Perturbation Theory

Abstract

Perturbation theory (PT) represents one of the bridges that takes us from a simpler, exactly solvable (unperturbed) problem to a corresponding real (perturbed) problem by expressing its solutions as a series expansion in a suitably chosen “small” parameter ε in such a way that the problem reduces to the unperturbed problem when ε = 0. It originated in classical mechanics and eventually developed into an important branch of applied mathematics enabling physicists and engineers to obtain approximate solutions of various systems of differential equations 1 ; 2 ; 3 ; 4 ; 5 . For the problems of atomic and molecular structure and dynamics, the perturbed problem is usually given by the time-independent or time-dependent Schrödinger equation 6 ; 7 ; 8 ; 9 ; 10 .

1 Matrix Perturbation Theory (PT)

A prototype of a time-independent PT considers an eigenvalue problem for the Hamiltonian H of the form

$$H=H_{0}+V\;,\quad V=\sum_{i=1}^{\infty}\varepsilon^{i}V_{i}\;,$$

(5.1)

acting in a (finite-dimensional) Hilbert space 𝒱_n, assuming that the spectral resolution of the unperturbed operator H₀ is known; i.e.,

$$H_{0}=\sum_{i}\omega_{i}P_{i}\;,\quad P_{i}P_{j}=\delta_{ij}P_{j}\;,\quad\sum_{i}P_{i}=I\;,$$

(5.2)

where ω_i are distinct eigenvalues of H₀, the P_i form a complete orthonormal set of Hermitian idempotents, and I is the identity operator on 𝒱_n. The PT problem for H can then be formulated within the Lie algebra 𝒜 (Sect. 3.2) generated by H₀ and V 11 ; 12 .

1.1 Basic Concepts

Define the diagonal part ⟨X⟩ of a general operator X ∈ 𝒜 by

$$\langle X\rangle=\sum_{i}P_{i}XP_{i}$$

(5.3)

and recall that the adjoint action of X ∈ 𝒜, ad X : 𝒜 → 𝒜 is defined by

$$\mathrm{ad}\,X(Y)=[X,Y]\;,\quad(\forall\;Y\in\mathcal{A})\;,$$

(5.4)

where the square bracket denotes the commutator. The key problem of PT is the “inversion” of this operation, i.e., the solution of the equation 11 ; 12 ; 13

$$\mathrm{ad}\,H_{0}(X)\equiv[H_{0},X]=Y\;.$$

(5.5)

Assuming that ⟨Y⟩ = 0, then

$$X=R(Y)+\langle A\rangle\;,$$

(5.6)

where A ∈ 𝒜 is arbitrary, and

$$R(Y)=\sum_{i\neq j}\Delta_{ij}^{-1}P_{i}YP_{j}\;,$$

(5.7)

with Δ_ij = ω_i − ω_j, represents the solution of Eq. (5.5) with the vanishing diagonal part ⟨R(Y)⟩ = 0.

1.2 Level-Shift Operators

To solve the PT problem for H, Eq. (5.1) we search for a unitary level-shift transformation U 11 ; 12 , U^†U = UU^† = I,

$$UHU^{\dagger}=U(H_{0}+V)U^{\dagger}=H_{0}+E\;,$$

(5.8)

where the level-shift operator E satisfies the condition

$$E=\langle E\rangle\;.$$

(5.9)

To guarantee the unitarity of U we express it in the form

$$U=\,\mathrm{e}^{G}\;,\quad G^{\dagger}=-G\;,\quad\langle G\rangle=0\;.$$

(5.10)

Using the Hausdorff formula

$$\,\mathrm{e}^{A}B\,\mathrm{e}^{-A}=\sum_{k=0}^{\infty}(k!)^{-1}(\mathrm{ad}\,A)^{k}B$$

(5.11)

and defining the operator

$$F=[H_{0},G]\;,$$

(5.12)

we find from Eq. (5.8) that

$$\begin{aligned}\displaystyle E+F&\displaystyle=V+\frac{1}{2}[G,V+E]\\ \displaystyle&\displaystyle\quad+\sum_{k=2}^{\infty}(k!)^{-1}B_{k}(\mathrm{ad}\,G)^{k}(V-E)\;,\end{aligned}$$

(5.13)

where we used the identity 14

$$\left(\sum_{k=0}^{\infty}\frac{B_{k}}{k!}X^{k}\right)\left(\sum_{k=0}^{\infty}\frac{1}{(k+1)!}X^{k}\right)=I\;,$$

(5.14)

and B_k designates the Bernoulli numbers 15

$$\begin{aligned}\displaystyle B_{0}&\displaystyle=1\;,\quad B_{1}=-\frac{1}{2}\;,\quad B_{2}=\frac{1}{6}\;,\\ \displaystyle B_{2k+1}&\displaystyle=0\quad(k\geq 1)\;,\\ \displaystyle B_{4}&\displaystyle=\frac{1}{30}\;,\quad B_{6}=\frac{1}{42}\;,\quad\text{{etc.}}\end{aligned}$$

(5.15)

1.3 General Formalism

Introducing the PT expansion for relevant operators,

$$X=\sum_{i=1}^{\infty}\varepsilon^{i}X_{i}\;,\quad X=E,F,G\;;\quad F_{i}=[H_{0},G_{i}]\;,$$

(5.16)

Eq. (5.13) leads to the following system of equations

$$\begin{aligned}\displaystyle E_{1}+F_{1}&\displaystyle=V_{1}\;,\\ \displaystyle E_{2}+F_{2}&\displaystyle=V_{2}+\frac{1}{2}[G_{1},V_{1}+E_{1}]\;,\\ \displaystyle E_{3}+F_{3}&\displaystyle=V_{3}+\frac{1}{2}[G_{1},V_{2}+E_{2}]\\ \displaystyle&\displaystyle\quad+\frac{1}{2}[G_{2},V_{1}+E_{1}]\\ \displaystyle&\displaystyle\quad+\frac{1}{12}[G_{1},[G_{1},V_{1}-E_{1}]]\;,\quad\text{{etc.}}\;,\end{aligned}$$

(5.17)

which can be solved recursively for E_i and G_i by taking their diagonal part and applying operator R, Eq. (5.7), since

$$\begin{aligned}\displaystyle\langle E_{i}\rangle&\displaystyle=E_{i}\;,&\displaystyle\langle G_{i}\rangle&\displaystyle=\langle F_{i}\rangle=0\;,\\ \displaystyle RF_{i}&\displaystyle=G_{i}\;,&\displaystyle RE_{i}&\displaystyle=0\;.\end{aligned}$$

(5.18)

We thus get

$$\begin{aligned}\displaystyle E_{1}&\displaystyle=\langle V_{1}\rangle\;,\\ \displaystyle E_{2}&\displaystyle=\langle V_{2}\rangle+\frac{1}{2}\langle[RV_{1},V_{1}]\rangle\;,\\ \displaystyle E_{3}&\displaystyle=\langle V_{3}\rangle+\langle[RV_{1},V_{2}]\rangle\\ \displaystyle&\displaystyle\quad+\frac{1}{6}\langle[RV_{1},[RV_{1},2V_{1}+E_{1}]]\rangle\;,\quad\text{{etc.}}\;,\end{aligned}$$

(5.19)

and

$$\begin{aligned}\displaystyle G_{1}&\displaystyle=RV_{1}\;,\\ \displaystyle G_{2}&\displaystyle=RV_{2}+\frac{1}{2}R[RV_{1},V_{1}+E_{1}]\;,\quad\text{{etc.}}\end{aligned}$$

(5.20)

Since

$$\begin{aligned}\displaystyle\langle R(X)\rangle&\displaystyle=R\langle X\rangle=0\;,&\displaystyle\langle R(X)Y\rangle&\displaystyle=-\langle XR(Y)\rangle\;,&\displaystyle\\ \displaystyle R(X\langle Y\rangle)&\displaystyle=R(X)\langle Y\rangle,&\displaystyle R(\langle X\rangle Y)&\displaystyle=\langle X\rangle R(Y)\;,&\displaystyle\end{aligned}$$

(5.21)

these relationships can be transformed to a more conventional form

$$\begin{aligned}\displaystyle E_{2}&\displaystyle=\langle V_{2}\rangle-\langle V_{1}RV_{1}\rangle\;,\\ \displaystyle E_{3}&\displaystyle=\langle V_{3}\rangle-\langle V_{1}RV_{2}\rangle-\langle V_{2}RV_{1}\rangle\\ \displaystyle&\displaystyle\quad+\frac{1}{6}\langle R(V_{1})R(V_{1})[2V_{1}+\langle V_{1}\rangle]\rangle\\ \displaystyle&\displaystyle\quad-\frac{1}{3}\langle R(V_{1})[2V_{1}+\langle V_{1}\rangle]R(V_{1})\rangle\\ \displaystyle&\displaystyle\quad+\frac{1}{6}\langle[2V_{1}+\langle V_{1}\rangle]R(V_{1})R(V_{1})\rangle\;,\quad\text{{etc.}}\end{aligned}$$

(5.22)

However, in this way certain nonphysical terms arise that exactly cancel when the commutator form is employed (Sect. 5.3.7).

