**6 Adiabatic-connection fluctuation-dissipation theory**

The adiabatic connection fluctuation-dissipation (AC-FDT) [12] technique will be explained in order to discover the exact exchange-correlation energy in RPA. It will serve as the starting point for introducing the random phase approximation because it provides a general formulation for the exact correlation energy.

#### **6.1 Adiabatic-connection (AC)**

The adiabatic connection (AC) is a way to express the exact exchangecorrelation energy function. The central idea in this approach is to build an interpolation Hamiltonian, which connects a Hamiltonian of an independent particle (reference Hamiltonian) *<sup>H</sup>*^ <sup>0</sup> <sup>¼</sup> *<sup>H</sup>*^ ð Þ *<sup>λ</sup>* <sup>¼</sup> <sup>0</sup> and the physical Hamiltonians (multibody Hamiltonian) *<sup>H</sup>*^ <sup>¼</sup> *<sup>H</sup>*^ ð Þ *<sup>λ</sup>* <sup>¼</sup> <sup>1</sup> , with *<sup>λ</sup>* being a connection parameter. The AC technique can be used to derive the total energy of the ground state of a Hamiltonian of multiple interacting bodies, in which a continuous set of Hamiltonians dependent on the coupling force (*λ*) is introduced by:

$$\begin{split} \hat{H}(\boldsymbol{\lambda}) &= \hat{H}\_0 + \lambda \hat{H}\_1(\boldsymbol{\lambda}) = \hat{T} + V\_{n\epsilon} + \hat{V}(\boldsymbol{\lambda}) + \lambda W\_{\epsilon\epsilon} \\ &= \sum\_{i=1}^{N} \left[ \frac{-1}{2} \nabla\_i^2 + \boldsymbol{\nu}\_{\boldsymbol{\lambda}}^{\text{ext}(i)} \right] + \sum\_{i>j=1}^{N} \frac{\boldsymbol{\lambda}}{|r\_i - r\_j|}. \end{split} \tag{42}$$

With N being the number of electrons, *vext <sup>λ</sup>* is an external potential with *vext <sup>λ</sup>*¼<sup>1</sup>ð Þ¼ *<sup>r</sup> vext*ð Þ*<sup>r</sup>* , being the external physical potential of the fully interactive system. Additionally, *vext <sup>λ</sup>* can be spatially non-local for *λ* 6¼ 1. Following that, the reference Hamiltonian, or the Hamiltonian for an independent particle specified by the Eq. (42) for *λ* ¼ 0, is of the mean field type, or is known in English as (Meanfield (MF)), i.e., a simple synthesis on a single-particle Hamiltonian:

$$\hat{H}\_0 = \sum\_{i=1}^{N} \left[ \frac{-\mathbf{1}}{2} \nabla\_i^2 + \boldsymbol{\nu}\_{k=0}^{\text{ext}}(r\_i) \right] = \sum\_{i=1}^{N} \left[ \frac{-\mathbf{1}}{2} \nabla\_i^2 + \boldsymbol{\nu}^{\text{ext}}(r\_i) + \boldsymbol{\nu}^{\text{MF}}(r\_i) \right]. \tag{43}$$

With *vMF* is an average field potential resulting from the electron–electron interaction. It can be the Hartree-Fock (HF) potential (*vHF*) or the Hartree-plus correlation-exchange potential (*vHxc*) in the DFT. According to the two Eqs. (42) and (43), the perturbative Hamiltonian becomes:

$$\begin{split} \hat{H}\_{1}(\lambda) &= \sum\_{i>j=1}^{N} \frac{\mathbf{1}}{|r\_{i} - r\_{j}|} + \frac{\mathbf{1}}{\lambda} \sum\_{i=1}^{N} \left[ v\_{\lambda}^{\text{ext}}(r\_{i}) - v\_{\lambda=0}^{\text{ext}}(r\_{i}) \right] \\ &= \sum\_{i>j=1}^{N} \frac{\mathbf{1}}{|r\_{i} - r\_{j}|} + \frac{\mathbf{1}}{\lambda} \sum\_{i=1}^{N} \left[ v\_{\lambda}^{\text{ext}}(r\_{i}) - v^{\text{ext}}(r\_{i}) - v^{\text{MF}}(r\_{i}) \right]. \end{split} \tag{44}$$

