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zPbjNAc<-#1@oe5PdV6T94Zqvc{ypb;hYh7LH~PPiI~FD)=Ko1F{U78~MMEcB=O5as zsJ)G=t(`LwJENkLsfneri@g&O^Uvt}Z+ui~F&!!ZD<`KWfQ6Y;#9I!)%EAUUhtQ;%=R(AHE0u}%Z3#T15(f{8Dx{T6xX7)e5Wg=4kKWMLP?2O9x_AW&K zVW%lGs#toN5;3#0{SOYTrn0FSqlOMM5i1e1?*BS#jE#VRDI_2ufFUS|tl=~8%f}w_ z0+sdz5 indice du 1er CFG qui a i SOMO -* Boucler sur toutes les CFG mono-excitees par rapport a toutes les autres - -* p,q,cfg -> 0|1|2|3 (Type d'excitation) - - -222200000000 -000010100101 -p->q -222200200000 -000000000101 - -SOMO->SOMO - -DOMO -> SOMO - -do p in DO - do q in not(DO or SO) - p->q + 2 DOMO->VMO - - -do p in DO - do q in not(DO or SO) - p->q + 2 DOMO->VMO -DOMO -> VMO -test si bit=1 dans DO -test si bit=1 dans VO diff --git a/Notes_CSF/Theory_CFG_CIPSI.org b/Notes_CSF/Theory_CFG_CIPSI.org deleted file mode 100644 index 511af0d1..00000000 --- a/Notes_CSF/Theory_CFG_CIPSI.org +++ /dev/null @@ -1,184 +0,0 @@ -#+TITLE: CFG CIPSI -#+AUTHOR: Vijay Gopal Chilkuri -#+EMAIL: vijay.gopal.c@gmail.com -#+DATE: 2020-12-08 Tue 08:27 -#+startup: latexpreview -#+STARTUP: inlineimages - - -#+LATEX_HEADER: \usepackage{braket} - -* Biblio -* Theoretical background - - Here we describe the main theoretical background and definitions of the - Configuration (CFG) based CIPSI algorithm. The outline of the document is as follows. - First, we give some definitions of the CFG many-particle basis follwed by the - definitions of the overlap, one-particle, and two-particle matrix-elements. Finally, - an algorithm is presented for the sigma-vector (\( \sigma \)-vector defined later) calculation using - the CFG basis. - - -** Definitinon of CI basis - - In CFG based CIPSI, the wavefunction is represented in CFG basis - as shown in Eq: [[Eq:definebasis1]]. - - #+LATEX: \newcommand{\Ncfg}{N_{\text{CFG}}} - #+LATEX: \newcommand{\Ncsf}{N_{\text{CSF}}} - #+LATEX: \newcommand{\Nsomo}{N_{\text{SOMO}}} - #+NAME: Eq:definebasis1 - \begin{equation} - \ket{\Psi} = \sum_{i=1}^{\Ncfg} \sum_{j=1}^{\Ncsf(i)} c_{ij} {^S\ket{\Phi^j_i}} - \end{equation} - - - where the \(\ket{\Phi^j_i}\) represent Configuration State Functions (CSFs) - which are expanded in terms of Bonded functions (BFs) as shown in - [[Eq:definebasis2]]. - #+NAME: Eq:definebasis2 - \begin{equation} - \ket{\Phi^j_i} = \sum_k O^{\Nsomo(i)}_{kj} \ket{^S\phi_k(i,j)} - \end{equation} - where the functions \(\ket{^S\phi_k(i,j)}\) represent the BFs for the CFG - \(\ket{^S\Phi_i}\). - The coefficients \(O^b_{a,k}\) depend only on the number of SOMOs - in \(\Phi_i\). - - Each CFG contains a list of CSFs related to it which describes the - spin part of the wavefunction (see Eq: [[Eq:definebasis3]]) which is - encoded in the BFs as shown below in Eq: [[Eq:definebasis5]]. - - - #+NAME: Eq:definebasis3 - \begin{equation} - \ket{^S\Phi_i} = \left\{ \ket{^S\Phi^1_i}, \ket{^S\Phi^2_i}, \dots, \ket{^s\Phi^{\Ncsf}_i} \right\} - \end{equation} - - - - #+NAME: Eq:definebasis4 - \begin{equation} - \mathbf{c}_i = \left\{ c^1_i, c^2_i, \dots, c^{\Ncsf}_i \right\} - \end{equation} - - - Each of the CSFs belonging to the CFG \(\ket{^S\Phi_i}\) have coefficients - associated to them as shown in Eq: [[Eq:definebasis4]]. Crucially, the bonded functions - defined in Eq: [[Eq:definebasis5]] are not northogonal to each other. - - - #+NAME: Eq:definebasis5 - \begin{equation} - \ket{^S\phi_k(i,j)} = (a\bar{a})\dots (b\ c) (d (e - \end{equation} - $i$ is the index of the CFG and $j$ determines the coupling. - - - The bonded functions are made up of products of