1.4 Nondegenerate Case

In the nondegenerate case, when $P_{i}=|i\rangle\langle i|$, with | i⟩ representing the eigenvector of H₀ associated with the eigenvalue ω_i, the level-shift operator is diagonal, and its explicit PT expansion (as well as that for the corresponding eigenvectors) is readily obtained from Eqs. (5.19) and (5.20). Writing x_ij for the matrix element ⟨i | X | j⟩, we get

$$\begin{aligned}\displaystyle(e_{1})_{ii}&\displaystyle=(v_{1})_{ii}\;,\\ \displaystyle(e_{2})_{ii}&\displaystyle=(v_{2})_{ii}-{\sum_{j}}^{\prime}\;\frac{(v_{1})_{ij}(v_{1})_{ji}}{\Delta_{ji}}\;,\\ \displaystyle(e_{3})_{ii}&\displaystyle=(v_{3})_{ii}-{\sum_{j}}^{\prime}\;\frac{(v_{1})_{ij}(v_{2})_{ji}+(v_{2})_{ij}(v_{1})_{ji}}{\Delta_{ji}}\\ \displaystyle&\displaystyle\quad+{\sum_{j,k}}^{\prime}\frac{(v_{1})_{ij}(v_{1})_{jk}(v_{1})_{ki}}{\Delta_{ji}\Delta_{ki}}\\ \displaystyle&\displaystyle\quad-\;(v_{1})_{ii}\;{\sum_{j}}^{\prime}\frac{(v_{1})_{ij}(v_{1})_{ji}}{\Delta^{2}_{ji}}\;,\quad\text{{etc.}},\end{aligned}$$

(5.23)

the prime on the summation symbols indicating that the terms with the vanishing denominator are to be deleted.

Note that in contrast to PT expansions, which directly expand the level-shift transformation $U,\;U=1+\varepsilon U_{1}+\varepsilon^{2}U_{2}+\cdots$, the above Lie algebraic formulation has the advantage that U stays unitary in every order of PT. This is particularly useful in spectroscopic applications, such as line broadening.

2 Time-Independent Perturbation Theory

For stationary problems, particularly those arising in atomic and molecular electronic structure studies relying on ab initio model Hamiltonians, the PT of Sect. 5.1 can be given a more explicit form that avoids a priori the nonphysical, size inextensive terms 7 ; 8 ; 9 ; 10 ; 16 ; 17 ; 18 .

2.1 General Formulation

We wish to find the eigenvalues and eigenvectors of the full (perturbed) problem

$$K|\Psi_{i}\rangle\equiv(K_{0}+W)|\Psi_{i}\rangle=k_{i}|\Psi_{i}\rangle\;,$$

(5.24)

assuming we know those of the unperturbed problem

$$K_{0}|\Phi_{i}\rangle=\kappa_{i}|\Phi_{i}\rangle,\quad\langle\Phi_{i}|\Phi_{j}\rangle=\delta_{ij}\;.$$

(5.25)

For simplicity, we restrict ourselves to the nondegenerate case (κ_i ≠ κ_j   if   i ≠ j) and consider only the first-order perturbation (Eq. (5.1), $\varepsilon V_{1}\equiv W,\;V_{i}=0$   for   i ≥ 2). Of course, K and K₀ are Hermitian operators acting in a Hilbert space, which, in ab initio applications, is finite-dimensional.

Using the intermediate normalization for | Ψ_i⟩,

$$\langle\Psi_{i}|\Phi_{i}\rangle=1\;,$$

(5.26)

the asymmetric energy formula gives

$$k_{i}=\kappa_{i}+\langle\Phi_{i}|W|\Psi_{i}\rangle\;.$$

(5.27)

The idempotent Hermitian projectors

$$P_{i}=|\Phi_{i}\rangle\langle\Phi_{i}|,\;Q_{i}=P_{i}^{\perp}=1-P_{i}=\!\!\sum_{j(\neq i)}\!|\Phi_{j}\rangle\langle\Phi_{j}|{}$$

(5.28)

commute with K₀, so that

$$(\lambda-K_{0})Q_{i}|\Psi_{i}\rangle=Q_{i}(\lambda-k_{i}+W)|\Psi_{i}\rangle\;,$$

(5.29)

λ being an arbitrary scalar (note that we write λI simply as λ). Since the resolvent (λ − K₀)⁻¹ of K₀ is nonsingular on the orthogonal complement of the i-th eigenspace, we get

$$Q_{i}|\Psi_{i}\rangle=|\Psi_{i}\rangle-|\Phi_{i}\rangle=R_{i}(\lambda)(\lambda-k_{i}+W)|\Psi_{i}\rangle\;,$$

(5.30)

where

$$\begin{aligned}\displaystyle R_{i}\equiv R_{i}(\lambda)&\displaystyle=(\lambda-K_{0})^{-1}Q_{i}\\ \displaystyle&\displaystyle=Q_{i}(\lambda-K_{0})^{-1}=\sum_{j(\neq i)}\frac{|\Phi_{j}\rangle\langle\Phi_{j}|}{\lambda-\kappa_{j}}\;,\end{aligned}$$

(5.31)

assuming (λ ≠ κ_j). Iterating this relationship, we get prototypes of the desired PT expansion for | Ψ_i⟩,

$$|\Psi_{i}\rangle=\sum_{n=0}^{\infty}\;[R_{i}(\lambda-k_{i}+W)]^{n}|\Phi_{i}\rangle\;,$$

(5.32)

and, from Eq. (5.27), for k_i,

$$k_{i}=\kappa_{i}+\sum_{n=0}^{\infty}\langle\Phi_{i}|W[R_{i}(\lambda-k_{i}+W)]^{n}|\Phi_{i}\rangle\;.$$

(5.33)

2.2 Brillouin–Wigner and Rayleigh–Schrödinger PT (RSPT)

So far, the parameter λ was arbitrary as long as λ ≠ κ_j  (j ≠ i). The following two choices lead to the two basic types of many-body perturbation theory (MBPT):

Brillouin–Wigner (BW) PT

Setting λ = k_i gives

$$\begin{aligned}k_{i} & =\kappa_{i}+\sum_{n=0}^{\infty}\langle\Phi_{i}|W\Big(R_{i}^{(\mathrm{{BW}})}W\Big)^{n}|\Phi_{i}\rangle\;,\end{aligned}$$

(5.34)

$$\begin{aligned}|\Psi_{i}\rangle & =\sum_{n=0}^{\infty}\left(R_{i}^{(\mathrm{{BW}})}W\right)^{n}|\Phi_{i}\rangle\;,\end{aligned}$$

(5.35)

where

$$R_{i}^{(\mathrm{{BW}})}=\sum_{j(\neq i)}\;\frac{|\Phi_{j}\rangle\langle\Phi_{j}|}{k_{i}-\kappa_{j}}\;.$$

(5.36)

Rayleigh–Schrödinger (RS) PT

Setting λ = κ_i gives

$$\begin{aligned}k_{i} & =\kappa_{i}+\sum_{n=0}^{\infty}\langle\Phi_{i}|W\Big[R_{i}^{(\mathrm{RS})}(\kappa_{i}-k_{i}+W)\Big]^{n}|\Phi_{i}\rangle\;,\end{aligned}$$

(5.37)

$$\begin{aligned}|\Psi_{i}\rangle & =\sum_{n=0}^{\infty}\left[R_{i}^{(\mathrm{RS})}(\kappa_{i}-k_{i}+W)\right]^{n}|\Phi_{i}\rangle\;,\end{aligned}$$

(5.38)

where now

$$R_{i}\equiv R_{i}^{(\mathrm{RS})}=\sum_{j(\neq i)}\;\frac{|\Phi_{j}\rangle\langle\Phi_{j}|}{\kappa_{i}-\kappa_{j}}\;.$$

(5.39)

The main distinction between these two PTs lies in the fact that the BW form has the exact eigenvalues appearing in the denominators and, thus, leads to polynomial expressions for k_i. Although these are not difficult to solve numerically, since the eigenvalues are separated, the resulting energies are never size extensive and, thus, unusable for extended systems. They are also unsuitable for finite systems when the particle number changes, as in various dissociation processes. From now on, we thus investigate only the RSPT, which yields a fully size-extensive theory.

2.3 Bracketing Theorem and RSPT

Expressions Eqs. (5.37) and (5.38) are not explicit, since they involve the exact eigenvalues k_i on the right-hand side. To achieve an order-by-order separation, set

$$k_{i}\equiv k=\sum_{j=0}^{\infty}k^{(j)}\;,\quad|\Psi_{i}\rangle\equiv|\Psi\rangle=\sum_{j=0}^{\infty}|\Psi^{(j)}\rangle\;,$$

(5.40)

where the superscript (j) indicates the j-th-order in the perturbation W. We only consider the eigenvalue expressions, since the corresponding eigenvectors are readily recovered from them by removing the bra state and the first interaction W (Eqs. (5.37) and (5.38)). We also simplify the mean value notation writing for a general operator X,

$$\langle X\rangle\equiv\langle\Phi_{i}|X|\Phi_{i}\rangle\;.$$

(5.41)

Substituting the first expansion Eq. (5.40) into Eq. (5.37) and collecting the terms of the same order in W, we get

$$\begin{aligned}\displaystyle k^{(0)}&\displaystyle=\kappa_{i}\;,\\ \displaystyle k^{(1)}&\displaystyle=\langle W\rangle\;,\\ \displaystyle k^{(2)}&\displaystyle=\langle WRW\rangle\;,\\ \displaystyle k^{(3)}&\displaystyle=\big\langle W(RW)^{2}\big\rangle-\langle W\rangle\big\langle WR^{2}W\big\rangle\;,\\ \displaystyle k^{(4)}&\displaystyle=\big\langle W(RW)^{3}\big\rangle\\ \displaystyle&\displaystyle\quad-\langle W\rangle\big(\big\langle WR(RW)^{2}\big\rangle+\big\langle(WR)^{2}RW\big\rangle\big)\\ \displaystyle&\displaystyle\quad+\langle W\rangle^{2}\big\langle WR^{3}W\big\rangle-\langle WRW\rangle\big\langle WR^{2}W\big\rangle\;,\quad\text{{etc.}}\end{aligned}$$

(5.42)

The general expression has the form

$$k^{(n)}=\big\langle W(RW)^{n-1}\big\rangle+\mathcal{R}^{(n)}\;,$$

(5.43)

the first term on the right-hand side being referred to as the principal n-th-order term, while ℛ⁽ⁿ⁾ designates the so-called renormalization terms that are obtained by the bracketing theorem 16 ; 19 as follows:

• 1. Insert the bracketings ⟨⋯⟩ around the W, WRW, …, WR⋯RW operator strings of the principal term in all possible ways.

