Many-Body Perturbation Theory (MBPT) and Time-Dependent Density-Functional Theory (TD-DFT): MBPT Insights About What Is Missing In, and Corrections To, the TD-DFT Adiabatic Approximation

Abstract

In their famous paper, Kohn and Sham formulated a formally exact density-functional theory (DFT) for the ground-state energy and density of a system of N interacting electrons, albeit limited at the time by certain troubling representability questions. As no practical exact form of the exchange-correlation (xc) energy functional was known, the xc-functional had to be approximated, ideally by a local or semilocal functional. Nowadays, however, the realization that Nature is not always so nearsighted has driven us up Perdew’s Jacob’s ladder to find increasingly nonlocal density/wavefunction hybrid functionals. Time-dependent (TD-) DFT is a younger development which allows DFT concepts to be used to describe the temporal evolution of the density in the presence of a perturbing field. Linear response (LR) theory then allows spectra and other information about excited states to be extracted from TD-DFT. Once again the exact TD-DFT xc-functional must be approximated in practical calculations and this has historically been done using the TD-DFT adiabatic approximation (AA) which is to TD-DFT very similar to what the local density approximation (LDA) is to conventional ground-state DFT. Although some of the recent advances in TD-DFT focus on what can be done within the AA, others explore ways around the AA. After giving an overview of DFT, TD-DFT, and LR-TD-DFT, this chapter focuses on many-body corrections to LR-TD-DFT as one way to build hybrid density-functional/wavefunction methodology for incorporating aspects of nonlocality in time not present in the AA.

Appendix: Order Analysis

We have presented the superoperator PP procedure as if we simply manipulated Feynman diagrams. In reality we expanded the matrices using Wick’s theorem with the help of a home-made FORTRAN program. The result was a series of algebraic expressions which were subsequently analyzed by drawing the corresponding Feynman diagrams. This leads to about 200 diagrams which we ultimately resum to give a more compact expression. It is the generation of this expression that we now wish to discuss.

Let us analyze this expression for the PP according to the order of excitation operator. Following Casida [58], we partition the space as

$$ -{\Pi}_{sr,qp}\left(\omega \right)=\left(\left({\widehat{p}}^{\dagger}\widehat{q}\Big|{\mathbf{T}}_1^{\dagger}\right) \left({\widehat{p}}^{\dagger}\widehat{q}\Big|{\mathbf{T}}_{2+}^{\dagger}\right)\right){\boldsymbol{\varGamma}}^{-1}\left(\omega \right)\left(\begin{array}{c}\hfill \left({\mathbf{T}}_1^{\dagger}\Big|{\widehat{r}}^{\dagger}\widehat{s}\right)\hfill \\ {}\hfill \left({\mathbf{T}}_{2+}^{\dagger}\Big|{\widehat{r}}^{\dagger}\widehat{s}\right)\hfill \end{array}\right) , $$

(143)

where ${\mathbf{T}}_{2+}^{\dagger }$ corresponds to the operator space of two-electron and higher excitations and

$$ {\boldsymbol{\varGamma}}^{-1}\left(\omega \right)={\left[\begin{array}{cc}\hfill {\boldsymbol{\varGamma}}_{1,1}\left(\omega \right)\hfill & \hfill {\boldsymbol{\varGamma}}_{1,2+}\hfill \\ {}\hfill {\boldsymbol{\varGamma}}_{2+,1}\hfill & \hfill {\boldsymbol{\varGamma}}_{2+,2+}\left(\omega \right)\hfill \end{array}\right]}^{-1} , $$

(144)

has been blocked:

$$ {\boldsymbol{\varGamma}}_{i,j}\left(\omega \right)=\left({\mathbf{T}}_i^{\dagger}\left|\omega \overset{\smile }{1}+\overset{\smile }{H}\right|{\mathbf{T}}_j^{\dagger}\right) . $$

(145)

Using the well-known expression for the inverse of a two-by-two block matrix allows us to transform (143) into

$$ \begin{array}{l}-{\Pi}_{sr,qp}\left(\omega \right)=\left[\left({\widehat{p}}^{\dagger}\widehat{q}\Big|{\boldsymbol{T}}_1^{\dagger}\right)-\left({\widehat{p}}^{\dagger}\widehat{q}\Big|{\boldsymbol{T}}_{2+}^{\dagger}\right){\boldsymbol{\varGamma}}_{2+,2+}^{-1}\left(\omega \right){\boldsymbol{\varGamma}}_{2+,1}\right]\\ {} \times {\boldsymbol{P}}^{-1}\left(\omega \right)\left[\left({\boldsymbol{T}}_1^{\dagger}\Big|{\widehat{r}}^{\dagger}\widehat{s}\right)-{\boldsymbol{\varGamma}}_{1,2+}{\boldsymbol{\varGamma}}_{2+,2+}^{-1}\left(\omega \right)\left({\boldsymbol{T}}_{2+}^{\dagger}\Big|{\widehat{r}}^{\dagger}\widehat{s}\right)\right]+\left({\widehat{p}}^{\dagger}\widehat{q}\Big|{\boldsymbol{T}}_{2+}^{\dagger}\right){\boldsymbol{\varGamma}}_{2+,2+}^{-1}\left(\omega \right)\left({\boldsymbol{T}}_{2+}^{\dagger}\Big|{\widehat{r}}^{\dagger}\widehat{s}\right) ,\end{array} $$