slater determinants. There are - three types of determinants, first, the closed shell pairs \((a\bar{a})\). Second, - the open-shell singlet pairs \((b\ c)\) which are expanded as - \((b\ c) = \frac{\ket{b\bar{c}}-\ket{\bar{b}c}}{\sqrt{2}}\). Third, the - open-shell SOMOs which are coupled parallel and account for the total spin of the - wavefunction \((l (m \dots\). They are shown as open brackets. - -** Overlap of the wavefunction - - Once the wavefunction has been expanded in terms of the CSFs, the most fundamental - operation is to calculate the overlap between two states. The overlap in the - basis of CSFs is defined as shown in Eq: [[Eq:defineovlp1]]. - - - #+NAME: Eq:defineovlp1 - \begin{equation} - \braket{^S\Phi_i|^S\Phi_j} = \sum_{kl} C_i C_j \braket{^S\Psi^k_i|^S\Psi^l_j} - \end{equation} - - - Where the sum is over the CSFs \(k\) and \(l\) corresponding to the \(i\) - and \(j\) CFGs respectively. The overlap between the CSFs can be expanded in terms - of the BFs using the definition given in Eq: [[Eq:definebasis2]] and - Eq: [[Eq:definebasis3]] as given in Eq: [[Eq:defineovlp2]]. - - - #+NAME: Eq:defineovlp2 - \begin{equation} - \braket{^S\Phi^k_i|^S\Phi^l_j} = \sum_m \sum_n \left( O^k_{i,m}\right)^{\dagger} \braket{^S\phi_m(i,k)|^S\phi_n(j,l)} O^l_{j,n} - \end{equation} - - - Therefore, the overlap between two CSFs can be expanded in terms of the overlap - between the constituent BFs. The overlap matrix \(S_{mn}\) is of dimension \(\left( N^k_{N_{BF}} , N^l_{N_{BF}} \right)\). - The equation shown above (Eq: [[Eq:defineovlp2]]) can be written in marix-form as - shown below in Eq: [[Eq:defineovlp3]]. - - #+NAME: Eq:defineovlp3 - \begin{equation} - \braket{^S\Phi_i|^S\Phi_j} = \left( C_{i,1} \right)^{\dagger} \mathbf{O}_i\cdot\mathbf{S}_{ij}\cdot\mathbf{O}_j C_{j,1} - \end{equation} - - - Note that the overlap between two CFGs does not depend on the orbital - labels. It only depends on the number of Singly Occupied Molecular Orbitals - (SOMOs) therefore it can be pretabulated. Actually, it is possible to - redefine the CSFs in terms of a linear combination of BFs such that - \(S_{ij}\) becomes the identity matrix. In this case, one needs to store the - orthogonalization matrix \(\mathbf{\tilde{O}}_i\) which is given by - \(\mathbf{O}_i\cdot S^{1/2}_i\) for a given CFG \(i\). Note that the a CFG - \(i\) is by definition of an orthonormal set of MOs automatically orthogonal - to a CFG \(j\) with a different occupation. - -** Definition of matrix-elements - - The matrix-element (ME) evaluation follows a similar logic as the evalulation of - the overlap. However, here the metric is the one-, or two-particle operator \(\hat{E}_{pq}\) - or \(\hat{E}_{pq}\hat{E}_{rs}\) as shown in Eq: [[Eq:defineme1]] and Eq: [[Eq:defineme2]]. - - #+NAME: Eq:defineme1 - \begin{equation} - \braket{^S\Phi^k_i|\hat{O}_{pq}|^S\Phi^l_j} = \left( C_{i,1} \right)^{\dagger} \mathbf{O}_i\cdot\mathbf{A}^{pq}_{ij}\cdot\mathbf{O}_j C_{j,1} - \end{equation} - - #+NAME: Eq:defineme2 - \begin{equation} - \braket{^S\Phi^k_i|\hat{O}_{pq,rs}|^S\Phi^l_j} = \sum_K \left( C_{i,1} \right)^{\dagger} \mathbf{O}_i\cdot\mathbf{A}^{pq}_{ik}\cdot\mathbf{O}_k \mathbf{O}_k\cdot\mathbf{A}^{rs}_{kj}\cdot\mathbf{O}_j C_{j,1} - \end{equation} - - - Where, \(\hat{O}_{pq}\) and \(hat{O}_{pq,rs}\) represent an arbitrary one-, and - two-particle operators respectively. Importantly, the one-, and two-particle - matrix-element evaluation can be recast into an effecient matrix multiplication - form which is crucial for a