• 2. Bracketings involving the rightmost and/or the leftmost interaction vanish.

• 3. The sign of each bracketed term is given by $(-1)^{n_{{\mathrm{B}}}}$, where n_B is the number of bracketings.

• 4. Bracketings within bracketings are allowed, e.g., ⟨WR⟨WR⟨W⟩RW⟩RW⟩ = ⟨W⟩⟨WR²W⟩².

• 5. The total number of bracketings (including the principal term) is $(2n-2)!/[n!(n-1)!]$.

3 Fermionic Many-Body Perturbation Theory (MBPT)

3.1 Time-Independent Wick's Theorem

The development of an explicit MBPT formalism is greatly facilitated by the exploitation of the time-independent version of Wick's theorem. This version of the theorem expresses an arbitrary product of creation (a ^†_μ ) and annihilation (a_μ) operators (Chap. 6) as a normal product (relative to | Φ₀⟩) and as normal products with all possible contractions of these operators 16 ; 17 ; 18 ,

$$\begin{aligned}\displaystyle&\displaystyle x_{1}x_{2}\cdots x_{k}=N[x_{1}x_{2}\cdots x_{k}]+\Sigma N[x_{1}x_{2}\cdots\cdots x_{k}]\;,\kern-52.068543pt\raisebox{7.0pt}{\vrule width 0.7pt depth 2.0pt height 0.7pt\vrule width 16.0pt depth 0.0pt height 0.7pt\vrule width 0.7pt depth 2.0pt height 0.7pt}\kern-7.113189pt\raisebox{9.0pt}{\vrule width 0.7pt depth 4.0pt height 0.7pt\vrule width 16.0pt depth 0.0pt height 0.7pt\vrule width 0.7pt depth 4.0pt height 0.7pt}\\ \displaystyle&\displaystyle(x_{i}=a_{\mu_{i}}^{\dagger}\;\;\text{ or}\;\;x_{i}=a_{\mu_{i}})\;,\end{aligned}$$

(5.44)

where

$$\begin{aligned}\displaystyle&\displaystyle a_{\mu}^{\dagger}a_{\nu}^{\dagger}={a_{\mu}a_{\nu}}=0\;,\kern-79.098661pt\raisebox{10.0pt}{\vrule width 0.7pt depth 2.0pt height 0.7pt\vrule width 11.0pt depth 0.0pt height 0.7pt\vrule width 0.7pt depth 2.0pt height 0.7pt}\kern 22.762205pt\raisebox{9.0pt}{\vrule width 0.7pt depth 2.0pt height 0.7pt\vrule width 10.0pt depth 0.0pt height 0.7pt\vrule width 0.7pt depth 2.0pt height 0.7pt}\\ \displaystyle&\displaystyle{a_{\mu}^{\dagger}a_{\nu}}=h(\mu)\delta_{\mu\nu}\;,\quad{a_{\mu}a_{\nu}^{\dagger}}=p(\mu)\delta_{\mu\nu}\;,\kern-159.335433pt\raisebox{10.0pt}{\vrule width 0.7pt depth 2.0pt height 0.7pt\vrule width 11.0pt depth 0.0pt height 0.7pt\vrule width 0.7pt depth 2.0pt height 0.7pt}\kern 73.40811pt\raisebox{9.0pt}{\vrule width 0.7pt depth 2.0pt height 0.7pt\vrule width 10.0pt depth 0.0pt height 0.7pt\vrule width 0.7pt depth 2.0pt height 0.7pt}\end{aligned}$$

(5.45)

and

$$\begin{aligned}\displaystyle&\displaystyle h(\mu)=1\;,\;p(\mu)=0\;\text{if }|\mu\rangle\text{ is occupied in }|\Phi_{0}\rangle\\ \displaystyle&\displaystyle\hbox{(hole states),}\\ \displaystyle&\displaystyle h(\mu)=0\;,\;p(\mu)=1\;\text{if }|\mu\rangle\text{ is unoccupied in }|\Phi_{0}\rangle\\ \displaystyle&\displaystyle\hbox{(particle states)\;.}\end{aligned}$$

(5.46)

The N-product with contractions is defined as a product of individual contractions times the N-product of uncontracted operators (defining N[∅] ≡ 1 for an empty set) with the sign given by the parity of the permutation reordering the operators into their final order.

Note that the Fermi vacuum mean value of an N-product vanishes unless all operators are contracted. Thus, ⟨x₁x₂⋯x_k⟩ is given by the sum over all possible fully contracted terms (vacuum terms). Similar rules follow for the expressions of the type (x₁x₂⋯x_k) | Φ⟩. Moreover, if some operators on the left-hand side of Eq. (5.44) are already in the N-product form, all the terms involving contractions between these operators vanish.

3.2 Normal Product Form of PT

Consider the eigenvalue problem for a general ab initio or semiempirical electronic Hamiltonian H with one and two-body components Z and V, namely,

$$\begin{aligned}\displaystyle H|\Psi_{i}\rangle&\displaystyle=E_{i}|\Psi_{i}\rangle\;,\\ \displaystyle H&\displaystyle=Z+V=\sum_{i}z(i)+\sum_{i<j}v(i,j)\;,\end{aligned}$$

(5.47)

and a corresponding unperturbed problem

$$\begin{aligned}\displaystyle H_{0}|\Phi_{i}\rangle&\displaystyle=\varepsilon_{i}|\Phi_{i}\rangle\;,\\ \displaystyle H_{0}&\displaystyle=Z+U\;,\quad\langle\Phi_{i}|\Phi_{j}\rangle=\delta_{ij}\;,\end{aligned}$$

(5.48)

with U representing some approximation to V. In the case that U is also a one-electron operator, U = Σ_iu(i), the unperturbed problem Eq. (5.48) is separable and reduces to a one-electron problem,

$$(z+u)|\mu\rangle=\omega_{\mu}|\mu\rangle\;,$$

(5.49)

which is assumed to be solved. Choosing the orthonormal spin orbitals { | μ⟩} as a basis of the second quantization representation Sect. 6.1, the N-electron solutions of Eq. (5.48) can be represented as

$$\begin{aligned}|\Phi_{i}\rangle & =a_{\mu_{1}}^{\dagger}\;a_{\mu_{2}}^{\dagger}\cdots a_{\mu_{N}}^{\dagger}|0\rangle\;,\end{aligned}$$

(5.50)

$$\begin{aligned}\varepsilon_{i} & =\sum_{j=1}^{N}\;\omega_{\mu_{j}}\;,\end{aligned}$$

(5.51)

the state label i representing the occupied spin orbital set $\{\mu_{1},\mu_{2},\ldots,\mu_{N}\}$, while the one and two-body operators take the form

$$\begin{aligned}X & =\sum_{\mu,\nu}\;\langle\mu|x|\nu\rangle a_{\mu}^{\dagger}a_{\nu}\;,\quad X=Z,U;\;x=z,u\;,\end{aligned}$$

(5.52)

$$\begin{aligned}V & =\frac{1}{2}\sum_{\mu,\nu,\sigma,\tau}\langle\mu\nu|v|\sigma\tau\rangle a_{\mu}^{\dagger}a_{\nu}^{\dagger}a_{\tau}a_{\sigma}\;.\end{aligned}$$

(5.53)

Considering, for simplicity, a nondegenerate ground state $|\Phi\rangle\equiv|\Phi_{0}\rangle=a_{1}^{\dagger}a_{2}^{\dagger}\cdots a_{N}^{\dagger}|0\rangle$, referred to as a Fermi vacuum, we define the normal product form of these operators relative to | Φ⟩

$$\begin{aligned} & \begin{aligned}\displaystyle X_{N}&\displaystyle\equiv X-\langle X\rangle=\sum_{\mu,\nu}\langle\mu|x|\nu\rangle N\big[a_{\mu}^{\dagger}a_{\nu}\big]\;,\\ \displaystyle&\displaystyle(X=Z,U,G;\quad x=z,u,g)\end{aligned}\end{aligned}$$

(5.54a)

$$\begin{aligned} & \begin{aligned}\displaystyle V_{N}&\displaystyle\equiv V-\langle V\rangle-G_{N}\\ \displaystyle&\displaystyle=\frac{1}{2}\sum_{\mu,\nu,\sigma,\tau}\;\langle\mu\nu|v|\sigma\tau\rangle N\big[a_{\mu}^{\dagger}a_{\nu}^{\dagger}a_{\tau}a_{\sigma}\big]\\ \displaystyle&\displaystyle=\frac{1}{4}\sum_{\mu,\nu,\sigma,\tau}\langle\mu\nu|v|\sigma\tau\rangle_{A}N\big[a_{\mu}^{\dagger}a_{\nu}^{\dagger}a_{\tau}a_{\sigma}\big]\;,\end{aligned}\end{aligned}$$

