Energy identity

Substitution of these expressions into (43) leads to the energy identity... 

The energy identity. We study the problem of stability of scheme (1) by the method of the energy inequalities involving as the necessary manipulations the inner product of both sides of equation (1) with 2rj/ = 2(y — y) ... 

Substituting (12) into (11) we obtain the energy identity for scheme (1) ... 

Sufficiency. Granted condition (14), the energy identity for problem (la) (with ip — 0)... 

Stability with respect to the initial data in Hb- Let us write down the second energy identity for scheme (la) assuming B also to be a self-adjoint operator 5 = 5 > 0. At the first stage we take the inner product of (la) and 2ry ... 

Remark One fails to prove Theorem 3 on account of the energy identity (20). However, the method developed in Section 6 may be of assistance in achieving this aim. In the case D — B conditions (36) are equivalent to the condition i — tB A < p or to being nonnegative of the functional... 

Proof The energy identity (13) is involved at the first stage. Estimation of it.s right-hand side 2r(9j, y ) is stipulated by successive use of the generalized Cauchy-Bunyakovskii inequality and the e-inequality... 

The energy identity (13) with A — A t) is the starting point in special investigations. To obtain a recurrence inequality, we should modify the expression... 

The basic energy identity. We will carry out the derivation of the energy identity for the three-layer scheme (1) with variable operators A = A(t), B = B t) and R = R t). This identity is aimed at achieving a priori estimates expressing the stability of a scheme with respect to the initial data and right-hand side. 

We are going to show that this scheme is stable for 7-11 4 < 1 and for it the energy identity holds true ... 

Proof The energy identity (13) for scheme (48) takes for now the form... 

Finally, let us note some interesting identities for other thermodynamic potentials that follow from Equation (6.31). From the energy identity U = TS — PV + ix tj and basic thermodynamic definitions, we can readily infer that... 

The initial effect in IR absorption is the transition of a molecule from a ground state to a vibration excited state by an absorption process of an IR photon with energy identical to the difference between the energies of the vibration ground and excited states. The opposite process, that is, IR emission, takes place when a molecule in the excited state emits a photon during the transition to a ground state. 

In most formulations a is treated as essentially an atomic parameter that is chosen to give a local density or Xa total energy identical to the Har-tree-Fock value for each atom of the molecule (Schwarz, 1972, 1974), although other criteria for choice of a have been considered (Smith and Sabin, 1978). The value of a obtained by matching to atomic Hartree-Fock energies is around 0.7 for all atoms, and small variations in a cause little change in properties. 

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