Orbital-optimized MP2.5 and its analytic gradients: Approaching CCSD(T) quality for noncovalent interactions

Bozkaya, Uğur; Sherrill, C. David

doi:10.1063/1.4902226

Orbital-optimized MP2.5 [or simply “optimized MP2.5,” OMP2.5, for short] and its analytic energy gradients are presented. The cost of the presented method is as much as that of coupled-cluster singles and doubles (CCSD) [O(N⁶) scaling] for energy computations. However, for analytic gradient computations the OMP2.5 method is only half as expensive as CCSD because there is no need to solve λ₂-amplitude equations for OMP2.5. The performance of the OMP2.5 method is compared with that of the standard second-order Møller–Plesset perturbation theory (MP2), MP2.5, CCSD, and coupled-cluster singles and doubles with perturbative triples (CCSD(T)) methods for equilibrium geometries, hydrogen transfer reactions between radicals, and noncovalent interactions. For bond lengths of both closed and open-shell molecules, the OMP2.5 method improves upon MP2.5 and CCSD by 38%–43% and 31%–28%, respectively, with Dunning's cc-pCVQZ basis set. For complete basis set (CBS) predictions of hydrogen transfer reaction energies, the OMP2.5 method exhibits a substantially better performance than MP2.5, providing a mean absolute error of 1.1 kcal mol⁻¹, which is more than 10 times lower than that of MP2.5 (11.8 kcal mol⁻¹), and comparing to MP2 (14.6 kcal mol⁻¹) there is a more than 12-fold reduction in errors. For noncovalent interaction energies (at CBS limits), the OMP2.5 method maintains the very good performance of MP2.5 for closed-shell systems, and for open-shell systems it significantly outperforms MP2.5 and CCSD, and approaches CCSD(T) quality. The MP2.5 errors decrease by a factor of 5 when the optimized orbitals are used for open-shell noncovalent interactions, and comparing to CCSD there is a more than 3-fold reduction in errors. Overall, the present application results indicate that the OMP2.5 method is very promising for open-shell noncovalent interactions and other chemical systems with difficult electronic structures.

I. INTRODUCTION

Orbital-optimized many-body perturbation theory and coupled-cluster methods have significant importance in modern computational chemistry as robust methods for the study of chemical systems where the orbital relaxation effects are important.^1–17 Previous studies demonstrated that the orbital-optimized methods are very beneficial for the molecular systems with challenging electronic structures, such as symmetry-breaking problems,^{3,8,9,12–14,18} transition states,^7,11,19 free radicals,^7,11,19 transition-metal complexes,^7,11 excited states,⁵ bond-breaking problems,^10,20,21 open-shell noncovalent interactions,^15,16 and the computation of ionization potentials²² and electron affinities.²³ More detailed discussions of the orbital-optimized methods can be found in our previous papers.^8–12

The accurate computation of noncovalent interaction energies is a great challenge for computational chemistry methods.^24–31 Second-order Møller–Plesset perturbation theory (MP2) is the most commonly employed wave function based electron correlation method for the study of weak interactions. On the other hand, third-order Møller–Plesset perturbation theory (MP3) has not been popular for investigating such interactions. The MP3 method [formally scaling as O(N⁶), where N is the number of basis functions] is computationally much expensive than MP2 [O(N⁵)] and its results are generally not much better than those of MP2.^32–38 Further, for dispersion-bound complexes, especially those with π–π interactions, MP2 generally overestimates interaction energies, while MP3 underestimates it.^32–38 It was suggested that the arithmetic mean of the MP2 and MP3 interaction energies may provide more accurate results than either method separately. This is the basic idea of the recently introduced MP2.5 and MP2.X methods.^36–38 The theoretical basis of the MP2.5 method was discussed by Hobza and co-workers in details.^36–38 Hence, we do not repeat that discussion. However, we note that the MP2.5 method is a special version of the spin scaled MP3 method³⁶ with the third-order energy scaling factor of 1/2.

In this study, the orbital-optimized MP2.5 method [or simply “optimized MP2.5,” OMP2.5, for short] and its analytic energy gradients are presented. The OMP2.5 method is applied to equilibrium geometries, hydrogen transfer reactions between free radicals, and noncovalent interactions. The equations presented have been implemented in a new computer code, OMP2.5, written by one of the authors (U.B.) and added to the PSI4 package.³⁹ The OMP2.5 implementation takes advantage of real Abelian point group symmetry utilizing the direct product decomposition (DPD) approach,^40,41 and both restricted and unrestricted Hartree–Fock (RHF and UHF) references are implemented.

II. THEORETICAL APPROACH

A. OMP2.5 energy

1. The MP2.5-Λ energy functional and amplitude equations

Let us consider the OMP2.5 energy first. Section II C will present the OMP2.5 analytic energy gradients. For the orbital indexing the usual notation is employed: i, j, k, l, m, n for occupied orbitals; a, b, c, d, e, f for virtual orbitals; and p, q, r, s, t, u, v, w for general spin orbitals. The MP2.5 correlation energy can be written as follows:

\begin{eqnarray}\Delta E_{MP2.5} &=& \langle 0| \big (\hat{W}_{N}\hat{T}_{2}^{(1)}\big )_{c} |0\rangle + \frac{1}{2}\langle 0| \big (\hat{W}_{N}\hat{T}_{2}^{(2)}\big )_{c} |0\rangle ,\end{eqnarray}

\begin{matrix} Δ E_{M P 2.5} & = & ⟨ 0 | {({\hat{W}}_{N}, {\hat{T}}_{2}^{(1)})}_{c} | 0 ⟩ + \frac{1}{2} ⟨ 0 | {({\hat{W}}_{N}, {\hat{T}}_{2}^{(2)})}_{c} | 0 ⟩, \end{matrix}