(5.54b)

where

$$\begin{aligned} & \langle\mu|g|\nu\rangle=\sum_{\sigma=1}^{N}\langle\mu\sigma|v|\nu\sigma\rangle_{A}\;,\end{aligned}$$

(5.55)

$$\begin{aligned} & \langle\mu\nu|v|\sigma\tau\rangle_{A}=\langle\mu\nu|v|\sigma\tau\rangle-\langle\mu\nu|v|\tau\sigma\rangle\;,\end{aligned}$$

(5.56)

$\langle X\rangle=\langle\Phi|X|\Phi\rangle$, and N[⋯] designates the normal product relative to | Φ⟩ 16 ; 17 ; 18 . (Recall that $N[x_{1}x_{2}\cdots x_{k}]=\pm b_{\mu_{1}}^{\dagger}\cdots b_{\mu_{i}}^{\dagger}b_{\mu_{i+1}}\cdots b_{\mu_{k}}$, where $x_{i}=b_{\mu_{i}}$ or $b_{\mu_{i}}^{\dagger}$ are the annihilation and creation operators of the particle-hole formalism relative to | Φ⟩, i.e., b_μ = a ^†_μ for μ ≤ N and b_μ = a_μ for μ > N, the sign being determined by the parity of the permutation p : j ↦ μ_j.)

Defining

$$K=H-\langle H\rangle,\quad K_{0}=H_{0}-\langle H_{0}\rangle=H_{0}-\varepsilon_{0}\;,$$

(5.57)

we can return to Eqs. (5.24) and (5.25), where now

$$\begin{aligned}\displaystyle k_{i}&\displaystyle=E_{i}-\langle H\rangle,\quad\kappa_{i}=\varepsilon_{i}-\varepsilon_{0},\\ \displaystyle\varepsilon_{0}&\displaystyle=\sum_{\mu=1}^{N}\omega_{\mu}\;,\end{aligned}$$

(5.58)

and

$$W=K-K_{0}=V-U-\langle V-U\rangle\;.$$

(5.59)

With this choice, ⟨W⟩ = 0, so that for the reference state | Φ⟩, Eq. (5.42) simplify to (we drop the subscript 0 for simplicity)

$$\begin{aligned}\displaystyle k^{(0)}&\displaystyle=0,\quad k^{(1)}=0\;,\\ \displaystyle k^{(2)}&\displaystyle=\langle WRW\rangle\;,\\ \displaystyle k^{(3)}&\displaystyle=\langle WRWRW\rangle\;,\\ \displaystyle k^{(4)}&\displaystyle=\big\langle W(RW)^{3}\big\rangle-\langle WRW\rangle\big\langle WR^{2}W\big\rangle\;,\quad\text{{etc.}}\end{aligned}$$

(5.60)

Note that W is also in the N-product form,

$$W=W_{1}+W_{2}\;,\quad W_{1}=G_{N}-U_{N}\;,\quad W_{2}=V_{N}\;.$$

(5.61)

3.3 Møller–Plesset and Epstein–Nesbet PT

Choosing U = G we have

$$H_{0}=Z+G\equiv F\;,$$

(5.62)

so that Eq. (5.49) represent Hartree–Fock (HF) equations, and ω_μ and | μ⟩ the canonical HF orbital energies and spin orbitals, respectively. Since $\langle H\rangle=\sum_{\mu=1}^{N}\;\big(\langle\mu|z|\mu\rangle+\frac{1}{2}\langle\mu|g|\mu\rangle\big)$ is the HF energy, k = k₀ gives directly the ground state correlation energy. (Note, however, that the N-product form of PT eliminates the first-order contribution k⁽¹⁾ = ⟨W⟩ in any basis, even when F is not diagonal.) With this choice, W₁ = 0, W = V_N, and the denominators in Eq. (5.39) are given by the differences of HF orbital energies

$$\kappa_{0}-\kappa_{j}=\sum_{i=1}^{\lambda}(\omega_{\mu_{i}}-\omega_{\nu_{i}})\equiv\Delta\langle\{\mu_{i}\};\{\nu_{j}\}\rangle\;,$$

(5.63)

assuming that | Φ_j⟩ is a λ-times excited configuration relative to | Φ⟩ obtained through excitations $\mu_{i}\rightarrow\nu_{i},\;i=1,\ldots,\lambda$. Using the Slater rules (or the second quantization algebra), we can express the second-order contribution in terms of the two-electron integrals and HF orbital energies as

$$k^{(2)}=\frac{1}{2}\;\sum_{a,b,r,s}\;\frac{\langle ab|v|rs\rangle\big(\langle rs|v|ab\rangle-\langle rs|v|ba\rangle\big)}{\omega_{a}+\omega_{b}-\omega_{r}-\omega_{s}}\;,$$

(5.64)

where the summations over a , b (r , s) extend over all occupied (unoccupied) spin orbitals in | Φ⟩. Obtaining the corresponding higher-order corrections becomes more and more laborious and, beginning with the fourth-order, important cancellations arise between the principal and renormalization terms, even when the N-product form is employed. These will be addressed in Sect. 5.3.7.

The above outlined PT with H₀ given by the HF operator is often referred to as the Møller–Plesset PT 20 and, when truncated to the n-th order, is designated by the acronym MPn, $n=2,3,\ldots$. In this version, the two-electron integrals enter the denominators only through the HF orbital energies. In an alternative, less often employed variant, referred to as the Epstein–Nesbet PT 21 ; 22 , the whole diagonal part of H is used as the unperturbed Hamiltonian, i.e.,

$$H_{0}=\sum_{i}\langle\Phi_{i}|H|\Phi_{i}\rangle P_{i}\;.$$

(5.65)

With this choice, the denominators are given as differences of the diagonal elements of the configuration interaction matrix.

3.4 Diagrammatic MBPT

To facilitate the evaluation of higher-order terms and, especially, to derive the general properties and characteristics of the MBPT, it is useful to employ a diagrammatic representation 7 ; 8 ; 9 ; 10 ; 16 ; 17 ; 18 . Representing all the operators in Eqs. (5.42) and (5.43) or (5.60) in the second quantized form, we have to deal with the reference state (i.e., the Fermi vacuum) mean values of the strings of annihilation and creation operators (or with these strings acting on the reference in the case of a wave function). This is efficiently done using Wick's theorem and its diagrammatic representation via a special form of Feynman diagrams. In this representation, we associate with various operators suitable vertices with incident oriented lines representing the creation (outgoing lines) and annihilation (ingoing lines) operators that are involved in their second quantization form. A few typical diagrams representing operators ( − U) ,  W₁ and V are shown in Fig. 5.1a, Fig. 5.1b and Fig. 5.1c, Fig. 5.1d, respectively. Using the N-product form of PT with HF orbitals (Sect. 5.3.3), we only need the two-electron operator V or V_N, which can be represented using either nonantisymmetrized vertices (Fig. 5.1c), leading to the Goldstone diagrams 23 , or antisymmetrized vertices (Fig. 5.1d), associated with antisymmetrized two-electron integrals Eq. (5.56) and yielding the Hugenholtz diagrams 24 .

Fig. 5.1

Diagrammatic representation of one and two-electron operators

3.5 Vacuum and Wave Function Diagrams

Applying Wick's theorem to the strings of operators involved, we represent the individual contractions, Eq. (5.45), by joining corresponding oriented lines. To obtain a nonvanishing contribution, only contractions preserving the orientation need be considered (Eq. (5.45)). The resulting internal lines have either the left–right orientation (hole lines) or the right–left one (particle lines). Only fully contracted terms, represented by the so-called vacuum diagrams (having only internal lines), can contribute to the energy, while those representing wave function contributions have uncontracted or free lines extending to the left. When the operators involved are in the N-product form, no contractions of oriented lines issuing from the same vertex are allowed. The projection-like operators R, Eq. (5.39), or their powers, lead to the denominators, Eq. (5.63), given by the difference of hole and particle orbital energies associated with, respectively, hole and particle lines passing through the interval separating the corresponding two neighboring vertices. Clearly, there must always be at least one pair of such lines lest the denominator vanish. Thus, for example, the second-order contribution ⟨WRW⟩ is represented either by the two Goldstone diagrams 23 (Fig. 5.2a,b) or by the single Hugenholtz diagram 24 (Fig. 5.2c). The rules for the energy (vacuum) diagram evaluation are as follows:

• 1. Associate appropriate matrix elements with all vertices and form their product. The outgoing (ingoing) lines on each vertex define the bra (ket) states of a given matrix element, and for the Goldstone diagrams, the oriented lines attached to the same node are associated with the same electron number, (e.g., for the leftmost vertex in diagram (a) of Fig. 5.2, we have $\langle ab|\hat{v}|rs\rangle\equiv\langle a(1)b(2)|v|r(1)s(2)\rangle$).

  Fig. 5.2

  

  The second-order Goldstone (a), (b) and Hugenholtz (c) diagrams

• 2. Associate a denominator, Eq. (5.63), or its appropriate power, with every neighboring pair of vertices (and, for the wave function diagrams, also with the free lines extending to the left of the leftmost vertex; with each pair of such free lines associate also the corresponding pair of particle creation and hole annihilation operators).

• 3. Sum over all hole and particle labels.

