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Pierre-Francois Loos 2020-05-25 08:11:03 +02:00
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BSEdyn.tex
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@ -39,8 +39,8 @@
\newcommand{\T}[1]{#1^{\intercal}}
% coordinates
\newcommand{\br}[1]{\mathbf{r}_{#1}}
\newcommand{\dbr}[1]{d\br{#1}}
\newcommand{\br}{\mathbf{r}}
\newcommand{\dbr}{d\br}
% methods
\newcommand{\evGW}{ev$GW$}
@ -336,7 +336,7 @@ Picking up the $e^{+i \Oms t_2 }$ component in $L(1,2; 1',2')$ and $L(6,2;5,2')$
\titou{For the lowest $\Oms$ excitation energies falling in the quasiparticle gap of the system}, $L_0(1,2;1',2')$ cannot contribute \titou{to?} since its lowest excitation energy is precisely the quasiparticle gap, namely the difference between the electronic affinity and the ionization potential.
\titou{T2: I think we should specify at which level of theory this quasiparticle gap is computed. What do you think?}
The Fourier components with respect to $t_1$ of $L_0$ reads, dropping the (space/spin) variables:
The Fourier components with respect to $t_1$ of $L_0$ reads, dropping the (space/spin) variables
\begin{align} \label{eq:iL0}
[iL_0]( \omega_1 )
= \int \frac{d\omega}{2\pi} \; G\qty(\omega - \frac{\omega_1}{2} ) G\qty( {\omega} + \frac{\omega_1}{2} )
@ -347,7 +347,9 @@ We now adopt the Lehman representation of the one-body Green's function in the q
\begin{equation}
G(\bx_1,\bx_2 ; \omega) = \sum_p \frac{ \phi_p(\bx_1) \phi_p^*(\bx_2) } { \omega - \e{p} + i \eta \times \text{sgn} (\e{p} - \mu) }
\end{equation}
where the $\lbrace \e{p} \rbrace$ are quasiparticle energies and $\lbrace \phi_p \rbrace$ the associated one-body molecular orbitals, namely $GW$ quasiparticle energies and input Hartree-Fock molecular orbitals in the present study. After projecting onto \titou{$\phi_a^*(\bx_1) \phi_i(\bx_{1'})$}, one obtains the $\omega_1= \Oms$ component
where \titou{$\mu$ is the chemical potential}.
The set $\lbrace \e{p} \rbrace$ are quasiparticle energies and $\lbrace \phi_p \rbrace$ is their associated one-body \titou{(spin)} orbitals, \titou{namely $GW$ quasiparticle energies and input Hartree-Fock molecular orbitals in the present study. (T2: shall we really mention this here?)}
After projecting onto \titou{$\phi_a^*(\bx_1) \phi_i(\bx_{1'})$}, one obtains the $\omega_1= \Oms$ component
\begin{multline}
\int d\bx_1 d\bx_{1'} \; \phi_a^*(\bx_1) \phi_i(\bx_{1'}) L_0(\bx_1,3;\bx_{1'},4; \Oms)
\\
@ -356,6 +358,7 @@ where the $\lbrace \e{p} \rbrace$ are quasiparticle energies and $\lbrace \phi_
\times \qty[ \theta( \tau ) e^{i ( \e{i} + \hOms) \tau } + \theta( - \tau ) e^{i (\e{a} - \hOms \tau) } ]
\end{multline}
with $\tau = \tau_{34}$.
\titou{We used chemist notations with $(i,j)$ indexing occupied orbitals and $(a,b)$ virtual ones. (T2: I wouldn't call that chemist notations...)}
Adopting now the $GW$ approximation for the exchange-correlation self-energy, \ie,
\begin{equation}
@ -366,26 +369,26 @@ leads to the following simplified BSE kernel
\Xi(3,5;4,6) = v(3,6) \delta(3,4) \delta(5,6) - W(3^+,4) \delta(3,6) \delta(4,5),
\end{equation}
where $W$ is its dynamically-screened Coulomb operator.