(1)

where

$\hat{W}_{N}$

{\hat{W}}_{N}

is the two-electron component of the normal ordered Hamiltonian operator,^41–43 |0⟩ is the reference determinant (Fermi-vacuum),

$\hat{T}_{2}^{(1)}$

{\hat{T}}_{2}^{(1)}

and

$\hat{T}_{2}^{(2)}$

{\hat{T}}_{2}^{(2)}

are the first- and second-order double excitation operator, and subscript c means only connected diagrams are included

\begin{eqnarray}\hat{T}_{2}^{(1)} &=& \frac{1}{4}\sum _{i,j}^{occ} \sum _{a,b}^{vir} t_{ij}^{ab(1)} \ \hat{a}^{\dagger } \hat{b}^{\dagger } \hat{j} \hat{i} ,\end{eqnarray}

\begin{matrix} {\hat{T}}_{2}^{(1)} & = & \frac{1}{4} \sum_{i, j}^{o c c} \sum_{a, b}^{v i r} t_{i j}^{a b (1)} {\hat{a}}^{†} {\hat{b}}^{†} \hat{j} \hat{i}, \end{matrix}

(2)

\begin{eqnarray}\hat{T}_{2}^{(2)} &=& \frac{1}{4}\sum _{i,j}^{occ} \sum _{a,b}^{vir} t_{ij}^{ab(2)} \ \hat{a}^{\dagger } \hat{b}^{\dagger } \hat{j} \hat{i} ,\end{eqnarray}

\begin{matrix} {\hat{T}}_{2}^{(2)} & = & \frac{1}{4} \sum_{i, j}^{o c c} \sum_{a, b}^{v i r} t_{i j}^{a b (2)} {\hat{a}}^{†} {\hat{b}}^{†} \hat{j} \hat{i}, \end{matrix}

(3)

where

$t_{ij}^{ab(1)}$

t_{i j}^{a b (1)}

and

$t_{ij}^{ab(2)}$

t_{i j}^{a b (2)}

are the first- and second-order double excitation amplitudes. We can write the MP2.5 energy more explicitly as follows:

\begin{eqnarray}E_{MP2.5} & = & E_{ref} + \frac{1}{4} \sum _{i,j}^{occ} \sum _{a,b}^{vir} t_{ij}^{ab} \langle ij||ab\rangle ,\end{eqnarray}

\begin{matrix} E_{M P 2.5} & = & E_{r e f} + \frac{1}{4} \sum_{i, j}^{o c c} \sum_{a, b}^{v i r} t_{i j}^{a b} ⟨ i j ‖ a b ⟩, \end{matrix}

(4)

\begin{eqnarray}t_{ij}^{ab} & = & t_{ij}^{ab(1)} + \frac{1}{2} t_{ij}^{ab(2)} ,\end{eqnarray}

\begin{matrix} t_{i j}^{a b} & = & t_{i j}^{a b (1)} + \frac{1}{2} t_{i j}^{a b (2)}, \end{matrix}

(5)

where E_MP2.5 is the MP2.5 energy, E_ref is the reference, self-consistent field (SCF), energy. The first- and second-order amplitude equations can be written as

\begin{eqnarray}\big\langle \Phi _{ij}^{ab}\big| \big (\hat{f}_{N}^{d}\hat{T}_{2}^{(1)} + \hat{W}_{N} \big )_{c} |0\rangle &=& 0 ,\end{eqnarray}

\begin{matrix} ⟨ Φ_{i j}^{a b} | {({\hat{f}}_{N}^{d} {\hat{T}}_{2}^{(1)} + {\hat{W}}_{N})}_{c} | 0 ⟩ & = & 0, \end{matrix}

(6)

\begin{eqnarray}\big\langle \Phi _{ij}^{ab}\big| \big (\hat{f}_{N}^{d}\hat{T}_{2}^{(2)} + \hat{W}_{N}\hat{T}_{2}^{(1)} \big )_{c} |0\rangle &=& 0 ,\end{eqnarray}

\begin{matrix} ⟨ Φ_{i j}^{a b} | {({\hat{f}}_{N}^{d} {\hat{T}}_{2}^{(2)} + {\hat{W}}_{N} {\hat{T}}_{2}^{(1)})}_{c} | 0 ⟩ & = & 0, \end{matrix}

(7)

where

$\hat{f}_{N}^{d}$

{\hat{f}}_{N}^{d}

is the diagonal part of

$\hat{f}_{N}$

{\hat{f}}_{N}

and

$\langle \Phi _{ij}^{ab}|$

⟨ Φ_{i j}^{a b} |

is doubly excited Slater determinant. The explicit forms of amplitude equations can be found in our previous study.⁹

Single excitations are not considered, although the Fock matrix will not be diagonal during OMP2.5 iterations, as in case of previously reported orbital-optimized methods.^8,9,13 It is well-known that the presence of the single excitations will destroy the convergence properties of the orbital-optimized methods.^2,7–9,13

In order to obtain a variational energy functional (⁠

${\cal L}$

L

⁠), it is convenient^44,45 to introduce a Lagrangian (OMP2.5-Λ functional) as follows:

\begin{eqnarray}{\cal L} &=& \langle 0| \hat{H} | 0 \rangle \nonumber \\&& + \, \langle 0| \big (\hat{W}_{N}\hat{T}_{2}^{(1)}\big )_{c} |0\rangle + \frac{1}{2} \langle 0| \big (\hat{W}_{N}\hat{T}_{2}^{(2)}\big )_{c} |0\rangle \nonumber \\&& + \, \langle 0| \big\lbrace \hat{\Lambda }_{2}^{(1)} \ \big (\hat{f}_{N}^{d}\hat{T}_{2}^{(1)} + \hat{W}_{N} \big )_{c}\big\rbrace _{c} |0\rangle \nonumber \\&& + \, \frac{1}{2} \langle 0| \big\lbrace \hat{\Lambda }_{2}^{(1)} \ \big (\hat{f}_{N}^{d}\hat{T}_{2}^{(2)} + \hat{W}_{N}\hat{T}_{2}^{(1)} \big )_{c}\big\rbrace _{c} |0\rangle \nonumber \\&& + \, \frac{1}{2} \langle 0| \big\lbrace \hat{\Lambda }_{2}^{(2)} \ \big (\hat{f}_{N}^{d}\hat{T}_{2}^{(1)} + \hat{W}_{N} \big )_{c}\big\rbrace _{c} |0\rangle ,\end{eqnarray}