(146)

where

$$ \boldsymbol{P}\left(\omega \right)={\boldsymbol{\varGamma}}_{1,1}\left(\omega \right)-{\boldsymbol{\varGamma}}_{1,2+}{\boldsymbol{\varGamma}}_{2+,2+}^{-1}\left(\omega \right){\boldsymbol{\varGamma}}_{2+,1} . $$

(147)

Although (146) is somewhat complicated, it turns out that P(ω) plays much the same role in the smaller T ^†₁ space that Γ(ω) plays in the full T ^† space. To see how this comes about, it is necessary to introduce the concept of order in the fluctuation operator – see (67) – and in M _xc – see (69). We can now perform an order-by-order expansion of (146). Through second order only the T ^†₂ part of ${\boldsymbol{T}}_{2+}^{\dagger }$ contributes, so we need not consider higher than double excitation operators. However, we make some additional approximations. In particular, we follow the usual practice and drop the last term in (146) because it contributes only at second order and appears to be small when calculating excitation energies and transitions moments using the Hartree–Fock approximation as zero-order [52, 106–109]. For response functions such as dynamic polarizabilities, their inclusion is more critical, improving the agreement with experiments [49]. We also have no need to consider the second term in

$$ \left({\widehat{p}}^{\dagger}\widehat{q}\Big|{\mathbf{T}}_1^{\dagger}\right)-\left({\widehat{p}}^{\dagger}\widehat{q}\Big|{\mathbf{T}}_{2+}^{\dagger}\right){\boldsymbol{\varGamma}}_{2+,2+}^{-1}\left(\omega \right){\boldsymbol{\varGamma}}_{2+,1} . $$

(148)

This means that for the purposes of this chapter we can treat the PP in the present work as given by

$$ -{\Pi}_{sr,qp}\left(\omega \right)=\left({\widehat{p}}^{\dagger}\widehat{q}\Big|{\boldsymbol{T}}_1^{\dagger}\right){\mathbf{P}}^{-1}\left(\omega \right)\left({\boldsymbol{T}}_1^{\dagger}\Big|{\widehat{r}}^{\dagger}\widehat{s}\right) . $$

(149)

Comparing with (82) substantiates our earlier claim that P(ω) plays the same role in the T ^†₁ space that Γ(ω) plays over the full T ^† space.

1.1 First-Order Exchange-Correlation Kernel

We now turn to the first-order exchange-correlation kernel. Our main motivation here is to verify that we obtain the same terms as in exact exchange (EXX) calculations when we evaluate $\boldsymbol{\varPi} -{\boldsymbol{\varPi}}_s$ [59, 60]. Because our approach is in some ways more general than previous approaches to the EXX kernel, this section may also provide some new insight into the meaning of the EXX equations.

Because we are limited to first order, only zero- and first-order wavefunction terms need be considered. This implies that all the contributions from the ${\mathbf{T}}_{2+}^{\dagger }$ space (the space of double- and higher-excitations) are zero and substantiates our claim that (149) is exact to first-order. An order-by-order expansion gives

$$ \begin{array}{l}-{\Pi}_{sr,qp}^{\left(0+1\right)}\left(\omega \right)={\left({\widehat{p}}^{\dagger}\widehat{q}\Big|{\boldsymbol{T}}_1^{\dagger}\right)}^{(1)}{\boldsymbol{P}}^{(0),-1}\left(\omega \right){\left({\boldsymbol{T}}_1^{\dagger}\Big|{\widehat{r}}^{\dagger}\widehat{s}\right)}^{(0)}+{\left({\widehat{p}}^{\dagger}\widehat{q}\Big|{\boldsymbol{T}}_1^{\dagger}\right)}^{(0)}{\boldsymbol{P}}^{(0),-1}\left(\omega \right){\left({\boldsymbol{T}}_1^{\dagger}\Big|{\widehat{r}}^{\dagger}\widehat{s}\right)}^{(1)}\\ {}+{\left({\widehat{p}}^{\dagger}\widehat{q}\Big|{\boldsymbol{T}}_1^{\dagger}\right)}^{(0)}{\boldsymbol{P}}^{(1),-1}\left(\omega \right){\left({\boldsymbol{T}}_1^{\dagger}\Big|{\widehat{r}}^{\dagger}\widehat{s}\right)}^{(0)}-{\Pi}_{sr,qp}^s\left(\omega \right),\end{array} $$

