{ "cells": [ { "cell_type": "markdown", "id": "73d4113e", "metadata": {}, "source": [ "# CC Amplitude\n", "\n", "## CC ground state lambda amplitudes\n", "\n", "The CC ground state lambda Lagrangian is given by\n", "\n", "$$\n", "L = \\langle \\Phi_0|(1+\\Lambda) \\bar{H} | \\Phi_0 \\rangle = 0\n", "$$\n", "\n", "The CC lambda equations are given by\n", "\n", "$l_1$:\n", "$$\n", "\\lambda_{a}^{i} = \\frac{\\partial L}{\\partial t_i^a} = \\frac{\\partial E_\\text{CC}}{\\partial t_i^a} + \\sum_{ck}\\lambda_{c}^{k}\\frac{\\partial \\Omega_{kc}}{\\partial t_i^a} + \\sum_{klcd}\\lambda_{cd}^{kl}\\frac{\\partial \\Omega_{klcd}}{\\partial t_i^a} = 0\n", "$$\n", "\n", "$l_2$:\n", "$$\n", "\\lambda_{ab}^{ij} = \\frac{\\partial L}{\\partial t_{ij}^{ab}} = \\frac{\\partial E_\\text{CC}}{\\partial t_{ij}^{ab}} + \\sum_{ck}\\lambda_{c}^{k}\\frac{\\partial \\Omega_{ck}}{\\partial t_{ij}^{ab}} + \\sum_{klcd}\\lambda_{cd}^{kl}\\frac{\\partial \\Omega_{klcd}}{\\partial t_{ij}^{ab}} = 0\n", "$$\n", "\n", "we define:\n", "\n", "$$\n", "A_{\\mu_i\\nu_j} = \\frac{\\partial\\Omega_i}{\\partial t_j}\n", "$$\n", "\n", "For CC2 $l_1$, we have:\n", "\n", "$$\n", "A_{\\mu_1\\nu_1} = \\langle\\Phi_i^a|[\\bar{H}_1+[\\bar{H}_1,T_2],a^\\dagger i]|0\\rangle\n", "$$\n", "$$\n", "A_{\\mu_2\\nu_1} = \\langle\\Phi_{ij}^{ab}|[\\bar{H}_1,a^\\dagger i]|0\\rangle\n", "$$\n", "\n", "and for $l_2$:\n", "\n", "$$\n", "A_{\\mu_1\\nu_2} = \\langle\\Phi_{i}^{a}|[\\bar{H}_1,a^\\dagger b^\\dagger ij]|0\\rangle\n", "$$\n", "\n", "$$\n", "A_{\\mu_2\\nu_2} = \\langle\\Phi_{ij}^{ab}|[F,a^\\dagger b^\\dagger ij]|0\\rangle\n", "$$\n", "\n", "where:\n", "\n", "$$\n", "\\bar{H}_1 = e^{-T_1}\\hat{H}e^{T_1}\n", "$$\n", "\n", "Compared to CCSD lambda amplitudes, all terms containing $t_2l_2$ are missing in the CC2 $\\lambda_1$ equations. All terms containing $t_2$ and $l_2\\langle pq||rs\\rangle$ are missing in the CC2 $\\lambda_2$ equations. Therefore, we can write the full expression for CC2 lambda amplitudes as:\n", "\n", "$l_1$:\n", "$$\n", "\\begin{aligned}\n", "l_a^i\\Delta_i^a = &\n", "F^{(2)}_{ia} + \\sum_c l_c^i F^{(2)}_{ca} - \\sum_k l_a^k F^{(2)}_{ki} \\\\\n", "& - \\frac{1}{2}\\sum_{jbc} l_{bc}^{ij} I^{(3a)}_{jabc} - \\sum_{kc} l_{c}^{k} I^{(1)}_{kaic} - \\frac{1}{2}\\sum_{jkb} l_{ab}^{jk} I^{(2b)}_{jkib}\n", "\\end{aligned}\n", "$$\n", "\n", "$l_2$:\n", "$$\n", "\\begin{aligned}\n", "l_{ab}^{ij}\\Delta_{ij}^{ab} = &\n", "\\langle