\begin{matrix} L & = & ⟨ 0 | \hat{H} | 0 ⟩ \\ + ⟨ 0 | {({\hat{W}}_{N}, {\hat{T}}_{2}^{(1)})}_{c} | 0 ⟩ + \frac{1}{2} ⟨ 0 | {({\hat{W}}_{N}, {\hat{T}}_{2}^{(2)})}_{c} | 0 ⟩ \\ + ⟨ 0 | {\{{\hat{Λ}}_{2}^{(1)}, {({\hat{f}}_{N}^{d} {\hat{T}}_{2}^{(1)} + {\hat{W}}_{N})}_{c}\}}_{c} | 0 ⟩ \\ + \frac{1}{2} ⟨ 0 | {\{{\hat{Λ}}_{2}^{(1)}, {({\hat{f}}_{N}^{d} {\hat{T}}_{2}^{(2)} + {\hat{W}}_{N} {\hat{T}}_{2}^{(1)})}_{c}\}}_{c} | 0 ⟩ \\ + \frac{1}{2} ⟨ 0 | {\{{\hat{Λ}}_{2}^{(2)}, {({\hat{f}}_{N}^{d} {\hat{T}}_{2}^{(1)} + {\hat{W}}_{N})}_{c}\}}_{c} | 0 ⟩, \end{matrix}

(8)

where

$\hat{\Lambda }_{2}^{(1)}$

{\hat{Λ}}_{2}^{(1)}

and

$\hat{\Lambda }_{2}^{(2)}$

{\hat{Λ}}_{2}^{(2)}

are the first- and second-order double de-excitation operators. One can readily verify that⁹

\begin{eqnarray}\hat{\Lambda }_{2}^{(1)} &=& \hat{T}_{2}^{(1)\dagger },\end{eqnarray}

\begin{matrix} {\hat{Λ}}_{2}^{(1)} & = & {\hat{T}}_{2}^{(1) †}, \end{matrix}

(9)

\begin{eqnarray}\hat{\Lambda }_{2}^{(2)} &=& \hat{T}_{2}^{(2)\dagger }.\end{eqnarray}

\begin{matrix} {\hat{Λ}}_{2}^{(2)} & = & {\hat{T}}_{2}^{(2) †} . \end{matrix}

(10)

The standard OMP2.5 t₂-amplitude equations are obtained by requiring that

${\cal L}$

L

be stationary with respect to λ₂-amplitudes, while the stationary requirement with respect to t₂-amplitudes leads to the λ₂-amplitude equations.

2. The parametrization of the OMP2.5 wave function

For the parametrization of OMP2.5 wave function, we will follow our previous formulations.^8–12 The orbital variations may be performed with an exponential unitary operator^46–49

\begin{eqnarray}\tilde{\hat{p}}^{\dagger } &=& e^{\hat{K}} \hat{p}^{\dagger } e^{-\hat{K}},\end{eqnarray}

\begin{matrix} {\tilde{\hat{p}}}^{†} & = & e^{\hat{K}} {\hat{p}}^{†} e^{- \hat{K}}, \end{matrix}

(11)

\begin{eqnarray}\tilde{\hat{p}} &=& e^{\hat{K}} \hat{p} e^{-\hat{K}},\end{eqnarray}

\begin{matrix} \tilde{\hat{p}} & = & e^{\hat{K}} \hat{p} e^{- \hat{K}}, \end{matrix}

(12)

\begin{eqnarray}| \tilde{p} \rangle &=& e^{\hat{K}} | p \rangle ,\end{eqnarray}

\begin{matrix} | \tilde{p} ⟩ & = & e^{\hat{K}} | p ⟩, \end{matrix}

(13)

where

$\hat{K}$

\hat{K}

is the orbital rotation operator

\begin{eqnarray}\hat{K} &=& \sum _{p,q}^{} K_{pq} \hat{p}^{\dagger } \hat{q} = \sum _{p>q}^{} \kappa _{pq} (\hat{p}^{\dagger } \hat{q} - \hat{q}^{\dagger } \hat{p}),\end{eqnarray}

\begin{matrix} \hat{K} & = & \sum_{p, q} K_{p q} {\hat{p}}^{†} \hat{q} = \sum_{p > q} κ_{p q} ({\hat{p}}^{†} \hat{q} - {\hat{q}}^{†} \hat{p}), \end{matrix}

(14)

\begin{eqnarray}{\bf K} &=& Skew({\bf \kappa }),\end{eqnarray}

\begin{matrix} K & = & S k e w (κ), \end{matrix}

(15)

where κ_pq are the orbital rotation parameters. The effect of the orbital rotations on the molecular orbital (MO) coefficients can be written as

\begin{eqnarray}{\bf C({\bf \kappa })} &=& {\bf C^{(0)}} e^{{\bf K}},\end{eqnarray}

\begin{matrix} C (κ) & = & C^{(0)} e^{K}, \end{matrix}

(16)

where C⁽⁰⁾ is the initial MO coefficient matrix and

${\bf C({\bf \kappa })}$

C (κ)

is the new MO coefficient matrix as a function of κ.