• 4. Multiply each diagram contribution by the weight factor given by the reciprocal value of the order of the group of automorphisms of the diagram (stripped of summation labels) and by the sign ( − 1)^h+ℓ, where h designates the number of internal hole lines, and ℓ gives the number of closed loops of oriented lines (for Hugenholtz diagrams, use any of its Goldstone representatives to determine the correct phase).

Applying these rules to diagrams (a) and (b) of Fig. 5.2 we clearly recover Eq. (5.64) or, using the Hugenholtz diagram of Fig. 5.2, the equivalent expression

$$k^{(2)}=\frac{1}{4}\sum_{a,b,r,s}\langle ab|v|rs\rangle_{A}\langle rs|v|ab\rangle_{A}\Delta^{-1}(a,b;r,s)\;.$$

(5.66)

The possible third-order Hugenholtz diagrams are shown in Fig. 5.3 with the central vertex involving particle–particle, hole–hole, and particle–hole interaction 16 ; 17 ; 18 .

Fig. 5.3

Hugenholtz diagrams for the third-order energy contribution

3.6 Hartree–Fock Diagrams

In the general case (non-HF orbitals and/or not normal product form of PT), the one-electron terms, as well as the contractions between operators associated with the same vertex, can occur (the latter are always the hole lines). Representing the W₁ and ( − U) operators as shown in Fig. 5.1, the one-body perturbation W₁ represents in fact the three diagrams as shown in Fig. 5.4. The second-order contribution of this type is then represented by the diagrams in Fig. 5.5, which in fact represents nine diagrams that result when each W₁ vertex is replaced by three vertices as shown in Fig. 5.4.

Fig. 5.4

Schematic representation of W₁ = G_N − U_N

Fig. 5.5

The second-order, one-particle contribution

Using HF orbitals, all these terms mutually cancel out, as can be seen above. For this reason, the diagrams involving contractions of lines issuing from the same vertex are referred to as Hartree–Fock diagrams. Note, however, that even when not employing the canonical HF orbitals, it is convenient to introduce W₁ vertices of the normal product form PT and replace all nine HF-type diagrams by a single diagram of Fig. 5.5. Clearly, this feature provides even greater efficiency in higher orders of PT.

3.7 Linked and Connected Cluster Theorems

Using the N-product form of PT, the first nonvanishing renormalization term occurs in the fourth-order (Eq. (5.60)). For a system consisting of N noninteracting species, the energy given by this nonphysical term is proportional to N² and, thus, violates the size extensivity of the theory. It was first shown by Brueckner 25 that in the fourth order, these terms are in fact exactly canceled by the corresponding contributions originating in the principal term. A general proof of this cancellation in an arbitrary order was then given by Goldstone 23 and Hubbard 26 using a time-dependent PT based on Gell-Mann and Low adiabatic theorem and evolution operator (Sect. 5.4) and by Hugenholtz 24 employing a time-independent PT, relying on Green's function or the resolvent operator.

To comprehend this cancellation, consider the fourth-order energy contribution arising from the so-called unlinked diagrams (no such contribution can arise in the second or third order) shown in Fig. 5.6. An unlinked diagram is defined as a diagram containing a disconnected vacuum diagram (for the energy diagrams, the terms unlinked and disconnected are synonymous). The numerators associated with both diagrams being identical, we only consider the denominators. Designating the denominator associated with the top and the bottom part by A and B, respectively, we find for the overall contribution

$$\begin{aligned}\displaystyle\frac{1}{B}&\displaystyle\cdot\frac{1}{A+B}\cdot\frac{1}{B}+\frac{1}{A}\cdot\frac{1}{A+B}\cdot\frac{1}{B}\\ \displaystyle&\displaystyle=\left(\frac{1}{B}+\frac{1}{A}\right)\frac{1}{(A+B)B}=\frac{1}{AB^{2}}\;.\end{aligned}$$

(5.67)

Thus, the contribution from these terms exactly cancels that from the renormalization term ⟨WRW⟩⟨WR²W⟩.

Fig. 5.6

The fourth-order unlinked diagrams

Generalizing Eq. (5.67) we obtain the factorization lemma of Frantz and Mills 27 , which implies the cancellation of renormalization terms by the unlinked terms originating from the principal term. This result holds for the energy as well as for the wave function contributions in every order of PT, as ascertained by the linked cluster theorem, which states that

$$\begin{aligned} & \Delta E=k=\sum_{n=0}^{\infty}\langle\Phi|W|\Psi^{(n)}\rangle=\sum_{n=0}^{\infty}\langle W(RW)^{n}\rangle_{\mathrm{L}}\;,\end{aligned}$$

(5.68)

$$\begin{aligned} & |\Psi\rangle=\sum_{n=0}^{\infty}\;|\Psi^{(n)}\rangle=\sum_{n=0}^{\infty}\big\{(RW)^{n}|\Phi\rangle\big\}_{\mathrm{L}}\;,\end{aligned}$$

(5.69)

where the subscript L indicates that only linked diagrams (or terms) are to be considered. This enables us to obtain general, explicit expressions for the n-th-order PT contributions by first constructing all possible linked diagrams involving n vertices and by converting them into the explicit algebraic expressions using the rules of Sect. 5.3.5. Note that linked energy diagrams are always connected, but the linked wave function diagrams are either connected or disconnected, each disconnected component possessing at least one pair of particle–hole free lines extending to the left.

To reveal a deeper structure of the result Eq. (5.69) define the cluster operator T that generates all connected wave function diagrams,

$$T|\Phi\rangle=\sum_{n=1}^{\infty}\big\{(RW)^{n}|\Phi\rangle\big\}_{\mathrm{C}}\;,$$

(5.70)

the subscript C indicating that only contributions from connected diagrams are to be included. Since the general component with r disconnected parts can be shown to be represented by the term (r!)⁻¹T^r | Φ⟩, the general structure of the exact wave function | Ψ⟩ is given by the connected cluster theorem 24 ; 26 , which states that

$$|\Psi\rangle=\,\mathrm{e}^{T}\;|\Phi\rangle\;,$$

(5.71)

assuming an intermediate normalization ⟨Φ | Ψ⟩ = 1. In other words, the wave operator 𝒲 that transforms the unperturbed independent particle model wave function | Φ⟩ into the exact one according to

$$|\Psi\rangle=\mathcal{W}|\Phi\rangle$$

(5.72)

is given by the exponential of the cluster operator T,

$$\mathcal{W}=\,\mathrm{e}^{T}\;,$$

(5.73)

which, in turn, is given by the connected wave function diagrams. This is in fact the basis of the coupled cluster methods 28 ; 29 ; 30 ; 31 ; 32 ; 33 (Sect. 5.3.8). For recent reviews, see, e.g., 10 ; 17 ; 18 ; 34 ; 35 ; 36 ; 37 ; 38 ; 39 ; 40 ; 41 ; for a historical perspective, see 42 ; 43 .

The contributions to T may be further classified by their excitation rank i,

$$T=\sum_{i=1}^{N}\;T_{i}\;,$$

(5.74)

where T_i designates connected diagrams with i pairs of free particle–hole lines, producing i-times excited components of | Ψ⟩ when acting on | Φ⟩.

3.8 Coupled Cluster Theory

To motivate coupled cluster (CC) approaches we recall that summing all HF diagrams (Sect. 5.3.6) is equivalent to solving the HF equations. Likewise, depending on the average electron density of the system, it may be essential to sum certain types of PT diagrams to infinite order at the post-HF level, namely the ring diagrams (random phase approximation) in the case of high-density electron gas and ladder diagrams for low-density systems. An ideal approach that can sum different kinds of diagrams and their combinations, and is thus capable of recovering a large part of the electronic correlation energy, is based on the connected cluster theorem (Sect. 5.3.7), referred to in this context as the exponential cluster ansatz Eq. (5.71) for the wave operator. Using this ansatz, one derives a system of energy-independent nonlinear coupled cluster (CC) equations 10 ; 17 ; 18 ; 30 ; 31 ; 32 ; 33 determining the cluster amplitudes of T. These CC equations can be regarded as recurrence relations generating the MBPT series 17 ; 18 , so that by solving these equations one, in fact, implicitly generates all the relevant MBPT diagrams and sums them to infinite order. Since the solution of the full CC equations is equivalent to the exact solution of the Schrödinger equation, we must—in all practical applications—introduce a suitable truncation scheme, which implies that only diagrams of certain types are summed. (For different possible truncation schemes, see 44 , Sect. 2.)

Generally, using the cluster expansion Eq. (5.71) in the N-product form of the Schrödinger equation,

$$\begin{aligned}\displaystyle H_{{\mathrm{N}}}|\Psi\rangle&\displaystyle\equiv(H-\langle H\rangle)|\Psi\rangle=\Delta E|\Psi\rangle\;,\\ \displaystyle\quad\Delta E&\displaystyle=E-E_{0}\;,\end{aligned}$$

(5.75)

premultiplying with the inverse of the wave operator, and using the Hausdorff formula Eq. (5.11) yields

$$\,\mathrm{e}^{-T}H_{{\mathrm{N}}}\,\mathrm{e}^{T}|\Phi\rangle=\sum_{n=0}^{\infty}\;\frac{[\mathrm{ad}\,(-T)]^{n}H_{{\mathrm{N}}}}{n!}=\Delta E|\Phi\rangle\;.$$

(5.76)

In fact, this expansion terminates, so that using Eq. (5.74) and projecting onto | Φ⟩ we obtain the energy expression

$$\Delta E=\langle H_{\mathrm{N}}T_{2}\rangle+\frac{1}{2}\big\langle H_{\mathrm{N}}T_{1}^{2}\big\rangle\;,$$

(5.77)

while the projection onto the manifold of excited states { | Φ_i⟩} relative to $|\Phi\rangle\equiv|\Phi_{0}\rangle$ gives the system of CC equations

$$\langle\Phi_{i}|H_{{\mathrm{N}}}+[H_{{\mathrm{N}}},T]+\frac{1}{2}[[H_{{\mathrm{N}}},T],T]+\cdots|\Phi\rangle=0\;.$$

(5.78)

Approximating, e.g., T by the most important pair cluster component T ≈ T₂ gives the so-called CCD (coupled clusters with doubles) approximation

$$\big\langle\Phi_{i}^{(2)}\big|H_{{\mathrm{N}}}+[H_{{\mathrm{N}}},T_{2}]+\frac{1}{2}[[H_{{\mathrm{N}}},T_{2}],T_{2}]|\Phi\rangle=0\;,$$

(5.79)

the superscript (2) indicating pair excitations relative to | Φ⟩.

