Virial tensor

Surface tensions have been calculated from the virial tensor as... 

In a simulation [19] the pressure tensor is obtained from the virial theorem [78]... 

Bulk phase fluid structure was obtained by solution of the Percus-Yevick equation (W) which is highly accurate for the Lennard-Jones model and is not expected to introduce significant error. This allows the pressure tensors to return bulk phase pressures, computed from the virial route to the equation of state, at the center of a drop of sufficiently large size. Further numerical details are provided in reference 4. 

A differential virial theorem represents an exact, local (at space point r) relation involving the external potential u(r), the (ee) interaction potential u r,r ), the diagonal elements of the 1st and 2nd order DMs, n(r) and n2(r,r ), and the 1st order DM p(ri r2) close to diagonal , for a particular system. As it will be shown, it is a very useful tool for establishing various exact relations for a many electron systems. The mentioned dependence on p may be written in terms of the kinetic energy density tensor, defined as... 

After a sufficiently long run in the network mode, the equilibrium value of the stress tensor ty was determined by application of the virial formula [5, 6]... 

Isobaric ensembles can also be generated in this way by requiring that df>/dt=0. In an inhomogeneous fluid, the three diagonal elements of the pressure tensor p should be considered separately, which means that one could have up to three constraints. Usually, only the two pressure elements px (perpendicular) and pn (parallel) to the surface need be considered. The virial expression for the pressure element in an inhomogeneous fluid can be written as [7] ... 

From eqn (6.30) it is clear that the virial of the electronic forces, which is the electronic potential energy, is totally determined by the stress tensor a and hence by the one-electron density matrix. The atomic statement of the virial theorem provides the basis for the definition of the energy of an atom in a molecule, as is discussed in the sections following Section 6.2.2. 

The term T (Q) is the virial of the forces exerted on the surface of the subsystem, a term expressible in terms of the stress tensor previously defined in eqns (8.173) and (6.12),... 

In addition, we used Bader s atoms-in-molecules (AIM) theory [56,57] to help analyze some of the results. For convenience, we give here a very brief overview of this approach. According to the AIM theory, every chemical bond has a bond critical point at which the first derivative of the charge density, p(r), is zero. The (> r) topology is described by a real, symmetric, second-rank Hessian-of-/3(r) tensor, and the tensor trace is related to the bond interaction energy by a local expression of the virial theorem ... 

Schweitz s representation of the quantum stress tensor a (r) in terms of the flux density operator acting on the momentum in Equation (19) makes clear its interpretation as a momentum flux density. Schweitz does not, however, consider how the surface flux virial in the quantum case, Zs or i ,s, may be related to the pv product. This, as demonstrated in the following section, has been accomplished using the atomic statement of the virial theorem [12]. 

A number of papers appeared in the 1980s by Nielsen and Martin [47-49] and one in 2002 by Pendas [50] that employ the classical approach in the definition of pressure as explored by Slater [14] and others and embodied in Equation (17). This approach identifies the pv product with the virial of the external forces acting on the nuclei relating the pressure, in "analogy to classical thinking" [50], to the trace of a stress tensor, Equation (29)... 

For anisotropic systems the stress tensor is the relevant observable to compute. Whereas the ideal gas contribution to the pressure still remains isotropic, the virial must be replaced by... 

Time constant for Hookean dumbbell model Time constants for Rouse chain model Solvent contnbution to thermal conductivity Tensor virial multiplied by 2 Momentum space distribution function Integration variable in Taylor series Stress tensor (momentum flux tensor) External force contribution to stress tensor Kinetic contribution to stress tensor Intramolecular contribution to stress tensor Intermolecular contribution to stress tensor Fluid density... 

The final conclusion is that the expression for the stress tensor developed in Sect. 7 is consistent with the tensor virial theorem. Furthermore, it is not permitted to add a divergenceless term to the stress tensor, and the partitioning of the external source term in Eq. (7.9) into an external force term G and the divergence of in Eq. (7.10) is correct. 

The viscosity can be computed by using molecular dynamics simulations from the virial form of the molecular pressure tensor P, which can be represented as a sum of four contributions ... 

For each of the force field contributions described above, there is a corresponding contribution to the total instantaneous stress tensor Lennard-Jones interactions for total energy calculations or for virial calculations of the stresses, long range corrections need to be included, as discussed by In t Veld et al. [26]. 

That is, the difference between the force equation and the differential virial theorem lies only in the definition of the stress tensor. The relationship between them is... 

Another necessary condition for equilibration is that the average pressure calculated during the simulation be close to the set pressure of the NPT simulation. Since in this MC simulation of polymer chains the bond lengths and angles are infinitely stiff, the determination of the stress tensor from interatomic forces is non-trivial. There are in fact many techniques for computing the stress tensor these are reviewed elsewhere [30]. Here we used both the so-called Molecular Virial method (see Appendix A of [37] and also [30]) and an inter-chain force-based method [30] to calculate the stress tensor and the pressure. We found the calculated pressure to be in excellent agreement with the set pressure for both methods within 1% for the Molecular Virial method and within 10% for the inter-chain force-based method. 

A generalization of the virial theorem to all components of the stress tensor has been given in Refs. 11 and 53 and termed the "stress... 

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