Now, let us define a variational energy functional (Lagrangian) as a function of κ,

\begin{eqnarray}{\cal L}({\bf \kappa }) &=& \langle 0| \hat{H}^{\kappa } | 0 \rangle \nonumber \\&& + \, \langle 0| \big (\hat{W}_{N}^{\kappa }\hat{T}_{2}^{(1)}\big )_{c} |0\rangle + \frac{1}{2} \langle 0| \big (\hat{W}_{N}^{\kappa }\hat{T}_{2}^{(2)}\big )_{c} |0\rangle \nonumber \\&& + \, \langle 0| \big\lbrace \hat{\Lambda }_{2}^{(1)} \ \big (\hat{f}_{N}^{\kappa }\hat{T}_{2}^{(1)} + \hat{W}_{N}^{\kappa } \big )_{c}\big\rbrace _{c} |0\rangle \nonumber \\&& + \, \frac{1}{2} \langle 0| \big\lbrace \hat{\Lambda }_{2}^{(1)} \ \big (\hat{f}_{N}^{\kappa }\hat{T}_{2}^{(2)} + \hat{W}_{N}^{\kappa }\hat{T}_{2}^{(1)} \big )_{c}\big\rbrace _{c} |0\rangle \nonumber \\&& + \, \frac{1}{2} \langle 0| \big\lbrace \hat{\Lambda }_{2}^{(2)} \ \big (\hat{f}_{N}^{\kappa }\hat{T}_{2}^{(1)} + \hat{W}_{N}^{\kappa } \big )_{c}\big\rbrace _{c} |0\rangle ,\end{eqnarray}

\begin{matrix} L (κ) & = & ⟨ 0 | {\hat{H}}^{κ} | 0 ⟩ \\ + ⟨ 0 | {({\hat{W}}_{N}^{κ}, {\hat{T}}_{2}^{(1)})}_{c} | 0 ⟩ + \frac{1}{2} ⟨ 0 | {({\hat{W}}_{N}^{κ}, {\hat{T}}_{2}^{(2)})}_{c} | 0 ⟩ \\ + ⟨ 0 | {\{{\hat{Λ}}_{2}^{(1)}, {({\hat{f}}_{N}^{κ} {\hat{T}}_{2}^{(1)} + {\hat{W}}_{N}^{κ})}_{c}\}}_{c} | 0 ⟩ \\ + \frac{1}{2} ⟨ 0 | {\{{\hat{Λ}}_{2}^{(1)}, {({\hat{f}}_{N}^{κ} {\hat{T}}_{2}^{(2)} + {\hat{W}}_{N}^{κ} {\hat{T}}_{2}^{(1)})}_{c}\}}_{c} | 0 ⟩ \\ + \frac{1}{2} ⟨ 0 | {\{{\hat{Λ}}_{2}^{(2)}, {({\hat{f}}_{N}^{κ} {\hat{T}}_{2}^{(1)} + {\hat{W}}_{N}^{κ})}_{c}\}}_{c} | 0 ⟩, \end{matrix}

(17)

where operators

$\hat{H}^{\kappa }$

{\hat{H}}^{κ}

⁠,

$\hat{f}_{N}^{\kappa }$

{\hat{f}}_{N}^{κ}

⁠, and

$\hat{W}_{N}^{\kappa }$

{\hat{W}}_{N}^{κ}

are defined as

\begin{eqnarray}\hat{H}^{\kappa } & = & e^{-\hat{K}} \hat{H} e^{\hat{K}},\end{eqnarray}

\begin{matrix} {\hat{H}}^{κ} & = & e^{- \hat{K}} \hat{H} e^{\hat{K}}, \end{matrix}

(18)

\begin{eqnarray}\hat{f}_{N}^{\kappa } & = & e^{-\hat{K}} \hat{f}_{N}^{d} e^{\hat{K}},\end{eqnarray}

\begin{matrix} {\hat{f}}_{N}^{κ} & = & e^{- \hat{K}} {\hat{f}}_{N}^{d} e^{\hat{K}}, \end{matrix}

(19)

\begin{eqnarray}\hat{W}_{N}^{\kappa } & = & e^{-\hat{K}} \hat{W}_{N} e^{\hat{K}}.\end{eqnarray}

\begin{matrix} {\hat{W}}_{N}^{κ} & = & e^{- \hat{K}} {\hat{W}}_{N} e^{\hat{K}} . \end{matrix}

(20)

The first and second derivatives of the Lagrangian with respect to the parameter κ at κ = 0 can be written as

\begin{eqnarray}w_{pq} &=& \frac{\partial {\cal L}}{\partial \kappa _{pq}} \bigg|_{{\bf \kappa }=0},\end{eqnarray}

\begin{matrix} w_{p q} & = & \frac{\partial L}{\partial κ_{p q}} |_{κ = 0}, \end{matrix}

(21)

\begin{eqnarray}A_{pq,rs} &=& \frac{\partial ^2{\cal L}}{\partial \kappa _{pq}\partial \kappa _{rs}} \bigg|_{{\bf \kappa }=0}.\end{eqnarray}

\begin{matrix} A_{p q, r s} & = & \frac{\partial^{2} L}{\partial κ_{p q} \partial κ_{r s}} |_{κ = 0} . \end{matrix}

(22)

Then the Lagrangian can be expanded up to second-order as follows:

\begin{equation}{\cal L}^{(2)}({\bf \kappa }) = {\cal L}^{(0)} + {\bf \kappa ^{\dagger } w} + \frac{1}{2} {\bf \kappa ^{\dagger } A \kappa },\end{equation}

L^{(2)} (κ) = L^{(0)} + κ^{†} w + \frac{1}{2} κ^{†} A κ,

(23)

where w is the MO gradient vector, κ is the MO rotation vector, and A is the MO Hessian matrix. Hence, minimizing the Lagrangian with respect to κ yields

\begin{eqnarray}{\bf \kappa } &=& -{\bf A^{-1}w}.\end{eqnarray}

\begin{matrix} κ & = & - A^{- 1} w . \end{matrix}

(24)

This final equation corresponds to the usual Newton-Raphson step. Hence, within an iterative procedure the optimized orbitals are obtained.