We further obtain the needed spectral representation of $\mel{N}{T \hpsi(3) \hpsi^{\dagger}(4)}{N,s}$ expanding the field operators over a complete orbital basis creation/destruction operators:
We further obtain the needed spectral representation of $\mel{N}{T \hpsi(3) \hpsi^{\dagger}(4)}{N,s}$ expanding the field operators over a complete orbital basis creation/destruction operators \titou{(T2: I don't understand why we need this spectral representation)}:
\begin{multline}
\mel{N}{T \hpsi(3) \hpsi^{\dagger}(4)}{N,s}
\\
= - \qty( e^{ -i \Omega_s t^{34} } ) \sum_{pq} \phi_p(\bx_3) \phi_q^*(\bx_4)
\mel{N}{\ha_q^{\dagger} \ha_p}{N,s}
\\
\times \qty[ \theta( \tau ) e^{- i ( \e{p} - \hOms ) \tau } + \theta( -\tau ) e^{ - i ( \e{q} + \hOms) \tau } ]
\times \qty[ \theta( \tau_{34} ) e^{- i ( \e{p} - \hOms ) \tau_{34} } + \theta( -\tau_{34} ) e^{ - i ( \e{q} + \hOms) \tau_{34} } ]
\end{multline}
with $\tau = \tau_{34}$ and where the $ \lbrace \eps_{n/m} \rbrace$ are proper addition/removal energies such that
where the $ \lbrace \eps_{p/q} \rbrace$ are proper addition/removal energies \titou{(T2: shall it be mentioned earlier around Eq. (14)?)} such that
\begin{equation}
e^{i {\hat H} \tau} \ha_p^{\dagger} \ket{N} = e^{ i (E_0^N + \e{p} ) \tau } \ha_p^{\dagger} \ket{N}
e^{i \hH \tau} \ha_p^{\dagger} \ket{N} = e^{ i (E_0^N + \e{p} ) \tau } \ha_p^{\dagger} \ket{N}
\end{equation}
with ${\hat H}$ the exact many-body Hamiltonian.
with $\hH$ the exact many-body Hamiltonian.
The $GW$ quasiparticle energies $\eGW{i/a}$ are good approximations to such removal/addition energies.
Selecting $(p,q)=(j,b)$ yields the largest components
$X_{jb}^{s} = \mel{N}{\ha_j^{\dagger} \ha_b}{N,s}$, while $(p,q)=(b,j)$ yields much weaker
$Y_{jb}^{s} = \mel{N}{\ha_b^{\dagger} \ha_j}{N,s}$ contributions. We used chemist notations with $(i,j)$ indexing occupied orbitals and $(a,b)$ virtual ones.
$Y_{jb}^{s} = \mel{N}{\ha_b^{\dagger} \ha_j}{N,s}$ contributions.
Neglecting the $Y_{jb}^{s}$ weights leads to the Tamm-Dancoff approximation (TDA).
Working out the same expansion for $\mel{N}{T \hpsi(5) \hpsi^{\dagger}(5)}{N,s}$ and $\mel{N}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1'})}{N,s}$, and projecting onto $\phi_a^*(\bx_1) \phi_i(\bx_{1'})$, one obtains after a few tedious manipulations (see {\SI}) the dynamical Bethe-Salpeter equation (dBSE) :
Working out the same expansion for $\mel{N}{T \hpsi(5) \hpsi^{\dagger}(5)}{N,s}$ and $\mel{N}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1'})}{N,s}$ \titou{(where do we need these terms?)}, and projecting onto $\phi_a^*(\bx_1) \phi_i(\bx_{1'})$, one obtains after a few tedious manipulations (see {\SI}) the dynamical Bethe-Salpeter equation (dBSE):
\begin{equation}
\begin{split}
( \e{a} - \e{i} - \Oms ) X_{ia}^{s}
@ -401,14 +404,16 @@ with an effective dynamically screened Coulomb potential (see Pina eq. 24):
\\
\times \qty[ \frac{1}{ \Omega_{ib}^s - \omega + i \eta } + \frac{1}{ \Omega_{ja}^{s} + \omega + i\eta } ]
\end{multline}
where $\Om{ib}{s} = \Oms - ( \e{b} - \e{i} )$ and $\Om{ja}{s} = \Oms - ( \e{a} - \e{j} )$. The Coulomb matrix elements are defined following the Mulliken notations :
\begin{align*}
v_{ai,bj} &= \int d{\bf r} d{\bf r}' \; \phi_a({\bf r}) \phi_i^*({\bf r}) v({\bf r} -{\bf r}')
\phi_b^*({\bf r}') \phi_j({\bf r}') \\
W_{ij,ab}({\omega}) &= \int d{\bf r} d{\bf r}' \; \phi_i({\bf r}) \phi_j^*({\bf r}) W({\bf r} ,{\bf r}'; \omega)
\phi_a^*({\bf r}') \phi_b({\bf r}')
\end{align*}
where we group together the indices of orbitals taken at the same space position, taking further as inner indices those associated with MO with complex conjugation.