(150)

where

$$ -{\Pi}_{sr,qp}^s\left(\omega \right)={\left({\widehat{p}}^{\dagger}\widehat{q}\Big|{\boldsymbol{T}}_1^{\dagger}\right)}^{(0)}{\left({\boldsymbol{T}}_1^{\dagger}\left|\omega \overset{\smile }{1}+{\overset{\smile }{h}}_{KS}\right|{\boldsymbol{T}}_1^{\dagger}\right)}^{(0),-1}{\left({\boldsymbol{T}}_1^{\dagger}\Big|{\widehat{r}}^{\dagger}\widehat{s}\right)}^{(0)} . $$

(151)

The evaluation of each of first-order block is straightforward using the basic definitions and Wick’s theorem.

Let us first consider the P parts. The zeroth-order contribution is

$$ {P}_{kc,ia}^{(0)}\left(\omega \right)=\left(\omega -{\varepsilon}_{i,a}\right){\delta}_{ik}{\delta}_{ac} $$

(152)

$$ {P}_{ck,ia}^{(0)}\left(\omega \right)=0 , $$

(153)

and the first-order contribution gives

$$ {P}_{kc,ia}^{(1)}=\left( ai\left|\right|kc\right)+{M}_{ac}{\delta}_{ik}-{M}_{ik}{\delta}_{ac} $$

(154)

$$ {P}_{ck,ia}^{(1)}=\left(ci\left|\right|ak\right) . $$

(155)

(It should be noted that P _kc,ia is part of the A block, whereas P _ck,ia is part of the B block.) The sum of ${P}^{(0)}+{P}^{(1)}$ gives the exact pole structure up to first-order in the SOPPA approach.

The zero-order contribution,

$$ {\left({\widehat{p}}^{\dagger}\widehat{q}\Big|{\mathbf{T}}_1^{\dagger}\right)}^{(0)}=\left({\mathbf{T}}_1^{\dagger}\Big|{\mathbf{T}}_1^{\dagger}\right) , $$

(156)

and the first-order contributions are given by

$$ {\left[\left({\widehat{p}}^{\dagger}\widehat{q}\Big|{\mathbf{T}}_1^{\dagger}\right)\right]}_{kc,ji}^{(1)}=-\frac{M_{jc}}{\varepsilon_{j,c}}{\delta}_{ik} $$

(157)

$$ {\left[\left({\widehat{p}}^{\dagger}\widehat{q}\Big|{\mathbf{T}}_1^{\dagger}\right)\right]}_{ck,ji}^{(1)}= \frac{M_{ic}}{\varepsilon_{i,c}}{\delta}_{kj} $$

(158)

$$ {\left[\left({\widehat{p}}^{\dagger}\widehat{q}\Big|{\mathbf{T}}_1^{\dagger}\right)\right]}_{kc, ba}^{(1)}= \frac{M_{ka}}{\varepsilon_{k,a}}{\delta}_{bc} $$

(159)

$$ {\left[\left({\widehat{p}}^{\dagger}\widehat{q}\Big|{\mathbf{T}}_1^{\dagger}\right)\right]}_{ck, ba}^{(1)}=-\frac{M_{kb}}{\varepsilon_{k,b}}{\delta}_{ca} . $$

(160)

The PP Π(ω) is now easily constructed by simple matrix multiplication according to (150). Applying the first approximation from Sect. 5 and expanding ${\boldsymbol{\varPi}}_s\left(\omega \right)-\boldsymbol{\varPi} \left(\omega \right)$ through first order allows us to recover Görling’s TD-EXX kernel [30]. The most convenient way to do this is to expand ${\boldsymbol{P}}^{(1),-1}$ using

$$ \begin{array}{l}{\left({\mathbf{T}}_1^{\dagger}\left|\omega \overset{\smile }{1}+\overset{\smile }{H}\right|{\mathbf{T}}_1^{\dagger}\right)}^{-1}\approx {\left({\mathbf{T}}_1^{\dagger}\left|\omega \overset{\smile }{1}+{\overset{\smile }{H}}^{(0)}\right|{\mathbf{T}}_1^{\dagger}\right)}^{-1}\\ {} +{\left({\mathbf{T}}_1^{\dagger}\left|\omega \overset{\smile }{1}+{\overset{\smile }{H}}^{(0)}\right|{\mathbf{T}}_1^{\dagger}\right)}^{-1}\left({\mathbf{T}}_1^{\dagger}\left|{\overset{\smile }{H}}^{(1)}\right|{\mathbf{T}}_1^{\dagger}\right){\left({\mathbf{T}}_1^{\dagger}\left|\omega \overset{\smile }{1}+{\overset{\smile }{H}}^{(0)}\right|{\mathbf{T}}_1^{\dagger}\right)}^{-1} .\end{array} $$