3. Response density matrices

Since the orbital gradient expression can be most conveniently presented in terms of response density matrices,^8,9 it is appropriate to first introduce unrelaxed response density matrices for the OMP2.5 Lagrangian (OMP2.5-Λ functional)

\begin{eqnarray}\gamma _{pq} &=& \langle 0|\hat{p}^{\dagger } \hat{q}|0 \rangle + \langle 0| \big \lbrace \hat{\Lambda }_{2}^{(1)} \ \big ( \lbrace \hat{p}^{\dagger } \hat{q}\rbrace ^{d} \ \hat{T}_{2}^{(1)}\big )_{c}\big \rbrace _{c} |0\rangle \nonumber \\&& + \, \frac{1}{2} \langle 0| \big \lbrace \hat{\Lambda }_{2}^{(2)} \ \big ( \lbrace \hat{p}^{\dagger } \hat{q}\rbrace ^{d} \ \hat{T}_{2}^{(1)}\big )_{c}\big \rbrace _{c} |0\rangle \nonumber \\&& +\, \ \frac{1}{2} \langle 0| \big \lbrace \hat{\Lambda }_{2}^{(1)} \ \big ( \lbrace \hat{p}^{\dagger } \hat{q}\rbrace ^{d} \ \hat{T}_{2}^{(2)}\big )_{c}\big \rbrace _{c} |0\rangle ,\end{eqnarray}

\begin{matrix} γ_{p q} & = & ⟨ 0 | {\hat{p}}^{†} \hat{q} | 0 ⟩ + ⟨ 0 | {\{{\hat{Λ}}_{2}^{(1)}, {({{\hat{p}}^{†} \hat{q}}^{d}, {\hat{T}}_{2}^{(1)})}_{c}\}}_{c} | 0 ⟩ \\ + \frac{1}{2} ⟨ 0 | {\{{\hat{Λ}}_{2}^{(2)}, {({{\hat{p}}^{†} \hat{q}}^{d}, {\hat{T}}_{2}^{(1)})}_{c}\}}_{c} | 0 ⟩ \\ + \frac{1}{2} ⟨ 0 | {\{{\hat{Λ}}_{2}^{(1)}, {({{\hat{p}}^{†} \hat{q}}^{d}, {\hat{T}}_{2}^{(2)})}_{c}\}}_{c} | 0 ⟩, \end{matrix}

(25)

\begin{eqnarray}\Gamma _{pqrs} &=& \frac{1}{4} \langle 0|\hat{p}^{\dagger } \hat{q}^{\dagger } \hat{s} \hat{r}|0 \rangle + \frac{1}{4} \langle 0| \big ( \lbrace \hat{p}^{\dagger } \hat{q}^{\dagger } \hat{s} \hat{r}\rbrace \ \hat{T}_{2}^{(1)}\big )_{c} |0\rangle \nonumber \\&& + \, \frac{1}{4} \langle 0| \big ( \hat{\Lambda }_{2}^{(1)} \ \lbrace \hat{p}^{\dagger } \hat{q}^{\dagger } \hat{s} \hat{r}\rbrace \big )_{c} |0\rangle \nonumber \\&&+\, \frac{1}{8} \langle 0| \big ( \lbrace \hat{p}^{\dagger } \hat{q}^{\dagger } \hat{s} \hat{r}\rbrace \ \hat{T}_{2}^{(2)}\big )_{c} |0\rangle \nonumber \\&& + \, \frac{1}{8} \langle 0| \big ( \hat{\Lambda }_{2}^{(2)} \ \lbrace \hat{p}^{\dagger } \hat{q}^{\dagger } \hat{s} \hat{r}\rbrace \big )_{c} |0\rangle \nonumber \\&&+\, \frac{1}{8} \langle 0| \big \lbrace \hat{\Lambda }_{2}^{(1)} \ \big ( \lbrace \hat{p}^{\dagger } \hat{q}^{\dagger } \hat{s} \hat{r}\rbrace \ \hat{T}_{2}^{(1)}\big )_{c}\big \rbrace _{c} |0\rangle ,\end{eqnarray}

\begin{matrix} Γ_{p q r s} & = & \frac{1}{4} ⟨ 0 | {\hat{p}}^{†} {\hat{q}}^{†} \hat{s} \hat{r} | 0 ⟩ + \frac{1}{4} ⟨ 0 | {({{\hat{p}}^{†} {\hat{q}}^{†} \hat{s} \hat{r}}, {\hat{T}}_{2}^{(1)})}_{c} | 0 ⟩ \\ + \frac{1}{4} ⟨ 0 | {({\hat{Λ}}_{2}^{(1)}, {{\hat{p}}^{†} {\hat{q}}^{†} \hat{s} \hat{r}})}_{c} | 0 ⟩ \\ + \frac{1}{8} ⟨ 0 | {({{\hat{p}}^{†} {\hat{q}}^{†} \hat{s} \hat{r}}, {\hat{T}}_{2}^{(2)})}_{c} | 0 ⟩ \\ + \frac{1}{8} ⟨ 0 | {({\hat{Λ}}_{2}^{(2)}, {{\hat{p}}^{†} {\hat{q}}^{†} \hat{s} \hat{r}})}_{c} | 0 ⟩ \\ + \frac{1}{8} ⟨ 0 | {\{{\hat{Λ}}_{2}^{(1)}, {({{\hat{p}}^{†} {\hat{q}}^{†} \hat{s} \hat{r}}, {\hat{T}}_{2}^{(1)})}_{c}\}}_{c} | 0 ⟩, \end{matrix}

(26)

where γ_pq and Γ_pqrs are the one- and two-particle response density matrices (OPDM and TPDM), respectively. Explicit equations for these density matrices can be readily obtained from those for MP3.^9,14 These PDMs are obtained multiplying the third-order OPDM and the second-order TPDM of the MP3 method,⁹ which correspond to the third-order energy correction, by a factor of