where $\Om{ib}{s} = \Oms - ( \e{b} - \e{i} )$ and $\Om{ja}{s} = \Oms - ( \e{a} - \e{j} )$.
Following Mulliken's notations, the Coulomb matrix elements are defined as
\begin{align}
v_{ai,bj}
& = \int d\br d\br' \; \phi_a(\br) \phi_i^*(\br) v(\br -\br') \phi_b^*(\br') \phi_j(\br'),
\\
W_{ij,ab}({\omega})
& = \int d\br d\br' \; \phi_i(\br) \phi_j^*(\br) W(\br ,\br'; \omega) \phi_a^*(\br') \phi_b(\br'),
\end{align}
where we group together the indices of orbitals taken at the same space position, taking further as inner indices those associated with orbitals with complex conjugation.
\xavier{A second coupled equation for the $(X_{ia}^{s}, Y_{ia}^{s} )$ vector can be obtained by projecting now onto the $\mel{N}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1'})}{N,s}$ left-hand side and right-hand-side of the BSE, leading to : }
@ -471,7 +476,7 @@ For a closed-shell system in a finite basis, to compute the BSE excitation energ
\label{eq:LR-dyn}
\begin{pmatrix}
\bA{}(\omega) & \bB{}(\omega) \\
-\bB{}(\omega) & -\bA{}(\omega) \\
-\bB{}(\titou{-}\omega) & -\bA{}(\titou{-}\omega) \\
\end{pmatrix}
\cdot
\begin{pmatrix}
@ -548,12 +553,13 @@ with
where the $\e{p}$'s are taken as the HF orbital energies in the case of $G_0W_0$ or as the $GW$ quasiparticle energies in the case of self-consistent schemes such as ev$GW$.
Now, let us decompose, using basic perturbation theory, the non-linear eigenproblem \eqref{eq:LR-dyn} as a zeroth-order static (hence linear) reference and a first-order dynamic (hence non-linear) perturbation, such that
\begin{equation}
\begin{multline}
\label{eq:LR-PT}
\begin{pmatrix}
\bA{}(\omega) & \bB{}(\omega) \\
-\bB{}(\omega) & -\bA{}(\omega) \\
-\bB{}(\titou{-}\omega) & -\bA{}(\titou{-}\omega) \\
\end{pmatrix}
\\
=
\begin{pmatrix}
\bA{(0)} & \bB{(0)} \\
@ -561,10 +567,10 @@ Now, let us decompose, using basic perturbation theory, the non-linear eigenprob
\end{pmatrix}
+
\begin{pmatrix}
\bA{(1)}(\omega) & \bB{(1)}(\omega) \\
-\bB{(1)}(\omega) & -\bA{(1)}(\omega) \\
\bA{(1)}(\omega) & \bB{(1)}(\omega) \\
-\bB{(1)}(\titou{-}\omega) & -\bA{(1)}(\titou{-}\omega) \\
\end{pmatrix}
\end{equation}
\end{multline}
with
\begin{subequations}
\begin{align}
@ -642,7 +648,7 @@ and, thanks to first-order perturbation theory, the first-order correction to th
\cdot
\begin{pmatrix}
\bA{(1)}(\Om{m}{(0)}) & \bB{(1)}(\Om{m}{(0)}) \\
-\bB{(1)}(\Om{m}{(0)}) & -\bA{(1)}(\Om{m}{(0)}) \\
-\bB{(1)}(\titou{-}\Om{m}{(0)}) & -\bA{(1)}(\titou{-}\Om{m}{(0)}) \\
\end{pmatrix}
\cdot
\begin{pmatrix}