(161)

The result is represented diagrammatically in Fig. 7. The corresponding expressions agree perfectly with the expanded expressions of the TD-EXX kernel obtained by Hirata et al. [59] which are equivalent to the more condensed form given by Görling [60]. The diagrammatic treatment makes clear the connection with the BSE approach. There are in fact just three time-unordered diagrams, shown in Fig. 11, whose various time orderings generate the diagrams in Fig. 7. However the “hanging parts” above and below the horizontal dotted lines now have the physical interpretation of initial and final state wave function correlation. Had we applied the second approximation of Sect. 5, then only diagrams in Fig. 7a–f would have survived.

Use of the Gonze–Scheffler relation (see further Sect. 5) then leads to

$$ \begin{array}{l}\omega ={\varepsilon}_{a,i}^{KS}+{f}_{\mathrm{xc}}\left({\varepsilon}_{a,i}^{KS}\right)\\ {} ={\varepsilon}_{a,i}^{KS}+\left\langle a\left|{\widehat{M}}_{\mathrm{xc}}\right|a\right\rangle -\left\langle i\left|{\widehat{M}}_{\mathrm{xc}}\right|i\right\rangle +\left( ai\left|\right|ia\right)\\ {} ={\varepsilon}_{a,i}^{HF}+\left( ai\left|\right|ia\right) ,\end{array} $$

(162)

which is exactly the configuration interaction singles (CIS, i.e., TDHF Tamm–Dancoff approximation) expression evaluated using Kohn–Sham orbitals. This agrees with a previous exact result obtained using Görling–Levy perturbation theory [82, 86, 87].

1.2 Second-Order Exchange-Correlation Kernel

Having verified some known results, let us go on to do the MBPT necessary to obtain the pole structure of the xc-kernel through second order in the second approximation. That is, we need to evaluate ${\boldsymbol{\varPi}}_s^{-1}\left(\omega \right)-{\boldsymbol{\varPi}}^{-1}\left(\omega \right)$ through second order in such a way that its pole structure is evident. The SOPPA/ADC strategy for this is to make a diagrammatic ${\boldsymbol{\varPi}}_s\left(\omega \right)-\boldsymbol{\varPi} \left(\omega \right)$ expansion of this quantity and then resum the expansion in an order-consistent way having the form

$$ {\left[{\boldsymbol{\varPi}}_s\left(\omega \right)-\boldsymbol{\varPi} \left(\omega \right)\right]}_{rs,qp}^{\left(0+1+\dots +n\right)}={\displaystyle \sum_{k=0}^n}{\displaystyle \sum_{i=0}^k}{\displaystyle \sum_{j=0}^{k-i}}{\left({\widehat{p}}^{\dagger}\widehat{q}\Big|{\mathbf{T}}_1^{\dagger}\right)}^{(i)}{\mathbf{P}}^{(j),-1}\left(\omega \right){\left({\mathbf{T}}_1^{\dagger}\Big|{\widehat{r}}^{\dagger}\widehat{s}\right)}^{\left(k-i-j\right)} , $$

when the Born approximation is applied to the P(ω) in the same way as in Sect. 5. The number of diagrams contributing to this expansion is large and, for the sake of simplicity, we only give the resumed expressions for each block. Evidently, after the calculation of each block there is an additional step matrix inversion in order to apply the second approximation to the xc-kernel.

It should be emphasized that although the treatment below may seem simple, application of Wick’s theorem is complicated and has been carried out using an in-house FORTRAN program written specifically for the purpose. The result before resummation is roughly 200 diagrams, which have been included as supplementary material.