$\frac{1}{2}$

\frac{1}{2}

⁠. The OPDM and TPDM can be decomposed into reference and correlation contributions as follows:

\begin{eqnarray}\gamma _{pq} &=& \gamma _{pq}^{ref} + \gamma _{pq}^{corr},\end{eqnarray}

\begin{matrix} γ_{p q} & = & γ_{p q}^{r e f} + γ_{p q}^{c o r r}, \end{matrix}

(27)

\begin{eqnarray}\nonumber \\\Gamma _{pqrs} &=& \Gamma _{pqrs}^{ref} + \Gamma _{pqrs}^{corr} \nonumber \\&& + \, \frac{1}{4} \ \delta _{pr}^{occ} \ \gamma _{qs}^{corr} + \frac{1}{4} \ \delta _{qs}^{occ} \ \gamma _{pr}^{corr} \nonumber \\& - & \frac{1}{4} \ \delta _{ps}^{occ} \ \gamma _{qr}^{corr} - \frac{1}{4} \ \delta _{qr}^{occ} \ \gamma _{ps}^{corr},\end{eqnarray}

\begin{matrix} Γ_{p q r s} & = & Γ_{p q r s}^{r e f} + Γ_{p q r s}^{c o r r} \\ + \frac{1}{4} δ_{p r}^{o c c} γ_{q s}^{c o r r} + \frac{1}{4} δ_{q s}^{o c c} γ_{p r}^{c o r r} \\ - & \frac{1}{4} δ_{p s}^{o c c} γ_{q r}^{c o r r} - \frac{1}{4} δ_{q r}^{o c c} γ_{p s}^{c o r r}, \end{matrix}

(28)

where

$\gamma _{pq}^{ref}$

γ_{p q}^{r e f}

and

$\Gamma _{pqrs}^{ref}$

Γ_{p q r s}^{r e f}

are the reference (SCF) contributions to PDMs, while

$\gamma _{pq}^{corr}$

γ_{p q}^{c o r r}

and

$\Gamma _{pqrs}^{corr}$

Γ_{p q r s}^{c o r r}

are the correlation contributions.

$\delta _{pr}^{occ}$

δ_{p r}^{o c c}

denotes Kronecker delta and superscript occ means that the orbital p must be an occupied orbital. The reference PDMs are given as follows:

\begin{eqnarray}\gamma _{pq}^{ref} &=& \delta _{pq}^{occ} ,\end{eqnarray}

\begin{matrix} γ_{p q}^{r e f} & = & δ_{p q}^{o c c}, \end{matrix}

(29)

\begin{eqnarray}\Gamma _{pqrs}^{ref} &=& \frac{1}{4} \ \big ( \delta _{pr}^{occ} \ \delta _{qs}^{occ} - \delta _{ps}^{occ} \ \delta _{qr}^{occ} \big ).\end{eqnarray}

\begin{matrix} Γ_{p q r s}^{r e f} & = & \frac{1}{4} (δ_{p r}^{o c c} δ_{q s}^{o c c} - δ_{p s}^{o c c} δ_{q r}^{o c c}) . \end{matrix}

(30)

It is noteworthy that the OPDM and TPDM have the same permutational symmetries as the one-electron and antisymmetrized two-electron integrals, respectively. The correlation contributions for non-zero blocks of response OPDMs can be written as follows:

\begin{eqnarray}\gamma _{ij}^{corr} &=& -\frac{1}{2} \sum _{m}^{occ} \sum _{e,f}^{vir} t_{im}^{ef(1)} \lambda _{ef}^{jm(1)} - \frac{1}{4} \sum _{m}^{occ} \sum _{e,f}^{vir} t_{im}^{ef(2)} \lambda _{ef}^{jm(1)} \nonumber \\&&-\, \frac{1}{4} \sum _{m}^{occ} \sum _{e,f}^{vir} t_{im}^{ef(1)} \lambda _{ef}^{jm(2)} ,\end{eqnarray}

\begin{matrix} γ_{i j}^{c o r r} & = & - \frac{1}{2} \sum_{m}^{o c c} \sum_{e, f}^{v i r} t_{i m}^{e f (1)} λ_{e f}^{j m (1)} - \frac{1}{4} \sum_{m}^{o c c} \sum_{e, f}^{v i r} t_{i m}^{e f (2)} λ_{e f}^{j m (1)} \\ - \frac{1}{4} \sum_{m}^{o c c} \sum_{e, f}^{v i r} t_{i m}^{e f (1)} λ_{e f}^{j m (2)}, \end{matrix}

(31)

\begin{eqnarray}\gamma _{ab}^{corr} &=& \frac{1}{2} \sum _{m,n}^{occ} \sum _{e}^{vir} t_{mn}^{be(1)} \lambda _{ae}^{mn(1)} + \frac{1}{4} \sum _{m,n}^{occ} \sum _{e}^{vir} t_{mn}^{be(2)} \lambda _{ae}^{mn(1)} \nonumber \\&&+\, \frac{1}{4} \sum _{m,n}^{occ} \sum _{e}^{vir} t_{mn}^{be(1)} \ \lambda _{ae}^{mn(2)} .\end{eqnarray}

\begin{matrix} γ_{a b}^{c o r r} & = & \frac{1}{2} \sum_{m, n}^{o c c} \sum_{e}^{v i r} t_{m n}^{b e (1)} λ_{a e}^{m n (1)} + \frac{1}{4} \sum_{m, n}^{o c c} \sum_{e}^{v i r} t_{m n}^{b e (2)} λ_{a e}^{m n (1)} \\ + \frac{1}{4} \sum_{m, n}^{o c c} \sum_{e}^{v i r} t_{m n}^{b e (1)} λ_{a e}^{m n (2)} . \end{matrix}

(32)

