The stress theorem

The choice of the parameter A as a scale of length in the physical system leads to a different class of theorems, which are of particular significance for our present work. As shown by Fock, ° the uniform scaling of the system in all dimensions leads to the quantum virial theorem  

This may be derived - - from (1) by replacing rj- (1 +A)rj and Pj- (1 - A)Pj and differentiating in the limit A- 0. The usual form of the virial theorem, with the left-hand side of (7) replaced by zero, applies for an isolated system. Hov/ever, if the system is in a fixed external potential (which is not scaled), then P is the pressure exerted upon the system by the external forces and is the volume of the system. The virial theorem is analogous to the force theorem in that it gives an expression for the external pressure (i.e. the force conjugate to the volume) in terms of the internal operators of the Hamiltonian. Unlike the force theorem, however, the virial theorem involves the kinetic energies and the interactions of all particles-nuclei and electrons. 

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 

The interpretation of the terms in the stress theorem are discussed in more detail in Ref. 13 and in papers to be published in Ref. 11. The basic idea is that the kinetic term is the quantum analogue of classical gas theory. It represents momentum p transferred by a flux p /m. The diagonal aa terms are always positive, i.e., expansive terms. The virial term is associated with potential forces in the a direction between particles displaced in the p direction. The diagonal terms are an energy density and must be negative (attractive) for system which is in equilbrium ( = 0) in order to cancel the expansive kinetic term, The off-diagonal terms may, of course, have either sign. 

The stress theorem is very useful in large macroscopic systems. In particular, since the structure of a crystal is completely specified by the size and shape of the unit ceil and positions of the atoms in the unit cell, the force and stress theorems give all the generalized forces conjugate to these variables. If the stress theorem is expressed in terms of relative coordinates and proper care is taken of the Coulomb interactions, then the stress theorem, Eqs. (8) and (8a), give the total stress in terms of the intrinsic bulk Hamiltonian and wave functions in the bulk of the crystal. Thus, the stress and force are sufficient to construct the complete equation of the state of any crystal. In general, the more complex and the lower the symmetry of the crystal, the more useful are the forces and stresses in the calculations. This is further described by Nielson in this proceedings and later in the present paper. 

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Thus, if the average a is --150MN/m (20,000psi), E/V ISOMJ/rn, i.e., a maximum temperature of --120K. In principal, any coil can probably be made self-protecting by this technique, except the energy store, which has cheated the stress theorem. 

It is most natural to consider next the elastic properties of the crystal because this is the dependence of the total energy upon the size and shape of the unit cell. As was discussed above, it is a great advantage to use the direct calculation of stress from the "stress theorem." This has been applied to accurate calculations of the elastic properties of the crystals Si, Ge, and GaAs, using methods which have general applicability to all solids. 

We have also investigated the stability of C in diamond, BC8 and another structure (MSG) discussed in Ref. 30. in Fig. 7 is shown the enthalpy H = E + PV for C in the region of the transition. Since P ia given by the stress theorem, H may be calculated directly. The main point is that for... 

The stress theorem determines the stress from the electronic ground state of any quantum system with arbitrary strains and atomic displacements. We derive this theorem in reciprocal space, within the local-density-functional approximation. The evaluation of stress, force and total energy permits, among other things, the determination of complete stress-strain relations including all microscopic internal strains. We describe results of ab-initio calculations for Si, Ge, and GaAs, giving the equilibrium lattice constant, all linear elastic constants Cy and the internal strain parameter t,. 

A different type of structural parameter was considered recently by the present authors, namely X representing a homogeneous macroscopic strain defined as the linear scaling of all particle positions as x- 1+e)x. The e is a constant 3x3 strain tensor, and e=0 corresponds to some reference configuration. The conjugate force is in this case defined as the macroscopic stress a, and an explicit general expression denoted the "stress theorem" is derived by Nielsen and Martin (1983). The result is a generalization of the quantum virial theorem (Born et al., 1926),... 

One application of the stress theorem is the study of elastic properties of solids, which becomes straightforward when a suitable finite macroscopic strain is applied to the solid. When the wavefunctions of the distorted solid are known, the stress tensor is evaluated with the stress theorem. In the harmonic approximation elastic constants are defined as the ratio of stress to strain, and it is furthermore possible to go to large strains to obtain all nonlinear elastic properties. In general it is necessary to be concerned with internal strains that may appear microscopically owing to the lower symmetry of the strained solid. In section 6 we show in detail how this problem is solved by combining the stress and force theorems. 

The present paper is organized as follows Section 2 deals with the pseudopotential technique for solving the Schrddinger equation, section 3 derives the stress theorem expressed in reciprocal space within the local-density approximation. Section 4 comments on a number of technical but nevertheless important points in ab-initio calculations. Section 5 deals with calculations on the semiconductors Si, Ge, and GaAs, whose elastic properties is the topic of section 6. 

The stress theorem relies upon the variational principle applied together with a strain-scaling of the quantum system, as discussed in detail by the present authors elsewhere (Nielsen and Martin, to be published). The strain scales particle positions as x- (1 + e)x, and by definition the macroscopic stress per volume n (a and B denote cartesian coordinates) is derived from the total energy by... 