It can be shown that the operator space may be truncated without loss of generality in a second-order treatment to only one- and two-electron excitation operators [52]. The wavefunction may also be truncated at second order. This truncation breaks the orthonormality of the T ^†₁ space:

$$ \left({\mathbf{T}}_1^{\dagger}\Big|{\mathbf{T}}_1^{\dagger}\right)\approx {\left({\mathbf{T}}_1^{\dagger}\Big|{\mathbf{T}}_1^{\dagger}\right)}^{(0)}+{\left({\mathbf{T}}_1^{\dagger}\Big|{\mathbf{T}}_1^{\dagger}\right)}^{(2)}\ne \left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill -1\hfill \end{array}\right) . $$

(163)

This complication is dealt with by orthonormalizing our operator space. The new operator set expressed in terms of the original set contains only second-order corrections:

$$ \begin{array}{l}{\left[{\widehat{a}}^{\dagger}\widehat{i}\right]}^{(2)}={\displaystyle \sum_b}\left(\frac{1}{4}{\displaystyle \sum_{kld}}\frac{\left(kd\left|\right| lb\right)\left(dk\left|\right| al\right)}{\varepsilon_{kl,bd}{\varepsilon}_{kl,da}}+{\displaystyle \sum_k}\frac{M_{kb}{M}_{ka}}{\varepsilon_{k,b}{\varepsilon}_{k,a}}\right){\widehat{b}}^{\dagger}\widehat{i}\\ {} +{\displaystyle \sum_j}\left(\frac{1}{4}{\displaystyle \sum_{mcd}}\frac{\left(md\left|\right|jc\right)\left(ci\left|\right| dm\right)}{\varepsilon_{mj, cd}{\varepsilon}_{im, cd}}+{\displaystyle \sum_d}\frac{M_{jd}{M}_{di}}{\varepsilon_{j,d}{\varepsilon}_{i,d}}\right){\widehat{a}}^{\dagger}\widehat{j} .\end{array} $$

(164)

(It should be noted that we have used the linked-cluster theorem to eliminate contributions from disconnected diagrams. For a proof for the EOM of the one- and two-particle the Green’s function, see [55].)

We may now proceed to calculate

$$ \begin{array}{l}-{\Pi}_{sr,qp}^{(2)}\left(\omega \right)={\left({\widehat{p}}^{\dagger}\widehat{q}\Big|{\mathbf{T}}_1^{\dagger}\right)}^{(1)}{\mathbf{P}}^{(1),-1}\left(\omega \right){\left({\mathbf{T}}_1^{\dagger}\Big|{\widehat{r}}^{\dagger}\widehat{s}\right)}^{(0)}\\ {} +{\left({\widehat{p}}^{\dagger}\widehat{q}\Big|{\mathbf{T}}_1^{\dagger}\right)}^{(0)}{\mathbf{P}}^{(1),-1}\left(\omega \right){\left({\mathbf{T}}_1^{\dagger}\Big|{\widehat{r}}^{\dagger}\widehat{s}\right)}^{(1)}\\ {} +{\left({\widehat{p}}^{\dagger}\widehat{q}\Big|{\mathbf{T}}_1^{\dagger}\right)}^{(1)}{\mathbf{P}}^{(0),-1}\left(\omega \right){\left({\mathbf{T}}_1^{\dagger}\Big|{\widehat{r}}^{\dagger}\widehat{s}\right)}^{(1)}\\ {} +{\left({\widehat{p}}^{\dagger}\widehat{q}\Big|{\mathbf{T}}_1^{\dagger}\right)}^{(0)}{\mathbf{P}}^{(2),-1}\left(\omega \right){\left({\mathbf{T}}_1^{\dagger}\Big|{\widehat{r}}^{\dagger}\widehat{s}\right)}^{(0)} .\end{array} $$

(165)

The only new contributions which arise at this level are from the block P ⁽²⁾, which is given by

$$ {\mathbf{P}}^{(2)}={\boldsymbol{\varGamma}}_{1,1}^{(2)}-{\boldsymbol{\varGamma}}_{1,2}^{(1)}{\boldsymbol{\varGamma}}_{2,2}^{(0),-1}\left(\omega \right){\boldsymbol{\varGamma}}_{2,1}^{(1)} . $$

(166)

(We are anticipating the ω-dependence of the various Γ-blocks which are derived below.) Because the block Γ ⁽²⁾_1,1 is affected by the orthonormalization procedure, it may be useful to provide a few more details. Expanding order-by-order,