fast evaluation of the action of the operators - \(\hat{O}_{pq}\) and \(hat{O}_{pq,rs}\). The matrix \(\mathbf{A}^{pq}_{ij}\) contains - the result of the operation \(\braket{^S\Phi^k_i|\hat{O}_{pq}|^S\Phi^l_j}\) in terms - of BFs and is therefore of size \(NCSF(i) \textit{x} NBF(i)\). In this formulation, the determinant basis is entirely avoided. - Note that the size and contents of the matrix \(\mathbf{A}^{pq}_{ij}\) depends - only on the total number of SOMOs and the total spin \(S\), therefore, an optimal - prototyping scheme can be deviced for a rapid calculaiton of these matrix contractions. - The resolution of identity (RI) is used to evaluate the two-particle operator since - this alleviates the necessacity to explicity construct two-particle matrix-elements - \(\braket{^S\Phi^k_i|\hat{O}_{pq,rs}|^S\Phi^l_j}\) directly. - -** Sigma-vector evaluation - - Once the \(\mathbf{A}^{pq}_{ij}\) matrices have been constructed for the given - selected list of CFGs, the prototype lists for the \(\mathbf{A}^{pq}_{ij}\) matrices - can be constructed. Following this, one can proceede to the evaluation of the sigma-vector - as defined in the Eq [[Eq:definesigma1]]. - - - #+NAME: Eq:definesigma1 - \begin{equation} - \sigma = \sum_{pq} \tilde{h}_{pq}\hat{E}_{pq}|\ket{^S\Phi^l_j} + \frac{1}{2}\sum_{pq,rs} g_{pq,rs} \hat{E}_{pq}\hat{E}_{rs}|\ket{^S\Phi^l_j} - \end{equation} - - The one-electron part of the sigma-vector can be calculated as shown in Eq: [[Eq:defineme1]] - and the two-electron part can be calculated using the RI as shown in Eq: [[Eq:defineme2]]. - The most expensive part involves the two-particle operator as shown on the RHS of Eq: [[Eq:definesigma1]]. - In this CFG formulation, the cost intensive part of the sigma-vector evaluation has been recast - into an efficient BLAS matrix multiplication operation. Therefore, this formulation is the most efficient - albeit at the cost of storing the prototype matrices \(\mathbf{A}^{pq}_{ij}\). However, where the total spin - is small and the largest number of SOMOs does not exceed 14, the \(\mathbf{A}^{pq}_{ij}\) matrices - can be stored in memory. - - diff --git a/RELEASE_NOTES.org b/RELEASE_NOTES.org index ab0d771a..e5fb24a7 100644 --- a/RELEASE_NOTES.org +++ b/RELEASE_NOTES.org @@ -73,7 +73,7 @@ - Added Shank function - Added utilities for periodic calculations - Added ~V_ne_psi_energy~ - - Added ~h_core_huess~ routine + - Added ~h_core_guess~ routine - Fixed Laplacians in real space (indices) - diff --git a/src/two_body_rdm/two_e_dm_mo.irp.f b/src/two_body_rdm/two_e_dm_mo.irp.f index 107161c8..19e8d75e 100644 --- a/src/two_body_rdm/two_e_dm_mo.irp.f +++ b/src/two_body_rdm/two_e_dm_mo.irp.f @@ -1,4 +1,4 @@ -BEGIN_PROVIDER [double precision, two_e_dm_mo, (mo_num,mo_num,mo_num,mo_num,1)] +BEGIN_PROVIDER [double precision, two_e_dm_mo, (mo_num,mo_num,mo_num,mo_num)] implicit none BEGIN_DOC ! two_e_dm_bb_mo(i,j,k,l,istate) = STATE SPECIFIC physicist notation for 2RDM of beta/beta electrons @@ -29,12 +29,12 @@ BEGIN_PROVIDER [double precision, two_e_dm_mo, (mo_num,mo_num,mo_num,mo_num,1)] jorb = list_core_inact_act(j) do i=1,mo_num iorb = list_core_inact_act(i) - two_e_dm_mo(iorb,jorb,korb,lorb,1) = state_av_full_occ_2_rdm_spin_trace_mo(i,j,k,l) + two_e_dm_mo(iorb,jorb,korb,lorb) = state_av_full_occ_2_rdm_spin_trace_mo(i,j,k,l) enddo enddo enddo enddo - two_e_dm_mo(:,:,:,:,:) = two_e_dm_mo(:,:,:,:,:) * 2.d0 + two_e_dm_mo(:,:,:,:) = two_e_dm_mo(:,:,:,:) * 2.d0 END_PROVIDER

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