as the LDA case, 3.57 \AA{} $\ensuremath{\le}a\ensuremath{\le}$ 3.73 \AA{}. The $2H$ phase also becomes as stable as the $3H$ phase as in the case of the LDA around the small $a$ value of 3.62 \AA{}, although the $3H$ phase energy is always lower than that of the $2H$ phase in the GGA calculation. Considering the chemical vapor deposition (CVD) growth process, we calculate the biaxial stress values to reduce the lattice constant of the substrate Si crystal to realize the small $a$ values, which can lead to the homoepitaxial growth of the $4H$ phase. The values obtained are 4.69 GPa and 8.14 GPa for the LDA and GGA, respectively. Next, we study the thermal effect on the relative stabilities of the 3 phases. We calculate the phonon dispersion, bulk modulus and thermal expansion including the vibrational and thermal effect. From the comparison with experimental values of the $3C$ phase, it is found that the LDA results show much better agreement with experiment than the GGA, indicating that the LDA calculation is reliable to predict thermal properties of the real systems. Finally, we derive the free energy as a function of $a$ under several designated temperatures. The relative stabilities of the $2H$ and $4H$ phases with respect to the $3H$ phase are found to be enhanced. At around $a=3.65$ \AA{}, stabilities of $2H$ and $4H$ phases relative to the $3H$ phase become most prominent. Although stabilities of the $2H$ and $4H$ phases relative to the $3H$ phase are reduced with increasing temperature at around $a$ = 3.60 \AA{}, $2H$ phase is found to be as stable as $3H$ and $4H$ phases at higher temperatures, indicating the possibility of the production of the $2H$ phase via high-temperature CVD process with the biaxial stress.

Impact of anharmonic effects on two phases of VO2 thin-film phase transition materials

Vanadium dioxide (VO2) is a strong correlation material, which can undergo transition from the monoclinic phase to the rutile phase near 340 K at ambient pressure. This phase transition can be controlled by extreme changes of optical, electrical and thermodynamic properties, and thus VO2 film possesses various potential device applications. Physically, lattice dynamics play a critical role in the investigation of the phase mechanism and properties of VO2. Given that widely adopted methods such as the density-functional perturbation theory and the quasiharmonic approximation (QHA) are typically only suitable to calculate the VO2 monoclinic phase, in this work we employ the temperature-dependent effective potential (TDEP) method to calculate the phonon properties of the two phases of VO2 by considering the anharmonicity of the lattice vibrations. Assisted by the experimental data, a resistance–temperature phase diagram has been constructed. The results show that the monoclinic phase is unstable because of the existence of the imaginary frequency. Compared with the thermal expansion, calculated by the QHA and TDEP methods, it is demonstrated that the TDEP method is reliable for the stable rutile phase, in which the imaginary frequency disappears. Simultaneously, it is shown that the calculated Gibbs free energy of VO2 agrees with the experimental results. Our work provides robust evidence tthat the anharmonicity in the lattice dynamics of VO2 cannot be ignored.

Phenomenological model for long-wavelength optical modes in transition metal dichalcogenide monolayer

Transition metal dichalcogenides (TMDs) are an exciting family of 2D materials; a member of this family, ${\mathrm{MoS}}_{2}$, became the first studied monolayer semiconductor. In this paper, a generalized phenomenological continuum approach for the optical vibrations of the monolayer TMDs valid in the long-wavelength limit is developed. The equation of motions for nonpolar and polar oscillations include the phonon dispersion up to a quadratic approximation in the phonon wave vector. On the other hand, the polar modes satisfy coupled equations for the displacement vector and the inner electric field. The two-dimensional phonon dispersion curves for in-plane and out-of-plane oscillations are thoroughly analyzed. The model parameters are fitted from density functional perturbation theory calculations. The current formalism provides an effective tool to describe the phonon dispersion curves around the $\mathrm{\ensuremath{\Gamma}}$ point of the Brillouin zone for a large group of members of the TMD monolayers. A detailed evaluation of the intravalley Pekar-Fr\"ohlich and the ${A}_{1}$-homopolar mode deformation potential coupling mechanisms is performed. The effects of metal ions and chalcogen atoms on polaron mass and binding energy are studied. It is argued that both mechanisms should be considered for a correct analysis of the properties of the polaron or of any process that involves the intraband transitions assisted by the electron-phonon interaction.