The equilibrium structure is diamond for Si and Ge, and zincblende for GaAs, as was verified in recent theoretical work (Yin and Cohen, 1982 Froyen and Cohen, 1982). With the given structure as the only input we have calculated the lattice constant, a, using the stress theorem. A first calculation of... 

The combined calculation of stress, forces and total energy have thus been shown to constitute a powerful and complete method for the study of structural, elastic and dynamic properties of solids. Using the stress theorem permits rapid determination of lattice parameters, for cubic as well as non-cubic crystals. This leads to large savings in computational effort, compared to direct calculations of the total energy with subsequent numerical differentiation. 

Stresses are usually related to strains through an effective modulus. If the components of stress are nondimensionalized by a suitable scalar modulus c, then they are also of order c. Using (A.94), (A.lOl), and the binomial theorem in (A.39), the relation between the normalized spatial stress s = s/c and the normalized referential stress S = S/c becomes... 

The end effects have been neglected here, including in the expression for change in reservoir entropy, Eq. (178). This result says in essence that the probability of a positive increase in entropy is exponentially greater than the probability of a decrease in entropy during heat flow. In essence this is the thermodynamic gradient version of the fluctuation theorem that was first derived by Bochkov and Kuzovlev [60] and subsequently by Evans et al. [56, 57]. It should be stressed that these versions relied on an adiabatic trajectory, macrovariables, and mechanical work. The present derivation explicitly accounts for interactions with the reservoir during the thermodynamic (here) or mechanical (later) work,... 

Stress Theorem in the Determination of Static Equilibrium by the Density Functional... 

In principle this integral could be applied directly to the Maxwell model to predict the decay of stress at any point in time. We can simplify this further with an additional assumption that is experimentally verified, i.e. that the function in the integral is continuous. The first value for the mean theorem for integrals states that if a function f(x) is continuous between the limits a and b there exists a value f(q) such that... 

It is important to stress the fact that in the proof of the MPC theorem, the laws of classical dynamics are never violated. One could summarize the signihcance of the MPC theorem by saying that, for a well-defined class of dynamical systems, the new formulation lays bare the arrow of time that is hidden in the illusorily deterministic formulation of these unstable systems. 

The evaluation of elements such as the M n,fin s is a very difficult task, which is performed with different levels of accuracy. It is sufficient here to mention again the so called sudden approximation (to some extent similar to the Koopmans theorem assumption we have discussed for binding energies). The basic idea of this approximation is that the photoemission of one-electron is so sudden with respect to relaxation times of the passive electron probability distribution as to be considered instantaneous. It is worth noting that this approximation stresses the one-electron character of the photoemission event (as in Koopmans theorem assumption). 

Take the results of the Gauss divergence theorem and evaluate the net force on the differential control volume using the divergence of the stress tensor,... 

In order to obtain Green s identities for the flow field (u,p), a vector z is defined as the dot product of the stress tensor a(u, p) and a second solenoidal vector field v (divergence-free). The divergence or Gauss Theorem (10.1.1) is applied to the vector z... 

The stress in the network can then be expressed by the virial theorem, Eq. (6), where in the molecular theory, the set of particles considered is not all the atoms of the system but only the end atoms of each chain, and they are regarded as subject to the force in the chain connecting them. That is, the stress ty is then given by... 

Applying the Divergence theorem to the volume integral, and adding an osmotic pressure IT to account for variations in ion concentration, yields the total force in terms of a stress tensor consisting of both osmotic and electrostatic components ... 

It is important to stress that the intersection theorem cannot be extended to three or more structures For... 

Alternatively, by substituting the strain in Eq. (11.66) sls a. linear function of the stress, the free energy can be represented as a quadratic function of Uy. Applying once more the Euler theorem, the corresponding counterpart of Eq. (17.64) in terms of the stress is obtained as... 

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 atomic statements of the Ehrenfest force law and of the virial theorem establish the mechanics of an atom in a molecule. As was stressed in the derivations of these statements, the mode of integration used to obtain an atomic average of an observable is determined by the definition of the subsystem energy functional i2]. It is important to demonstrate that the definition of this functional is not arbitrary, but is determined by the requirement that the definition of an open system, as obtained from the principle of stationary action, be stated in terms of a physical property of the total system. This requirement imposes a single-particle basis on the definition of an atom, as expressed in the boundary condition of zero flux in the gradient vector field of the charge density, and on the definition of its average properties. 

The opening section of this chapter stressed the importance of the presence of the surface integral in the hypervirial theorem for an open system, eqn (6.2). Unlike the theorem for a total system, eqn (6.4), in which case the average of the commutator of any observable G with the Hamiltonian H vanishes, the corresponding result for an atom in a molecule is proportional to the flux in the effective single-particle vector current density of the property G through the atomic surface. As a result, the hypervirial theorem plays an important role in determining the properties of an atom in a molecule. It also enables one to relate an atomic property to a sum of bond contributions, as is now demonstrated. 

This expression insures that our integrations are organized such that the integrands are well behaved in the regions of interest. If we use the fact that the stress fields satisfy the equilibrium equations (i.e. Oijj = 0) in conjunction with the divergence theorem, this expression may be rewritten as... 

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