$$ \begin{array}{l}{\boldsymbol{\varGamma}}_{1,1}^{(2)}=\left\langle {0}^{(1)}\left|\left[{\mathbf{T}}_1^{\dagger },\left[\omega \overset{\smile }{1}+{\overset{\smile }{H}}^{(0)},{\mathbf{T}}_1^{\dagger}\right]\right]\right|{0}^{(1)}\right\rangle \\ {} +\left\langle {0}^{(0)}\left|\left[{\mathbf{T}}_1^{\dagger },\left[\omega \overset{\smile }{1}+{\overset{\smile }{H}}^{(0)},{\mathbf{T}}_1^{\dagger}\right]\right]\right|{0}^{(2)}\right\rangle \\ {} +\left\langle {0}^{(2)}\left|\left[{\mathbf{T}}_1^{\dagger },\left[\omega \overset{\smile }{1}+{\overset{\smile }{H}}^{(0)},{\mathbf{T}}_1^{\dagger}\right]\right]\right|{0}^{(0)}\right\rangle \\ {} +\left\langle {0}^{(0)}\left|\left[{\mathbf{T}}_1^{\dagger (2)},\left[\omega \overset{\smile }{1}+{\overset{\smile }{H}}^{(0)},{\mathbf{T}}_1^{\dagger}\right]\right]\right|{0}^{(0)}\right\rangle \\ {} +\left\langle {0}^{(0)}\left|\left[{\mathbf{T}}_1^{\dagger },\left[\omega \overset{\smile }{1}+{\overset{\smile }{H}}^{(0)},{\mathbf{T}}_1^{\dagger (2)}\right]\right]\right|{0}^{(0)}\right\rangle \\ {} +\left\langle {0}^{(1)}\right|\left[{\mathbf{T}}_1^{\dagger },\left[{\widehat{H}}^{(1)},{\mathbf{T}}_1^{\dagger}\right]\right|{0}^{(0)}\Big\rangle \\ {} +\left\langle {0}^{(0)}\right|\left[{\mathbf{T}}_1^{\dagger },\left[{\widehat{H}}^{(1)},{\mathbf{T}}_1^{\dagger}\right]\right|{0}^{(1)}\Big\rangle ,\end{array} $$

(167)

where T ^† (2)₁ is the vector of second-order operators defined in (164). It is easily shown that the first term cancels with the contributions coming from the second-order operators, and that the contributions from second-order wave function are exactly zero. Hence, that block is simply

$$ {\boldsymbol{\varGamma}}_{1,1}^{(2)}=\left\langle {0}^{(1)}\right|\left[{\mathbf{T}}_1^{\dagger },\left[{\widehat{H}}^{(1)},{\mathbf{T}}_1^{\dagger}\right]\right|{0}^{(0)}\left\rangle +\right\langle {0}^{(0)}\left|\right[{\mathbf{T}}_1^{\dagger },\left[{\widehat{H}}^{(1)},{\mathbf{T}}_1^{\dagger}\right]\left|{0}^{(1)}\right\rangle, $$

(168)

which makes it frequency-independent. Its calculation gives

$$ {\left[{\varGamma}_{1,1}^{(2)}\right]}_{kc,ia}={\delta}_{ac}{\displaystyle \sum_d}\frac{M_{kd}{M}_{di}}{\varepsilon_{i,d}}+{\delta}_{ik}{\displaystyle \sum_l}\frac{M_{la}{M}_{lc}}{\varepsilon_{l,a}}+\frac{\delta_{ac}}{2}{\displaystyle \sum_{lde}}\frac{\left(le\left|\right|kd\right)\left( dl\left|\right|ei\right)}{\varepsilon_{im,de}}-\frac{\delta_{ik}}{2}{\displaystyle \sum_{lmd}}\frac{\left(ld\left|\right|mc\right)\left( dl\left|\right|ma\right)}{\varepsilon_{lm, ad}}, $$

(169)

$$ \begin{array}{l}{\left[{\varGamma}_{1,1}^{(2)}\right]}_{ck,ia}=\frac{M_{ak}{M}_{id}}{\varepsilon_{i,d}}+\frac{M_{ci}{M}_{ka}}{\varepsilon_{k,a}}\\ {} +2{\displaystyle \sum_d}\frac{M_{dk}\left( ad\left|\right|ci\right)}{\varepsilon_{k,d}}+2{\displaystyle \sum_l}\frac{M_{lc}\left(lk\left|\right| ai\right)}{\varepsilon_{l,c}}\\ {} -{\displaystyle \sum_{md}}\frac{\left(ce\left|\right| ad\right)\left(di\left|\right|em\right)}{\varepsilon_{im,de}}-{\displaystyle \sum_{me}}\frac{\left(ce\left|\right|mi\right)\left(ak\left|\right| me\right)}{\varepsilon_{km,ae}}\\ {} -\frac{1}{2}{\displaystyle \sum_{de}}\frac{\left(ce\left|\right| ad\right)\left(dk\left|\right|ei\right)}{\varepsilon_{ik,de}}-\frac{1}{2}{\displaystyle \sum_{ml}}\frac{\left(ik\left|\right| ml\right)\left(ac\left|\right| ml\right)}{\varepsilon_{lm,ac}} .\end{array} $$

(170)