Similarly, the correlation contributions for unique non-zero blocks of response TPDMs can be written as

\begin{eqnarray}\Gamma _{ijkl}^{corr} &=& \frac{1}{16} \sum _{e,f}^{vir} t_{ij}^{ef(1)} \ \lambda _{ef}^{kl(1)} ,\end{eqnarray}

\begin{matrix} Γ_{i j k l}^{c o r r} & = & \frac{1}{16} \sum_{e, f}^{v i r} t_{i j}^{e f (1)} λ_{e f}^{k l (1)}, \end{matrix}

(33)

\begin{eqnarray}\Gamma _{abcd}^{corr} &=& \frac{1}{16} \sum _{m,n}^{occ} t_{mn}^{cd(1)} \ \lambda _{ab}^{mn(1)} ,\end{eqnarray}

\begin{matrix} Γ_{a b c d}^{c o r r} & = & \frac{1}{16} \sum_{m, n}^{o c c} t_{m n}^{c d (1)} λ_{a b}^{m n (1)}, \end{matrix}

(34)

\begin{eqnarray}\Gamma _{iajb}^{corr} &=& - \frac{1}{8} \sum _{m}^{occ} \sum _{e}^{vir} t_{im}^{be(1)} \ \lambda _{ae}^{jm(1)},\end{eqnarray}

\begin{matrix} Γ_{i a j b}^{c o r r} & = & - \frac{1}{8} \sum_{m}^{o c c} \sum_{e}^{v i r} t_{i m}^{b e (1)} λ_{a e}^{j m (1)}, \end{matrix}

(35)

\begin{eqnarray}\Gamma _{ijab}^{corr} &=& \frac{1}{4} t_{ij}^{ab(1)} + \frac{1}{8} t_{ij}^{ab(2)} .\end{eqnarray}

\begin{matrix} Γ_{i j a b}^{c o r r} & = & \frac{1}{4} t_{i j}^{a b (1)} + \frac{1}{8} t_{i j}^{a b (2)} . \end{matrix}

(36)

Now, the correlation (⁠

$\Delta {\cal L}$

Δ L

⁠) and total energy (⁠

${\cal L}$

L

⁠) of the OMP2.5-Λ functional can be expressed in terms of PDMs as follows:

\begin{eqnarray}\Delta {\cal L} &=& \sum _{p,q} \gamma _{pq}^{corr} f_{pq} + \sum _{p,q,r,s} \Gamma _{pqrs}^{corr} \langle pq||rs \rangle ,\end{eqnarray}

\begin{matrix} Δ L & = & \sum_{p, q} γ_{p q}^{c o r r} f_{p q} + \sum_{p, q, r, s} Γ_{p q r s}^{c o r r} ⟨ p q ‖ r s ⟩, \end{matrix}

(37)

\begin{eqnarray}{\cal L} &=& \sum _{p,q} \gamma _{pq} h_{pq} + \sum _{p,q,r,s} \Gamma _{pqrs} \langle pq||rs \rangle .\end{eqnarray}

\begin{matrix} L & = & \sum_{p, q} γ_{p q} h_{p q} + \sum_{p, q, r, s} Γ_{p q r s} ⟨ p q ‖ r s ⟩ . \end{matrix}

(38)

4. Generalized-Fock and orbital gradient

The orbital gradient is expressed in terms of the generalized-Fock matrix (GFM), also called as the orbital Lagrangian, as follows:^8,9,12–14

\begin{eqnarray}w_{pq} &=& 2 (F_{pq}-F_{qp}) ,\end{eqnarray}

\begin{matrix} w_{p q} & = & 2 (F_{p q} - F_{q p}), \end{matrix}

(39)

where F is the GFM. The MO gradient is determined by the asymmetry of the GFM and at convergence the GFM is symmetric. The GFM can be written as follows:^8,9,12,13

\begin{eqnarray}F_{pq} &=& \sum _{r} h_{pr} \gamma _{rq} + 2 \sum _{rst} \langle rs||tp \rangle \Gamma _{rstq}.\end{eqnarray}

\begin{matrix} F_{p q} & = & \sum_{r} h_{p r} γ_{r q} + 2 \sum_{r s t} ⟨ r s ‖ t p ⟩ Γ_{r s t q} . \end{matrix}

(40)

As it shown in our recent study,¹³ in energy computations we do not need to build the entire GFM when all orbitals are correlated. Hence, we need only to consider the occupied-virtual (OV) and virtual-occupied (VO) blocks in such a case. Further, one may avoid forming Γ_abcd; instead, its contributions can be added directly to the GFM in energy computations.¹³ By this way, the cost of TPDMs can be significantly reduced.

5. The orbital optimization procedure

The OMP2.5 wave function is defined by a set of orbital rotation parameters κ, and the first- and second-order double excitation amplitudes

${\bf t_{2}^{(1)}}$

t_{2}^{(1)}

and

${\bf t_{2}^{(2)}}$

t_{2}^{(2)}

⁠. For computational efficiency,

${\bf t_{2}^{(1)}}$

t_{2}^{(1)}

⁠,

${\bf t_{2}^{(2)}}$

t_{2}^{(2)}

⁠, and κ are optimized simultaneously, as discussed in previous studies.^3,8–10,13 Direct inversion of the iterative subspace (DIIS) extrapolation technique⁵⁰ is implemented for orbital rotations as described in our recent study.¹³

B. Geometry dependent transformations

Before evaluating the gradients of the energy, we discuss the geometry dependent transformation of orbitals, hence molecular integrals.^12–14 Let us consider a molecular system at some reference geometry x₀. The optimized MOs at the reference geometry can be written in terms of atomic orbitals (AO) χ(x₀)

\begin{eqnarray}\phi _{p}({\bf x}_{0}) &=& \sum _{\mu } C_{\mu p}({\bf x}_{0}) \chi _{\mu } ({\bf x}_{0}),\end{eqnarray}

\begin{matrix} ϕ_{p} (x_{0}) & = & \sum_{μ} C_{μ p} (x_{0}) χ_{μ} (x_{0}), \end{matrix}