In the construction of the total energy, the ground state wave function ∣Ψ*λ*i is introduced for the system *λ*, such that

$$H(\lambda)|\Psi\_{\lambda}\rangle = E(\lambda)|\Psi\_{\lambda}\rangle. \tag{45}$$

Adopt the normalization condition, h i Ψ*λ*jΨ*<sup>λ</sup>* ¼ 1, the total interacting energy of the ground state can then be obtained using the theorem of Hellmann-Feynman [13]

$$E(\lambda = 1) = E\_0 + \int\_0^1 d\lambda \times \left\langle \Psi\_{\lambda} | \left( \hat{H}\_1(\lambda) + \lambda \frac{d\hat{H}\_1(\lambda)}{d\lambda} \right) | \Psi\_{\lambda} \right\rangle,\tag{46}$$

The energy of order zero is E0. It should be noted that the adiabatic connecting path chosen in Eq. (46) is not unique. In DFT, the path is chosen so that the electron density remains constant throughout the journey. This suggests a *λ* -dependency that *<sup>H</sup>*^ <sup>1</sup>ð Þ*<sup>λ</sup>* is not aware of.

### **6.2 The random phase approximation in the framework of adiabaticconnection fluctuation-dissipation theory**

We will quickly discuss the concept of RPA in the context of DFT, which has served as the foundation for current RPA computations. The total ground state energy for an interacting N electron system is a (implicit) function of the electron density n(r) in the Kohn-Sham approximation (KS-DFT) and can be divided into four terms:

$$E[n(r)] = T\_s[\psi(r)] + E\_{\text{ext}}[n(r)] + E\_H[n(r)] + E\_{\text{xc}}[\psi\_i(r)].\tag{47}$$

In the KS framework, the electron density is obtained from the single particle *<sup>ψ</sup>i*ð Þ*<sup>r</sup>* orbitals via *n r*ð Þ¼ <sup>P</sup>*occ <sup>i</sup> ψ<sup>i</sup>* j j ð Þ*r* 2 . Among the four terms of the Eq. (47) only *Eext*½ � *n r*ð Þ and *EH*½ � *n r*ð Þ are explicit functions of n(r). *Ts* is treated exactly in KS-DFT in terms of single particle *ψi*ð Þ*r* orbitals which are themselves functional of n(r). The unknown correlation-exchange (XC) energy term, which is approximated as an explicit functionality of n(r) (and its local gradients) in conventional functional functions (LDA and GGA) and as a function of *ψi*ð Þ*r* in more advanced functions, contains the complete complexity of many bodies (hybrid density functions, RPA, etc.). In DFT, several existing approximations of *Exc* can be categorized using a hierarchical approach called Jacob's scale [14]. But what if we wish to improve the accuracy of *Exc* in a larger number of systems? To that purpose, starting with the technically accurate manner of generating *Exc* using the AC technique mentioned above is instructive. As previously stated, the AC path is used in KS-DFT in order to maintain the correct electron density. Reducing the Eq. (46) for the total energy of the exact ground state E = E (*λ* = 1) to:

$$E = E\_0 + \int\_0^1 d\lambda \left< \Psi\_{\lambda} \frac{1}{2} \sum\_{i \neq j=1}^N \frac{1}{|r\_i - r\_j|} |\Psi\_{\lambda}\rangle \right> + \int\_0^1 d\lambda \left< \Psi\_{\lambda} \right| \sum\_{i=1}^N \frac{d}{d\lambda} v\_{\lambda}^{\text{ext}}(r\_i) |\Psi\_{\lambda}\rangle$$

$$= E\_0 + \frac{1}{2} \int\_0^1 d\lambda \int \left[ \int dr d\mathbf{r}' \times \left< \Psi\_{\lambda} |\dot{\mathbf{r}}(r) \left[ \frac{\hat{n}(r') - \delta(r - r')}{|r - r'|} \right] |\Psi\_{\lambda}\rangle + \int dr m(r) \left[ v\_{\lambda}^{\text{ext}}(r) - v\_{\lambda = 0}^{\text{ext}}(r) \right] \right]. \tag{48}$$

$$
\hat{m}(r) = \sum\_{i=1}^{N} \delta(r - r\_i),
\tag{49}
$$

*n r* ^ð Þ is the electron density operator and *n r*ð Þ¼ h i Ψ*λ*j*n r* ^ð ÞjΨ*<sup>λ</sup>* , for any 0⩽*λ*⩽1. For the reference state ∣Ψ0i of KS (given by the slater determinant of orbitals Ψ*i*ð Þ*r* occupied by a single particle, we get