Equivalently, Eqs. (5.77) and (5.78) can be written in the form

$$\displaystyle\Delta E=\big\langle H_{{\mathrm{N}}}\,\mathrm{e}^{T}\big\rangle_{{\mathrm{C}}}\;,$$

(5.80)

$$\displaystyle\big\langle\Phi_{i}\big|\big(H_{{\mathrm{N}}}\,\mathrm{e}^{T}\big)_{{\mathrm{C}}}\big|\Phi\big\rangle=0\;,$$

(5.81)

the subscript C again indicating that only connected diagrams are to be considered. The general explicit form of CC equations is

$$a_{i}+\sum_{j}b_{ij}t_{j}+\sum_{j\leq k}c_{ijk}t_{j}t_{k}+\cdots=0\;,$$

(5.82)

where $a_{i}=\langle\Phi_{i}|H_{{\mathrm{N}}}|\Phi_{0}\rangle,\quad b_{ij}=\langle\Phi_{i}|H_{{\mathrm{N}}}|\Phi_{j}\rangle_{{\mathrm{C}}}$, $c_{ijk}=\langle\Phi_{i}|H_{{\mathrm{N}}}|\Phi_{j}\otimes\Phi_{k}\rangle_{{\mathrm{C}}}$, etc. Writing the diagonal linear term b_ii in the form

$$b_{ii}=\Delta_{i}+b_{ii}^{\prime}\;,$$

(5.83)

this system can be solved iteratively by rewriting it in the form

$$\begin{aligned}\displaystyle t_{i}^{(n+1)}=\Delta_{i}^{-1}&\displaystyle\Bigl(a_{i}+b_{ii}^{\prime}t_{i}^{(n)}+\sum_{j}{{}^{\prime}}b_{ij}t_{j}^{(n)}\Bigr.\\ \displaystyle&\displaystyle\quad+\Bigl.\sum_{j\leq k}c_{ijk}t_{j}^{(n)}t_{k}^{(n)}+\cdots\Bigr)\;.\end{aligned}$$

(5.84)

Starting with the zeroth approximation t ⁽⁰⁾_i  = 0, the first iteration is

$$t_{i}^{(1)}=\Delta_{i}^{-1}a_{i}\;,$$

(5.85)

which yields the second-order PT energy when used in Eq. (5.77). Clearly, the successive iterations generate higher and higher orders of the PT. At any truncation level, a size extensive result is obtained.

The single reference (SR) CC methods continue to be remarkably efficient in handling the dynamic correlation effects and are, thus, widely exploited in computations of molecular electronic structure. Several general-purpose codes are available for this purpose (for reviews: 10 ; 18 ; 34 ; 35 ; 36 ; 38 ). The basic workhorse that provides the required T₁ and T₂ cluster amplitudes results by setting T_i = 0 for i > 2 in Eq. (5.74) (the CC with singles and doubles (CCSD) method 45 ). A small, but often important, T₃ component is usually accounted for perturbatively via the CCSD(T) method 46 ; 47 , which is often referred to as the “gold standard” of quantum chemistry, since it provides very accurate and reliable results. Unfortunately, this is no longer the case in the presence of quasidegeneracy when handling strongly-correlated systems or when breaking chemical bonds as in generating potential energy curves or surfaces. This can to some extent be avoided by employing one of the renormalized versions of CCSD(T) 48 but, in general, requires multireference-type approaches (see below).

In order to compute the excitation, ionization, or electron attachment energies one employs the equation-of-motion (EOM) and the linear-response formalisms relying on the generalized cluster ansatz 49 $|\Psi\rangle=R\,\mathrm{e}^{T}|\Phi\rangle$, where R yields an appropriate zero (or first)-order wave function of the open-shell system considered when acting on | Φ⟩. In general, this leads to a non-Hermitian eigenvalue problem, which requires a computation of the right and left eigenvectors 10 . These developments also enabled calculation of other properties than the energy (dipole and quadrupole moments, polarizabilities, etc.), 10 ; 18 ; 34 ; 35 ; 36 . Here, we should also mention a possibility of using the cluster ansatz based on the unitary group approach (UGA) 50 ; 51 (Sect. 4.3.6).

At this stage it is important to recall that the standard SR MBPT and CC approaches pertain to nondegenerate, lowest-lying closed-shell states of a given symmetry species. Although the CC methods are often used even for open-shell states by relying on the unrestricted HF (UHF) reference (of the different-orbitals-for-different-spins (DODS) type), a proper description of such states requires a multireference (MR) generalization based on the effective Hamiltonian formalism 7 ; 18 ; 36 ; 52 ; 53 ; 54 . Unfortunately, such a generalization is not unambiguous. The two viable formulations, the so-called valence universal (VU) 7 ; 53 and state universal (SU) 54 MR CC methods have enjoyed continued following (see 55 for recent developments), yet are computationally demanding and often plagued with the intruder state and other problems 17 ; 52 . For these reasons, no general-purpose codes have yet been developed, and only a few actual applications have been carried out 18 ; 36 (see, however, the recently formulated SU CC approach for general model spaces 56 ; 57 ; 58 ; 59 ).

In view of the ambiguity, complexity, and computational demands of genuine MR CC approaches it is tempting to design SR CC-type methods that are capable of accommodating both the nondynamic and dynamic correlation effects. This may be achieved by exploiting the complementarity of perturbative (i.e., CC) and variational (CI, CAS SCF, etc.)-type approaches, leading to the so-called externally corrected (ec) ecCCSD or ecCCSD(T) methods (for an overview, see 44 ; see also 60 ; 61 and the so-called state selective or state specific (SS) approaches 62 ). The basic idea here is to extract a suitable approximation T ⁽⁰⁾₃ and T ⁽⁰⁾₄ of, respectively, the T₃ and T₄ clusters from some independent source and account for them in the CCD or CCSD equations, i.e.,

$$\left\langle\Phi_{i}\right\rvert H_{{\mathrm{N}}}\left(T_{2}+\frac{1}{2}T_{2}^{2}+T_{3}^{(0)}+T_{4}^{(0)}\right)\left\lvert\Phi\right\rangle_{{\mathrm{C}}}=0\,,$$

(5.86)

thus achieving a more meaningful decoupling of the CC chain by simply evaluating the T ⁽⁰⁾₃ and T ⁽⁰⁾₄ -dependent terms and adding them to the absolute term. Note that using the exact (i.e., full configuration interaction (FCI)) T₃ and T₄ amplitudes the ecCCSD equations will recover the exact FCI energy. The most promising version of such an ecCCSD approach employs the three- and four-body amplitudes obtained by a cluster analysis of a small MRCI wave function and is referred to as the reduced MR CCSD (RMR CCSD) method 63 ; 64 ; 65 (see 44 for a list of numerous applications). Very recently, the required approximate three and four-body clusters were extracted by relying on the (G)UGA based ((graphical) UGA) (see 66 ; 67 ; 68 for recent reviews) FCI quantum Monte Carlo (FCI-QMC) or i-FCI-QMC codes 69 ; 70 ; 71 yielding encouraging results 72 ; 61 ; 73 ; 74 . Very recently, Chan et al. 75 exploited for the same purpose the density matrix renormalization group (DMRG) approach 76 ; 77 ; 78 in the ecCCSD formalism 44 .

The same goal, which simultaneously leads to more efficient SR CC formalisms, may also be achieved via an effective implicit account of higher-order clusters by relying on the role of the EPV (exclusion principle violating) diagrams that are separable over the hole lines, leading to the so-called internally corrected (ic) methods, such as approximate coupled pair (ACP-D45) approach, approximate CC with doubles (ACCD) or with approximate quads (ACPQ) methods (see 44 for relevant references). Very recently, similar ideas to the icCCSD approaches 44 were exploited by Kats et al. 79 ; 80 ; 81 in the so-called distinguished cluster (DC) approximations, such as DC with doubles (DCD), DC with singles and doubles (DCSD) and DCSD corrected for triples (DCSD(T)) methods, leading to numerous exploitations in actual systems 82 ; 83 ; 84 ; 85 , including transcorrelated CC methods 86 .

A very recent development 87 exploits the quantum Monte Carlo (MC) formalism in solving the CC equations, resulting in a stochastic CCMC method 87 ; 88 , which was also extended to the equation of motion CC (EOMCC) type approaches 89 enabling handling the excited states energetics. This way of solving the CC equations should allow applications to larger systems than do the currently available standard codes.

Finally, note that the CC approach has also been used to handle bosonic-type problems of the vibrational structure in molecular spectra and, generally, multimode dynamics 90 .