ij||ab\\rangle + P(ab)\\sum_k I^{(6)}_{ijka}l_{b}^{k} + P(ab)\\sum_{c} l_{ac}^{ij}F^{(2a)}_{cb} \\\\\n", "&+ P(ij)\\sum_c I^{(7)}_{icab}l_{c}^{j} + P(ij)\\sum_{k} l_{ab}^{jk}F^{(2a)}_{ki} + P(ab)P(ij)l_a^iF^{(2)}_{jb} \\\\\n", "\\end{aligned}\n", "$$\n", "\n", "The intermediates are defined as:\n", "$$\n", "\\begin{aligned}\n", "& F^{(1)}_{ca} = f_{ca} + \\sum_{kd}\\langle kc||da\\rangle-\\frac{1}{2}\\sum_{kld}\\langle kl||ad\\rangle t_{kl}^{cd}\n", "\\\\\n", "& F^{(2)}_{ia} = f_{ia} + \\sum_{jb}\\langle ij||ab\\rangle t_j^b\n", "\\\\\n", "& F^{(2)}_{ca} = F^{(1)}_{ca} - f_{ca} - \\sum_{k}f_{ka}t_k^c - \\sum_{kld}\\langle kl||ad\\rangle t_k^ct_l^d\n", "\\\\\n", "& F^{(2)}_{ik} = f_{ik} + \\sum_af_{ka}t_i^a + \\sum_{la}\\langle kl||ia\\rangle t_l^a + \\sum_{lab}\\langle kl||ab\\rangle t_i^at_l^b + \\frac{1}{2}\\sum_{lab}\\langle kl||ab\\rangle t_{il}^{ab}\n", "\\\\\n", "&F^{(2a)}_{ki} = f_{ki} + \\sum_a f_{ia}t_k^a\n", "\\\\\n", "&F^{(2a)}_{cb} = f_{cb} - \\sum_k f_{kb}t_k^c\n", "\\\\\n", "&I^{(1)}_{kaic}= \\langle ka||ic\\rangle - \\sum_{d}\\langle ic||ad\\rangle t_{k}^{d} - \\sum_{ld}\\langle il||ad\\rangle t_{kl}^{cd} + \\sum_{ld}\\langle il||ad\\rangle t_{k}^{d}t_l^c-\\sum_l\\langle il||ka\\rangle t_l^c\n", "\\\\\n", "&I^{(2b)}_{jkib} = \\langle jk||ib\\rangle - \\sum_l I^{(4)}_{jkil}t_l^b - P(jk)\\sum_c\\langle ib||kc\\rangle t_j^c + \\frac{1}{2}\\sum_{cd}\\langle ib || cd\\rangle \\tilde{t}_{jk}^{cd}\\\\\n", "\\\\\n", "&I^{(3a)}_{jabc} = \\langle ja||bc\\rangle - \\sum_d I^{(5)}_{bcad}t_j^d + P(bc)\\sum_k[\\langle kb||ja\\rangle -\\frac{1}{2}\\sum_l\\langle kl || ja\\rangle t_l^b]t_k^c\\\\\n", "\\\\\n", "&I^{(6)}_{ijka} = \\langle ij||ka\\rangle - \\sum_{d}\\langle ij||ad\\rangle t_k^d\n", "\\\\\n", "&I^{(7)}_{icab} = \\langle ic||ab\\rangle - \\sum_{k}\\langle ik||ab\\rangle t_c^k\n", "\\\\\n", "&\\tilde{t}_{ij}^{ab} = \\frac{1}{2}P(ij)P(ab)t_i^at_j^b\n", "\\end{aligned}\n", "$$\n", "\n", "For RI-CC2, the lambda equations are given by:\n", "\n", "$l_1$:\n", "$$\n", "\\begin{aligned}\n", "l_a^i\\Delta_i^a = &\n", "F^{(2)}_{ia} + \\sum_c l_c^i F^{(2)}_{ca} - \\sum_k l_a^k