The block Γ _1,2 and its adjoint is of at least first order because the space is orthonormal. For that reason, it is not affected by the orthonormalization at this level of approximation. Its calculation gives

$$ \begin{array}{l}{\left[{\varGamma}_{2,1}^{(2)}\right]}_{kc, jbia}=-{\delta}_{ik}\left(bc\left|\right|aj\right)+{\delta}_{jk}\left(bc\left|\right| ai\right)-{\delta}_{bc}\left( ai\left|\right|kj\right)+{\delta}_{ac}\left( bi\left|\right|kj\right)\\ {}{\left[{\varGamma}_{2,1}^{(2)}\right]}_{ck, jbia}=0 .\end{array} $$

(171)

Finally, the block Γ _2,2(ω) gives

$$ \begin{array}{l}{\left[{\varGamma}_{2,2}^{(2)}\left(\omega \right)\right]}_{ldkc, jbia}=\left(\omega -{\varepsilon}_{ij, ab}\right){\delta}_{jl}{\delta}_{ik}{\delta}_{ca}{\delta}_{db}\\ {}{\left[{\varGamma}_{2,2}^{(2)}\left(\omega \right)\right]}_{ckdl, jbia}=0 \end{array} $$

(172)

It should be noted that double excitations are treated only to zeroth-order in a second-order approach. To obtain a consistent theory with first-order corrections to double excitations, one should go at least to third order. This however becomes computationally quite heavy.

It is interesting to speculate what would happen if we were to include the first-order doubles correction within the present second-order theory. There are, in fact, indications that this can lead to improved agreement between calculated and experimental double excitations, though the quality of the single excitations is simultaneously decreased because of an imbalanced treatment [110, 111].

We can now construct the PP necessary to construct the second approximation of the xc-kernel (142) according to (149). Because the localizers of both left- and right-sides are constructed from the noninteracting KS PP, we are only concerned with ph and hp contributions. This means that the blocks involving pp or hh indices, corresponding to density shift operators, can be ignored at this level of approximation. This simplifies the construction of P(ω) in (149), which, up to second order, gives

$$ {\boldsymbol{\varPi}}^{\left(0+1+2\right),-1}\left(\omega \right)={\left({\boldsymbol{T}}_1^{\dagger}\Big|{\boldsymbol{T}}_1^{\dagger}\right)}^{-1}{\boldsymbol{P}}^{\left(0+1+2\right)}\left(\omega \right){\left({\boldsymbol{T}}_1^{\dagger}\Big|{\boldsymbol{T}}_1^{\dagger}\right)}^{-1} . $$

(173)

Separating ph and hp contributions, the PP takes the form of a 2 × 2 block-matrix in the same spirit as the LR-TD-DFT formulation of Casida,

$$ \begin{array}{l}{\boldsymbol{\varPi}}^{\left(0+1+2\right),-1}\left(\omega \right)\\ {} =\left(\begin{array}{cc}\hfill \mathbf{1}\hfill & \hfill \mathbf{0}\hfill \\ {}\hfill \mathbf{0}\hfill & \hfill -\mathbf{1}\hfill \\ {}\hfill \hfill & \hfill \hfill \end{array}\right)\left(\begin{array}{cc}\hfill {\boldsymbol{P}}^{\left(0+1+2\right)}\left(\omega \right)\hfill & \hfill {\boldsymbol{P}}^{\left(0+1+2\right)}\left(\omega \right)\hfill \\ {}\hfill {\boldsymbol{P}}^{\left(0+1+2\right)}\left(\omega \right)\hfill & \hfill {\boldsymbol{P}}^{\left(0+1+2\right)}\left(\omega \right)\hfill \\ {}\hfill \hfill & \hfill \hfill \end{array}\right)\left(\begin{array}{cc}\hfill \mathbf{1}\hfill & \hfill \mathbf{0}\hfill \\ {}\hfill \mathbf{0}\hfill & \hfill -\mathbf{1}\hfill \\ {}\hfill \hfill & \hfill \hfill \end{array}\right)\\ {} =\left(\begin{array}{rr}\hfill {\boldsymbol{P}}^{\left(0+1+2\right)}\left(\omega \right)& \hfill -{\boldsymbol{P}}^{\left(0+1+2\right)}\left(\omega \right)\\ {}\hfill -{\boldsymbol{P}}^{\left(0+1+2\right)}\left(\omega \right)& \hfill {\boldsymbol{P}}^{\left(0+1+2\right)}\left(\omega \right)\\ {}\hfill & \hfill \end{array}\right) .\end{array} $$

(174)

It follows that

$$ {\boldsymbol{\varPi}}_s^{-1}\left(\omega \right)-{\boldsymbol{\varPi}}^{\left(0+1+2\right),-1}\left(\omega \right)=\left(\begin{array}{rr}\hfill {\boldsymbol{P}}^{\left(1+2\right)}\left(\omega \right)& \hfill -{\boldsymbol{\varGamma}}_{1,1}^{\left(1+2\right)}\\ {}\hfill -{\boldsymbol{\varGamma}}_{1,1}^{\left(1+2\right)}& \hfill {\boldsymbol{P}}^{\left(1+2\right)}\left(\omega \right)\\ {}\hfill & \hfill \end{array}\right) . $$