4 Time-Dependent Perturbation Theory

4.1 Evolution Operator PT Expansion

By introducing the evolution operator U(t , t₀)

$$|\Psi(t)\rangle=U(t,t_{0})|\Psi(t_{0})\rangle\;,$$

(5.87)

the time-dependent Schrödinger equation

$$\mathrm{i}\hbar\frac{\partial}{\partial t}|\Psi(t)\rangle=H|\Psi(t)\rangle$$

(5.88)

becomes

$$\mathrm{i}\hbar\frac{\partial}{\partial t}U(t,t_{0})=HU(t,t_{0})\;.$$

(5.89)

Clearly,

$$\begin{aligned}\displaystyle U(t_{0},t_{0})&\displaystyle=1\;,\\ \displaystyle U(t,t_{0})&\displaystyle=U(t,t^{\prime})U(t^{\prime},t_{0})\;,\\ \displaystyle U(t,t_{0})^{-1}&\displaystyle=U(t_{0},t)=U^{\dagger}(t,t_{0})\;.\end{aligned}$$

(5.90)

If the Hamiltonian is time-independent, then

$$U(t,t_{0})=\exp\left[-\frac{\mathrm{i}}{\hbar}H(t-t_{0})\right]\;.$$

(5.91)

In the interaction picture (subscript I),

$$|\Psi(t)\rangle_{\mathrm{I}}=\exp\left(\frac{\mathrm{i}}{\hbar}H_{0}t\right)|\Psi(t)\rangle\;,$$

(5.92)

where now

$$H=H_{0}+V\;,$$

(5.93)

the Schrödinger equation becomes

$$\mathrm{i}\hbar\frac{\partial}{\partial t}|\Psi(t)\rangle_{\mathrm{I}}=V(t)_{\mathrm{I}}|\Psi(t)\rangle_{\mathrm{I}}\;.$$

(5.94)

This is known as the Tomonaga–Schwinger equation 91 . Analogously, the evolution operator in this picture (we drop the subscript I from now on) satisfies

$$\mathrm{i}\hbar\frac{\partial}{\partial t}U(t,t_{0})=V(t)U(t,t_{0})\;,$$

(5.95)

with the initial condition U(t₀ , t₀) = 1. This differential equation is equivalent to an integral equation

$$U(t,t_{0})=1-\frac{\mathrm{i}}{\hbar}\int\limits_{t_{0}}^{t}V(t_{1})U(t_{1},t_{0})\,\mathrm{d}t_{1}\;.$$

(5.96)

Iterating we get 92 ; 93

$$\begin{aligned}\displaystyle U(t,t_{0})&\displaystyle=\sum_{n=0}^{\infty}\left(-\frac{\mathrm{i}}{\hbar}\right)^{n}\\ \displaystyle&\displaystyle\quad\times\int\limits_{t_{0}}^{t}\,\mathrm{d}t_{1}\int\limits_{t_{0}}^{t_{1}}\,\mathrm{d}t_{2}\cdots\int\limits_{t_{0}}^{t_{n-1}}\,\mathrm{d}t_{n}V(t_{1})V(t_{2})\cdots V(t_{n})\\ \displaystyle&\displaystyle=\sum_{n=0}^{\infty}\frac{(-\mathrm{i}/\hbar)^{n}}{n!}\\ \displaystyle&\displaystyle\quad\times\int\limits_{t_{0}}^{t}\,\mathrm{d}t_{1}\cdots\int\limits_{t_{0}}^{t}\,\mathrm{d}t_{n}T[V(t_{1})\cdots V(t_{n})]\;,\end{aligned}$$

(5.97)

where T[⋯] designates the time-ordering or chronological operator.

4.2 Gell-Mann and Low Formula

For a time-independent perturbation, one introduces the so-called adiabatic switching by writing

$$H_{\alpha}(t)=H_{0}+\lambda\,\mathrm{e}^{-\alpha|t|}V,\quad\alpha> 0\;,$$

(5.98)

so that $H_{\alpha}(t\rightarrow\pm\infty)=H_{0}$ and $H_{\alpha}(t\rightarrow 0)=H=H_{0}+\lambda V$. Then

$$|\Psi(t)\rangle_{{\mathrm{I}}}=U_{\alpha}(t,-\infty|\lambda)|\Phi_{0}\rangle\;,$$

(5.99)

with $U_{\alpha}(t,-\infty|\lambda)$ obtained with $V_{\alpha}(t)=\lambda\,\mathrm{e}^{-\alpha|t|}V$ (all in the interaction picture). The desired energy is then given by the Gell-Mann and Low formula 94

$$\Delta E=\lim_{\alpha\rightarrow 0+}\mathrm{i}\hbar\alpha\lambda\frac{\partial}{\partial\lambda}\ln\langle\Phi_{0}|U_{\alpha}(0,-\infty|\lambda)|\Phi_{0}\rangle\;,{}$$

(5.100a)

or

$$\Delta E=\frac{1}{2}\lim_{\alpha\rightarrow 0+}\mathrm{i}\hbar\alpha\lambda\frac{\partial}{\partial\lambda}\ln\langle\Phi_{0}|U_{\alpha}(\infty,-\infty|\lambda)|\Phi_{0}\rangle\;,$$

(5.100b)

which result from the asymmetric energy formula Eq. (5.27). One can similarly obtain the perturbation expansion for the one or two-particle Green's functions, e.g.,

$$G_{\mu\nu}\big(t,t^{\prime}\big)=\!\lim_{\alpha\rightarrow 0+}\!\frac{\big\langle T\big\{a_{\mu}(t)a_{\nu}^{\dagger}(t^{\prime})U_{\alpha}(\infty,-\infty|\lambda)\big\}\big\rangle}{\langle U_{\alpha}(\infty,-\infty|\lambda)\rangle}\;,{}$$

(5.101)

with the operators in the interaction representation and the expectation values in the noninteracting ground state | Φ₀⟩. Analogous expressions result for G(rt , r^′t^′), etc., when the creation and annihilation operators are replaced by the corresponding field operators.

4.3 Potential Scattering and Quantum Dynamics

The Schrödinger equation for a free particle of energy E = ℏ²k² ∕ 2m, moving in the potential V(r),

$$\begin{aligned}\displaystyle\big(\nabla^{2}+k^{2}\big)\psi(k,\boldsymbol{r})&\displaystyle=v(\boldsymbol{r})\psi(k,\boldsymbol{r})\;,\\ \displaystyle v(\boldsymbol{r})&\displaystyle=\big(2m/\hbar^{2}\big)V(\boldsymbol{r})\;,\end{aligned}$$

(5.102)

has the formal solution

$$\psi(k,\boldsymbol{r})=\Phi(k,\boldsymbol{r})+\int G_{0}\big(k,\boldsymbol{r},\boldsymbol{r}^{\prime}\big)v\big(\boldsymbol{r}^{\prime}\big)\psi\big(k,\boldsymbol{r}^{\prime}\big)\,\mathrm{d}\boldsymbol{r}^{\prime}\;,{}$$

(5.103)

where Φ(k , r) is a solution of the homogeneous equation [v(r) ≡ 0], and G₀(k , r , r^′) is a classical Green's function

$$\big(\nabla^{2}+k^{2}\big)G_{0}\big(k,\boldsymbol{r},\boldsymbol{r}^{\prime}\big)=\delta\big(\boldsymbol{r}-\boldsymbol{r}^{\prime}\big)\;.$$

(5.104)

For an in-going plane wave $\Phi(k,\boldsymbol{r})\equiv\Phi_{\boldsymbol{k}_{i}}(\boldsymbol{r})=(2\pi)^{3/2}\exp(\mathrm{i}{\boldsymbol{k}_{i}}\cdot\boldsymbol{r})$ with the initial wave vector k_i and appropriate asymptotic boundary conditions (outgoing spherical wave with positive phase velocity), when $G_{0}(k,\boldsymbol{r},\boldsymbol{r}^{\prime})\equiv G_{0}^{(+)}(|\boldsymbol{r}-\boldsymbol{r}^{\prime}|)=-(4\pi|\boldsymbol{r}-\boldsymbol{r}^{\prime}|)^{-1}\,\mathrm{e}^{ik|\boldsymbol{r}-\boldsymbol{r}^{\prime}|}$, Eq. (5.103) is referred to as the Lippmann–Schwinger equation 95 . It can be equivalently transformed into the integral equation for Green's function

$$\begin{aligned}\displaystyle G^{(+)}\big(\boldsymbol{r},\boldsymbol{r}^{\prime}\big)&\displaystyle=G_{0}^{(+)}\big(\boldsymbol{r},\boldsymbol{r}^{\prime}\big)+\int G_{0}^{(+)}\big(\boldsymbol{r},\boldsymbol{r}^{\prime\prime}\big)v\big(\boldsymbol{r}^{\prime\prime}\big)\\ \displaystyle&\displaystyle\quad\times G^{(+)}\big(\boldsymbol{r}^{\prime\prime},\boldsymbol{r}^{\prime}\big)\;\,\mathrm{d}\boldsymbol{r}^{\prime\prime}\;,\end{aligned}$$

(5.105)

representing a special case of the Dyson equation.