F^{(2)}_{ki} -\\sum_{jbcP} l_{bc}^{ij}M_{ac}^PM_{jbP}^{2}\n", "\\\\\n", "& - \\sum_{kc} l_{c}^{k} \\left\\{\\sum_PM_{ik}^PM_{ac}^P-\\sum_PM_{kc}^PB_{ia}^P+\\sum_{ld}t_{kl}^{cd}\\left(\\sum_PB_{id}^PB_{la}^P\\right)\\right\\}\n", "\\\\\n", "& - \\frac{1}{2}\\sum_{jkb} l_{ab}^{jk} I^{(2b)}_{jkib}\n", "\\end{aligned}\n", "$$\n", "\n", "$l_2$:\n", "$$\n", "\\begin{aligned}\n", "l_{ab}^{ij}\\Delta_{ij}^{ab} = &\n", "P(ab)\\left\\{\\sum_PB_{ia}^PB_{jb}^P+\\sum_c F_{cb}^{(2a)}l_{ac}^{ij}\\right\\} + P(ij)\\left\\{\\sum_kF_{ki}^{(2a)}l_{ab}^{jk}+F_{jb}^{(2)}l_a^i\\right\\} \\\\\n", "& + P(ab)\\left\\{\\sum_PB_{ia}^P\\left(\\sum_cM_{bc}^Pl_c^j-\\sum_kM^P_{jk}l_b^k\\right)\\right\\}\n", "\\end{aligned}\n", "$$\n", "\n", "The intermediates are defined as:\n", "$$\n", "\\begin{aligned}\n", "&F^{(2)}_{ia}=f_{ia}+\\sum_{P}B_{ia}^PM^P - \\sum_{jP}B_{ja}^P(M_{ij}^P - B_{ij}^P)\n", "\\\\\n", "&F^{(2)}_{bc}=f_{bc}-\\sum_{kP}B_{kc}^P(M^{2T}_{kbP}+M^{3T}_{kbP}-M^{2TT}_{kbP})+\\sum_P M^PM^P_{cb}-\\sum_k f_{kc}t_k^b\n", "\\\\\n", "&F^{(2)}_{jk}=f_{jk}+\\sum_c f_k^ct_j^c + \\sum_PM_{kj}^PM^P - \\sum_{lP}M_{lj}^P(M_{kl}^P-B_{kl}^P) + \\sum_{cP}B_{kc}^PM_{jcP}^{2T}\n", "\\\\\n", "&F_{bc}^{(2a)}=f_{bc}-\\sum_k f_{kc}t_k^b\n", "\\\\\n", "&F_{jk}^{(2a)}=f_{jk}+\\sum_c f_k^ct_j^c\n", "\\\\\n", "&F_{ia}^{(2)} = f_{ia}+\\sum_PB_{ia}^PM^P - \\sum_{jP}B_{ja}^P(M_{ij}^P - B_{ij}^P)\n", "\\\\\n", "&I^{(2b)}_{ijka} = P(ij)\\sum_P M_{kj}^P\\left\\{M_{iaP}^{1T}+ M_{iaP}^{2TT}-B_{ia}^P-M^{3T}_{iaP}\\right\\}\n", "\\\\\n", "&M_{ij}^P = B_{ij}^P + \\sum_cB_{ic}^Pt_j^c\n", "\\\\\n", "&M_{ab}^P = B_{ab}^P - \\sum_iB_{ia}^Pt_i^b\n", "\\\\\n", "&M_{ia}^P = \\sum_c B_{ac}^P t_i^c - \\sum_k B_{ik}^P t_k^a - \\sum_k t_k^a(M_{ki}^P - B_{ki}^P) + B_{ia}^P + \\sum_{ld}B_{ld}^Pt_{il}^{ad}\n", "\\\\\n", "&M_{iaP}^{2} = \\sum_c B_{ac}^P t_i^c - \\sum_k B_{ik}^P t_k^a - \\sum_k t_k^a(M_{ki}^P - B_{ki}^P) + B_{ia}^P \n", "\\\\\n", "&M^P = \\sum_{ia}B_{ia}^Pt_i^a\n", "\\\\\n", "&M_{iaP}^{1T}=\\sum_{k}B_{ik}^Pt_{k}^{a}\n", "\\\\\n", "&M_{iaP}^{2T}=\\sum_{ld}B_{ld}^Pt_{il}^{ad}\n", "\\\\\n", "&M_{iaP}^{3T}=\\sum_{c}B_{ac}^Pt_{i}^{c}\n", "\\\\\n", "&M_{iaP}^{2TT}=\\sum_kt_k^a(M_{ki}^P - B_{ki}^P)\n", "\\\\\n", "\\end{aligned}\n", "$$" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.11.4" } }, "nbformat": 4, "nbformat_minor": 5 }