Legendre transformation of the energy

We inspect now the Legendre transformation of the energy with respect to volume, the free energy, or Helmholtz energy F ... 

The reason for this classification arises from the various Legendre transformations of the energy. However, these transformations allow for a more subtle classification. 

It becomes now clear that the susceptibilities describe the response of the extensive variable on a change of the associated intensive variable. At first glance, it seems to be sufficient to place only one index, because the nature of the extensive variable can be found out. The generalized susceptibilities cannot be derived readily from the energy equation, but are accessible from the Legendre transformations of the energy. For instance, for... 

Equation (6.27) is the Legendre transformation of the energy with respeet to entropy, volume, and water. Taking the constraints dV = 0 and d 2 = 0 into consideration, in equilibrium dB = 0. Erom this condition, immediately dT = 0 and d/Xj = 0 follow. This is the same, as obtained in Eq. (6.25), however in a more simple way. 

We recall that the Gibbs - Duhem equation is the complete Legendre transformation of the energy equation, U S, V, n). A relation between the variables T, p, pL emerges. Therefore, the set of the independent variables consists of any two variables in the set T, p, pL. For example, we could rearrange Eq. (7.15) into... 

The inverse of H determines the geometric compliance matrix (Nalewajski, 1993, 1995, 1997, 1999, 2000, 2002b, 2006a,b Nalewajski and Korchowiec, 1997 Nalewajski et al., 1996, 2008) describing the open system in the Qi,F)-representation. The relevant thermodynamic potential is defined by the total Legendre transform of the system BO potential, which replaces the state-parameters (N, Q) with their energy conjugates (/a, F), respectively ... 

Finally, the remaining (/a, Q) representation describing the equilibrium state of an externally open molecular system with the frozen nuclear framework is examined. The relevant partial Legendre transform of the total electronic energy, which replaces N by /a in the list of independent state-parameters, defines the BO grand-potential ... 

Here (N, F) (see Equation 30.17) stands for the Legendre transform of the BO potential energy surface W(N, Q), in which the nuclear-position coordinates Q are replaced by the corresponding forces F in the list of the parameters of state. Indeed, for the fixed number of electrons N,... 

Legendre transforms of the electronic energy and their derivatives 141... 

LEGENDRE TRANSFORMS OF THE ELECTRONIC ENERGY AND THEIR DERIVATIVES... 

Step 2. Use the total differential of specific enthalpy in terms of its natural variables, via Legendre transformation of the internal energy from classical thermodynamics, to re-express the pressure gradient in the momentum balance in terms of enthalpy, entropy, and mass fractions. Then, write the equation of change for kinetic energy in terms of specific enthalpy and entropy. 

Finally, let us examine the relevant transformations in the remaining (jj, Q)-representation, the "thermodynamic" potential for which is given by the following Legendre transform of the system energy ... 

In order to study the effect of the external magnetic field on the swelling equilibrium we rewrite Equation [5.13] by introducing a new function G-/i HM, which is a Legendre transformation of the Gibbs free energy function of G. 

A familiar example of Legendre transformation is the relationship that exists between the Lagrangian and Hamiltonian functions of classical mechanics [17]. In thermodynamics the simplest application is to the internal energy function for constant mole number U(S, V), with the differentials... 

The use of the electron density and the number of electrons as a set of independent variables, in contrast to the canonical set, namely, the external potential and the number of electrons, is based on a series of papers by Nalewajski [21,22]. A.C. realized that this choice is problematic because one cannot change the number of electrons while the electron density remains constant. After several attempts, he found that the energy per particle possesses the convexity properties that are required by the Legendre transformations. When the Legendre transform was performed on the energy per particle, the shape function immediately appeared as the conjugate variable to the external potential, so that the electron density was split into two pieces that can be varied independent the number of electrons and their distribution in space. 

We adopt an alternative route to the distribution function theory. The approach is based on the density-functional theory. In this approach, the change of variables is conducted through Legendre transform from the solute-solvent interaction potential function to the solute-solvent distribution function or the solvent density around the solute. The (solvation) free energy is then expressed approximately by expanding the corresponding Legendre-transformed function with respect to the distribution function to some low order. 

The successive Legendre transformations of E yield a state function, G, for which the natural variables p and T, are both intensive properties (independent of the size of the system). Furthermore, for dp = 0 and dT = 0 (isobaric, isothermal system), the state of the system is characterized by dG. This is clearly convenient for chemical applications under atmospheric pressure, constant-temperature conditions (or at any other isobaric, isothermal conditions). Then, in place of equation (21) for internal energy variation, we state the conditions for irreversible or reversible processes in terms of the Gibbs energy as... 

For T, P = constant, the Gibbs energy or Gibbs function, G, representing another partial Legendre transform of U, is properly used ... 

The natnral variables S and V of [/ are transformed to T and V by Legendre transformation of U to the energy fnnction A, called the Helmholtz free energy, or Helmholtz energy. By definition. 

The corresponding differentials of the system electronic energy in the subsystem resolution and its Legendre transforms of equations (66)-(68) are ... 

By this rearrangement, S is the function and U is one argument. Note that the entropy S(U,V) is a. function of the energy and the volume. Next, we perform the Legendre transformation with respect to U ... 

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