$$E\_0 = \left\langle \Psi\_0 \Big| \sum\_{i=1}^N \left[ -\frac{1}{2} \nabla^2 + \nu\_{\lambda=0}^{ext}(r\_i) \right] \middle| \Psi\_0 \right\rangle = T\_s[\Psi\_i(r)] + \int dr n(r) \nu\_{\lambda=0}^{ext}(r). \tag{50}$$

$$E = T\_s[\Psi\_i(r)] + \left[ dr n(r) v\_{\lambda=1}^{\text{ext}} + \frac{1}{2} \int\_0^1 d\lambda \int \left[ \int dr dr' \langle \Psi\_{\lambda} | \frac{\hat{n}(r)[\hat{n}(r') - \delta(r - r')]}{|r - r'|} | \Psi\_{\lambda}. \tag{51} \right]$$

From the Eqs. (47) and (51), we obtained:

$$E\_H[n(r)] = \frac{1}{2} \int dr dr' \frac{n(r)n(r')}{|r - r'|}. \tag{52}$$

$$E\_{\rm ext}[n(r)] = \int dr n(r) v\_{\lambda=1}^{\rm ext}(r). \tag{53}$$

We get the formally exact correlation-exchange energy expression XC;

$$E\_{\rm xc} = \frac{1}{2} \int d\lambda \int \int dr dr' \frac{n\_{\rm xc}^{\lambda}(r, r')n(r)}{|r - r'|},\tag{54}$$

with *n<sup>λ</sup> xc r*,*r*<sup>0</sup> ð Þ is defined by

$$m\_{\infty}^{\lambda}(r, r') = \frac{\langle \Psi\_{\lambda} | \delta \hat{n}(r) \delta \hat{n}(r') | \Psi\_{\lambda} \rangle}{n(r)} - \delta(r - r'). \tag{55}$$

The mathematical expression for the so-called XC-hole is 55, with *δn*^ ¼ *n r* ^ð Þ� *n r*ð Þ denoting the fluctuation of the density operator *n r* ^ð Þ around its expectation value n(r). The hole (XC) is also related to the density-density correlation function, as shown by the Eq. (55). It illustrates how the presence of an electron at point r reduces the density of all other electrons at point *r*<sup>0</sup> in physical terms. The temperature fluctuation-dissipation (FDT) theorem is used to relate the densitydensity correlations (fluctuations) in the Eq. (55) to the response (dissipation) features of the system in the second step. In statistical physics, FDT is a powerful approach. It shows that the reaction of a system in thermodynamic equilibrium to a tiny external disturbance is the same as the response to spontaneous internal fluctuations in the absence of disturbance [15]. FDT is applicable to both thermal and quantum mechanical fluctuations and shows itself in a variety of physical

phenomena. A good example of the latter is the dielectric formulation of the manybody problem by Nozières and Pines [16]. The FDT at zero temperature performed at [16] is relevant in this situation.

$$
\langle \Psi\_{\lambda} | \delta \hat{n}(r) \delta \hat{n}(r') | \Psi\_{\lambda} \rangle = -\frac{1}{\pi} \int\_{0}^{\infty} d\alpha I m \chi^{\lambda}(r, r', \alpha), \tag{56}
$$

with *<sup>χ</sup><sup>λ</sup> <sup>r</sup>*,*r*<sup>0</sup> ð Þ ,*<sup>ω</sup>* , is the linear density-response function of the system. Using the Eqs. (54) and (55) and *v r*,*r*<sup>0</sup> ð Þ¼ <sup>1</sup> ∣*rr*0 ∣ , we arrive at the renamed ACFD expression for XC energy in DFT

$$\begin{split} E\_{\rm{xx}} &= \frac{1}{2} \int\_{0}^{1} d\lambda \left[ \int dr dr' v(r, r') \times \left[ -\frac{1}{\pi} \int\_{0}^{\infty} da l m \chi^{\lambda}(r, r', a) - \delta(r - r') n(r) \right] = \frac{1}{2\pi} \int\_{0}^{1} d\lambda \int \left[ \int dr dr' \right] \\ &\times v(r, r') \times \left[ -\frac{1}{\pi} \int\_{0}^{\infty} da^{\lambda}(r, r', i\nu) - \delta(r - r') n(r) \right]. \end{split} \tag{57}$$