(175)

Note that the off-diagonal (ph,hp)- and (hp,ph)-blocks are frequency-independent and that the diagonal blocks are given by (166). Ignoring localization for the moment, we may now cast the present Kohn–Sham based second-order polarization propagator approximation (SOPPA/KS) into the familiar form of (27) with

$$ \begin{array}{l}{A}_{ia,jb}\left(\omega \right)={\delta}_{i,j}{\delta}_{a,b}{\varepsilon}_{a,i}+{P}_{ia,jb}^{\left(1+2\right)}\left(\omega \right)\\ {}{B}_{ia,bj}\left(\omega \right)=-{\left({\varGamma}_{1,1}^{\left(1+2\right)}\right)}_{ia,bj} .\end{array} $$

(176)

Localization – see (142) –complicates these formulae by mixing the ${\boldsymbol{P}}^{\left(1+2\right)}\left(\omega \right)$ and ${\boldsymbol{\varGamma}}_{1,1}^{\left(1+2\right)}$ terms,

$$ \begin{array}{l}{A}_{ia,jb}\left(\omega \right)={\delta}_{i,j}{\delta}_{a,b}\left({\varepsilon}_a-{\varepsilon}_i\right)\\ {} +{\left[{\left({\boldsymbol{\varLambda}}_s\right)}_{hp, hp}\left(\omega \right){\boldsymbol{P}}^{\left(1+2\right)}\left(\omega \right){\left({\boldsymbol{\varLambda}}_s^{\dagger}\right)}_{hp, hp}\left(\omega \right)\right]}_{ia,jb}\\ {} +{\left[{\left({\boldsymbol{\varLambda}}_s\right)}_{hp, ph}\left(\omega \right){\boldsymbol{P}}^{\left(1+2\right)}\left(\omega \right){\left({\boldsymbol{\varLambda}}_s^{\dagger}\right)}_{ph, hp}\left(\omega \right)\right]}_{ia,jb}\\ {} -{\left[{\left({\boldsymbol{\varLambda}}_s\right)}_{hp, ph}\left(\omega \right){\boldsymbol{\varGamma}}^{\left(1+2\right)}{\left({\boldsymbol{\varLambda}}_s^{\dagger}\right)}_{hp, hp}\left(\omega \right)\right]}_{ia,jb}\\ {} -{\left[{\left({\boldsymbol{\varLambda}}_s\right)}_{hp, hp}\left(\omega \right){\boldsymbol{\varGamma}}^{\left(1+2\right)}{\left({\boldsymbol{\varLambda}}_s^{\dagger}\right)}_{ph, hp}\left(\omega \right)\right]}_{ia,jb}\\ {}{B}_{ia,bj}\left(\omega \right)={\left[{\left({\boldsymbol{\varLambda}}_s\right)}_{hp, hp}{\boldsymbol{P}}^{\left(1+2\right)}\left(\omega \right){\left({\boldsymbol{\varLambda}}_s^{\dagger}\right)}_{hp, ph}\right]}_{ia,bj}\\ {} +{\left[{\left({\boldsymbol{\varLambda}}_s\right)}_{hp, ph}{\boldsymbol{P}}^{\left(1+2\right)}\left(\omega \right){\left({\boldsymbol{\varLambda}}_s^{\dagger}\right)}_{ph, ph}\right]}_{ia,bj}\\ {} -{\left[{\left({\boldsymbol{\varLambda}}_s\right)}_{hp, ph}\left(\omega \right){\boldsymbol{\varGamma}}^{\left(1+2\right)}{\left({\boldsymbol{\varLambda}}_s^{\dagger}\right)}_{hp, ph}\left(\omega \right)\right]}_{ia,bj}\\ {} -{\left[{\left({\boldsymbol{\varLambda}}_s\right)}_{hp, hp}\left(\omega \right){\boldsymbol{\varGamma}}^{\left(1+2\right)}{\left({\boldsymbol{\varLambda}}_s^{\dagger}\right)}_{ph, ph}\left(\omega \right)\right]}_{ia,bj} .\end{array} $$

(177)

Of course, this extra complication is unnecessary if all we want to do is to calculate improved excitation energies and transition amplitudes by means of DFT-based many-body perturbation theory. It is only needed when our goal is to study the effect of localization on purely TDDFT quantities such as the xc-kernel and the TDDFT vectors X and Y.