In the time-dependent case, considering the scattering of a spinless massive particle by a time-dependent potential V(r , t), we get similarly

$$\begin{aligned}\displaystyle\psi(\boldsymbol{r},t)&\displaystyle=\Phi(\boldsymbol{r},t)\\ \displaystyle&\displaystyle\quad+\int G_{0}\big(\boldsymbol{r},\boldsymbol{r}^{\prime};t,t^{\prime}\big)V\big(\boldsymbol{r}^{\prime},t^{\prime}\big)\psi\big(\boldsymbol{r}^{\prime},t^{\prime}\big)\,\mathrm{d}\boldsymbol{r}^{\prime}\,\mathrm{d}t^{\prime}\;,\end{aligned}$$

(5.106)

where the zero-order time-dependent Green's function now satisfies the equation

$$\left(\mathrm{i}\hbar\frac{\partial}{\partial t}-H_{0}\right)G_{0}\big(\boldsymbol{r},\boldsymbol{r}^{\prime};t,t^{\prime}\big)=\delta\big(\boldsymbol{r}-\boldsymbol{r}^{\prime}\big)\delta\big(t-t^{\prime}\big)\;.$$

(5.107)

Again, for causal propagation one chooses the time-retarded or causal Green's function or propagator $G_{0}^{(+)}(\boldsymbol{r},\boldsymbol{r}^{\prime};t,t^{\prime})$.

4.4 Born Series

Iteration of Eq. (5.106) gives the Born sequence

$$\begin{aligned} & \begin{aligned}\displaystyle\psi_{0}(\boldsymbol{r},t)&\displaystyle=\Phi(\boldsymbol{r},t)\;,\\ \displaystyle\psi_{1}(\boldsymbol{r},t)&\displaystyle=\Phi(\boldsymbol{r},t)+\int G_{0}^{(+)}\big(\boldsymbol{r},\boldsymbol{r}^{\prime};t,t^{\prime}\big)\\ \displaystyle&\displaystyle\quad\times V\big(\boldsymbol{r}^{\prime},t^{\prime}\big)\Phi\big(\boldsymbol{r}^{\prime},t^{\prime}\big)\,\mathrm{d}\boldsymbol{r}^{\prime}\,\mathrm{d}t^{\prime}\;,\end{aligned}\end{aligned}$$

(5.108a)

$$\begin{aligned} & \begin{aligned}\displaystyle\psi_{2}(\boldsymbol{r},t)&\displaystyle=\Phi(\boldsymbol{r},t)+\int G_{0}^{(+)}\big(\boldsymbol{r},\boldsymbol{r}^{\prime};t,t^{\prime}\big)\\ \displaystyle&\displaystyle\quad\times V\big(\boldsymbol{r}^{\prime},t^{\prime}\big)\psi_{1}\big(\boldsymbol{r}^{\prime},t^{\prime}\big)\,\mathrm{d}\boldsymbol{r}^{\prime}\,\mathrm{d}t^{\prime}\;,\end{aligned}\end{aligned}$$

(5.108b)

and, generally

$$\begin{aligned}\displaystyle\psi_{n}(\boldsymbol{r},t)&\displaystyle=\Phi(\boldsymbol{r},t)+\int G_{0}^{(+)}\big(\boldsymbol{r},\boldsymbol{r}^{\prime};t,t^{\prime}\big)\\ \displaystyle&\displaystyle\quad\times V\big(\boldsymbol{r}^{\prime},t^{\prime}\big)\psi_{n-1}\big(\boldsymbol{r}^{\prime},t^{\prime}\big)\,\mathrm{d}\boldsymbol{r}^{\prime}\,\mathrm{d}t^{\prime}\;.\end{aligned}$$

(5.109)

Summing individual contributions gives the Born series for $\psi(\boldsymbol{r},t)\equiv\psi^{(+)}(\boldsymbol{r},t)$,

$$\psi(\boldsymbol{r},t)=\sum_{n=0}^{\infty}\chi_{n}(\boldsymbol{r},t)\;,{}$$

(5.110)

where

$$\begin{aligned}\displaystyle\chi_{0}(\boldsymbol{r},t)&\displaystyle=\Phi(\boldsymbol{r},t)\;,\\ \displaystyle\chi_{n}(\boldsymbol{r},t)&\displaystyle=\int\mathcal{G}_{n}\big(\boldsymbol{r},\boldsymbol{r}^{\prime};t,t^{\prime}\big)\Phi\big(\boldsymbol{r}^{\prime},t^{\prime}\big)\,\mathrm{d}\boldsymbol{r}^{\prime}\,\mathrm{d}t^{\prime}\;,\end{aligned}$$

(5.111)

with

$$\begin{aligned}\displaystyle\mathcal{G}_{n}\big(\boldsymbol{r},\boldsymbol{r}^{\prime};t,t^{\prime}\big)&\displaystyle=\int\mathcal{G}_{1}\big(\boldsymbol{r},\boldsymbol{r}^{\prime\prime};t,t^{\prime\prime}\big)\\ \displaystyle&\displaystyle\qquad\times\mathcal{G}_{n-1}\big(\boldsymbol{r}^{\prime\prime},\boldsymbol{r}^{\prime};t^{\prime\prime},t^{\prime}\big)\,\mathrm{d}\boldsymbol{r}^{\prime\prime}\,\mathrm{d}t^{\prime\prime}\;,\;(n> 1)\\ \displaystyle\mathcal{G}_{1}\big(\boldsymbol{r},\boldsymbol{r}^{\prime};t,t^{\prime}\big)&\displaystyle=G_{0}^{(+)}\big(\boldsymbol{r},\boldsymbol{r}^{\prime};t,t^{\prime}\big)V\big(\boldsymbol{r}^{\prime},t^{\prime}\big)\;.\end{aligned}$$

(5.112)

In a similar way we obtain the Born series for the scattering amplitudes or transition matrix elements.

4.5 Variation of Constants Method

An alternative way of formulating the time-dependent PT is the method of variation of the constants 96 ; 97 . Start again with the time-dependent Schrödinger equation (5.88) with H = H₀ + V and assume that H₀ is time-independent, while V is a time-dependent perturbation. Designating the eigenvalues and eigenstates of H₀ by ε_i and | Φ_i⟩, respectively (Eq. (5.48)), the general solution of the unperturbed time-dependent Schrödinger equation

$$\mathrm{i}\hbar\frac{\partial}{\partial t}|\Psi_{0}\rangle=H_{0}|\Psi_{0}\rangle$$

(5.113)

has the form

$$|\Psi_{0}\rangle=\sum_{j}c_{j}|\Phi_{j}\rangle\exp\left(-\frac{\mathrm{i}}{\hbar}\varepsilon_{j}t\right)\;,{}$$

(5.114)

with c_j representing arbitrary constants, and the sum indicating both the summation over the discrete part and the integration over the continuum part of the spectrum of H₀.

In the spirit of the general variation of constants procedure, write the unknown perturbed wave function | Ψ(t)⟩, Eq. (5.88), in the form

$$|\Psi(t)\rangle=\sum_{j}C_{j}(t)|\Phi_{j}\rangle\exp\left(-\frac{\mathrm{i}}{\hbar}\varepsilon_{j}t\right)\;,{}$$

(5.115)

where the C_j(t) are now functions of time. Substituting this ansatz into the time-dependent Schrödinger equation (5.88) gives

$$\dot{C}_{j}(t)=(\mathrm{i}\hbar)^{-1}\sum_{k}C_{k}(t)V_{jk}\exp[(\mathrm{i}/\hbar)\Delta_{jk}t]\;,{}$$

(5.116)

where

$$\Delta_{jk}=\varepsilon_{j}-\varepsilon_{k}\;,\quad V_{jk}=\langle\Phi_{j}|V|\Phi_{k}\rangle\;.$$

(5.117)

Introducing again the “small” parameter λ by writing the Hamiltonian H in the form

$$H=H_{0}+\lambda V(t){}$$

(5.118)

and expanding the “coefficients” C_j(t) in powers of λ,

$$C_{j}\equiv C_{j}(t)=\sum_{k=0}^{\infty}C_{j}^{(k)}(t)\lambda^{k}\;,$$

(5.119)

gives the system of first-order differential equations

$$\begin{aligned}\displaystyle\dot{C}_{j}^{(n+1)}&\displaystyle=(\mathrm{i}\hbar)^{-1}\sum_{k}C_{k}^{(n)}V_{jk}\exp\left[(\mathrm{i}/\hbar)\Delta_{jk}t\right]\;,\\ \displaystyle n&\displaystyle=0,1,2,\ldots\;,\end{aligned}$$

(5.120)

with the initial condition $\dot{C}_{j}^{(0)}=0$, which implies that C ⁽⁰⁾_j are time independent, so that C ⁽⁰⁾_j  = c_j, obtaining Eq. (5.114) in the zeroth order. The system Eq. (5.120) can be integrated to any prescribed order. For example, if the system is initially in a stationary state | Φ_i⟩, then set

$$C_{j}^{(0)}=\begin{cases}\delta_{ji}\hphantom{(j-i)}\quad\text{for discrete states}\;,\\ \delta(j-i)\hphantom{{}_{ji}}\quad\text{for continuous states}\;,\end{cases}{}$$

(5.121)

so that

$$C_{j(i)}^{(1)}(t)=(\mathrm{i}\hbar)^{-1}\int_{-\infty}^{t}V_{ji}\exp\left[(\mathrm{i}/\hbar)\Delta_{ji}t^{\prime}\right]\,\mathrm{d}t^{\prime}\;,$$

(5.122)

assuming C ⁽¹⁾_j(i) ( − ∞) = 0. Clearly, | C ⁽¹⁾_j(i) (t)|² gives the first-order transition probability for the transition from the initial state | Φ_i⟩ to a particular state | Φ_j⟩. These in turn will yield the first-order differential cross sections 98 .