The analytical structure of *<sup>χ</sup><sup>λ</sup> <sup>r</sup>*,*r*<sup>0</sup> ð Þ , *<sup>ω</sup>* and the fact that it becomes real on the imaginary axis are the reasons why the above frequency integration can be conducted along the imaginary axis. The problem of computing the energy XC on one of the response functions of a succession of fictional systems along the path AC is transformed by the expression ACFD in the Eq. (57), which must also be tackled in practice. RPA is a particularly basic approximation of the response function in this context:

$$\chi^{i}\_{\rm RPA}(r, r', i\alpha) = \chi^{0}(r, r', i\alpha) + \int dr\_{1} dr\_{2} \chi^{0}(r, r\_{1}, i\alpha) \times \mathcal{w}(r\_{1} - r\_{2}) \chi^{i}\_{\rm RPA}(r\_{2}, r', \alpha), \tag{58}$$

*<sup>χ</sup>*<sup>0</sup>ð Þ *<sup>r</sup>*,*r*1, *<sup>i</sup><sup>ω</sup>* , is the response function of independent particles of KS of the reference system *λ* ¼ 0 and is known explicitly in terms of orbitals *ψi*ð Þ*r* single particle (KS), orbital energies *ε<sup>i</sup>* and occupancy factors *fi* :

$$\chi^{0}(r,r',i\alpha) = \sum\_{\vec{\eta}} \frac{\left(f\_i - f\_j\right)\psi\_i^\*\left(r\right)\psi\_j(r)\psi\_j^\*\left(r'\right)\psi\_i(r')}{e\_i - e\_j - i\alpha}.\tag{59}$$

From the Eqs. (57) and (58), the energy XC in RPA can be split into an exchange-exact (EX) and the correlation term RPA:

$$E\_{\rm xc}^{\rm RPA} = E\_{\rm x}^{\rm EX} + E\_{\rm RPA}^{\rm c}. \tag{60}$$

$$E\_x^{\to} = -\sum\_{\vec{\eta}} f\_i f\_{\vec{\eta}} \int \left[ dr dr' \psi\_i^\*(r) \psi\_j(r) v(r, r') \psi\_j^\*(r') \psi\_i(r'). \tag{61}$$

$$E\_c^{\rm RPA} = \frac{1}{2\pi} \int\_0^\infty d\alpha \,\mathrm{Tr} \left[ \ln \left( 1 - \chi^0(i\alpha)\nu \right) + \chi^0(i\alpha)\nu \right]. \tag{62}$$

### **7. Approximation of pseudo-potentials**

The goal is to study the ground state of a system made up of nuclei, core electrons and valence electrons. The heart electrons are often closely linked to *The Density Functional Theory and Beyond: Example and Applications DOI: http://dx.doi.org/10.5772/intechopen.100618*

**Figure 1.**

*Schematic illustration of all-electron potential (solid lines) and pseudo-potential (broken lines) and their corresponding wave functions.*

nuclei, they are considered (frozen). This approximation makes it possible to develop the valence wave functions on a reduced number of plane waves having a kinetic energy lower than the energy of the cut-off (E *cut* E*cut* ⩾ <sup>ℏ</sup><sup>2</sup> <sup>2</sup>*<sup>m</sup>* j j *<sup>K</sup>* <sup>þ</sup> *<sup>G</sup>* <sup>2</sup> ), which allows correct treatment of the problem depends on the pseudo-potential used and the system studied. It consists in replacing the ionic potential *Vel*,*nu* by a pseudopotential *Vps* (see **Figure 1**) which acts on a set of wave pseudo-functions instead and places true wave functions and having the same eigenstates in the atomic Schrdinger equation. This idea has been developing since the end of the 1950s. This potential is constructed so as to reproduce the scattering properties for the true valence wave functions, while ensuring that the pseudo-wave function does not have a node in the core region defined by a cutoff radius *rc* which is optimized for each orbital. Beyond the core region, the pseudopotential is reduced to the ionic potential so that the pseudo-wave function is equal to the true wave function. The use of a pseudo-potential reduces on the one hand the number of electrons considered in the problem by taking into account only the valence electrons and on the other hand makes it possible to restrict the base of plane waves for the electrons of valence by eliminating most of the oscillations of the wave functions in the heart region.