(41)

where C(x₀) is the MO coefficient matrix. The MOs are orthonormal at reference geometry satisfying

\begin{eqnarray}\langle \phi _{p}({\bf x}_{0}) | \phi _{q}({\bf x}_{0}) \rangle &=& \delta _{pq}.\end{eqnarray}

\begin{matrix} ⟨ ϕ_{p} (x_{0}) | ϕ_{q} (x_{0}) ⟩ & = & δ_{p q} . \end{matrix}

(42)

It is convenient to define a new orbital basis ϕ(x) called as unmodified molecular orbital (UMO) basis in order to describe variations of the orbitals at perturbed geometries.^51,52 The MOs in the UMO basis are expressed in terms of the optimized MO coefficients C(x₀) at the reference geometry^51,52

\begin{eqnarray}\phi _{p}({\bf x}) &=& \sum _{\mu } C_{\mu p}({\bf x}_{0}) \chi _{\mu } ({\bf x}).\end{eqnarray}

\begin{matrix} ϕ_{p} (x) & = & \sum_{μ} C_{μ p} (x_{0}) χ_{μ} (x) . \end{matrix}

(43)

The UMO basis is orthonormal only at x₀. Hence, it is appropriate to define a connection matrix T(x), which orthonormalizes the UMO basis at any geometry^44,51,52

\begin{eqnarray}{\bf T(x)^{\dagger } \ S(x) \ T(x)} &=& {\bf 1}.\end{eqnarray}

\begin{matrix} T {(x)}^{†} S (x) T (x) & = & 1 . \end{matrix}

(44)

The most popular choice is the Löwdin's symmetric connection^44,51,52

\begin{eqnarray}{\bf T(x)} &=& {\bf S(x)^{-1/2}}.\end{eqnarray}

\begin{matrix} T (x) & = & S {(x)}^{- 1 / 2} . \end{matrix}

(45)

The symmetric orbital connection matrix introduces a new geometry-dependent basis

${\bf \tilde{\phi } (x)}$

\tilde{ϕ} (x)

⁠, which is called as orthonormalized molecular orbital (OMO) basis

\begin{eqnarray}{\bf \tilde{\phi } (x)} &=& {\bf \phi (x) \ S(x)^{-1/2}} .\end{eqnarray}

\begin{matrix} \tilde{ϕ} (x) & = & ϕ (x) S {(x)}^{- 1 / 2} . \end{matrix}

(46)

The OMO basis is orthonormal at all geometries; hence, one can readily verify that

\begin{eqnarray}{\bf \tilde{S}(x)} &=& {\bf 1}.\end{eqnarray}

\begin{matrix} \tilde{S} (x) & = & 1 . \end{matrix}

(47)

In the OMO basis, the Hamiltonian operator can be written as follows:

\begin{eqnarray}\hat{H}({\bf x}) &=& \sum _{p,q} \tilde{h}_{pq}({\bf x}) \hat{p}^{\dagger } \hat{q} + \frac{1}{4} \sum _{p,q,r,s} \tilde{g}_{pqrs}({\bf x}) \hat{p}^{\dagger } \hat{q}^{\dagger } \hat{s} \hat{r},\end{eqnarray}

\begin{matrix} \hat{H} (x) & = & \sum_{p, q} {\tilde{h}}_{p q} (x) {\hat{p}}^{†} \hat{q} + \frac{1}{4} \sum_{p, q, r, s} {\tilde{g}}_{p q r s} (x) {\hat{p}}^{†} {\hat{q}}^{†} \hat{s} \hat{r}, \end{matrix}

(48)

where

$\tilde{g}_{pqrs}$

{\tilde{g}}_{p q r s}

is the antisymmetrized two-electron integral in the OMO basis

\begin{eqnarray}\tilde{g}_{pqrs} &=& \langle \tilde{p}\tilde{q}||\tilde{r}\tilde{s}\rangle .\end{eqnarray}

\begin{matrix} {\tilde{g}}_{p q r s} & = & ⟨ \tilde{p} \tilde{q} ‖ \tilde{r} \tilde{s} ⟩ . \end{matrix}

(49)

One- and two-electron integrals

$\tilde{h}_{pq}({\bf x})$

{\tilde{h}}_{p q} (x)

and

$\tilde{g}_{pqrs}({\bf x})$

{\tilde{g}}_{p q r s} (x)

in the OMO basis can be expressed in the UMO basis as follows:

\begin{eqnarray}{\bf \tilde{h}(x)} &=& {\bf S(x)^{-1/2} \ h(x) \ S(x)^{-1/2}} ,\end{eqnarray}

\begin{matrix} \tilde{h} (x) & = & S {(x)}^{- 1 / 2} h (x) S {(x)}^{- 1 / 2}, \end{matrix}

(50)

\begin{eqnarray}\tilde{g}_{pqrs}({\bf x}) &=& \sum _{tuvw} S_{tp}({\bf x})^{-1/2} S_{uq}({\bf x})^{-1/2} \nonumber \\&&\times\, S_{vr}({\bf x})^{-1/2} S_{ws}({\bf x})^{-1/2} g_{tuvw}({\bf x}).\end{eqnarray}

\begin{matrix} {\tilde{g}}_{p q r s} (x) & = & \sum_{t u v w} S_{t p} {(x)}^{- 1 / 2} S_{u q} {(x)}^{- 1 / 2} \\ \times S_{v r} {(x)}^{- 1 / 2} S_{w s} {(x)}^{- 1 / 2} g_{t u v w} (x) . \end{matrix}

(51)

Gradients of molecular integrals

$\tilde{h}_{pq}({\bf x})$

{\tilde{h}}_{p q} (x)

and

$\tilde{g}_{pqrs}({\bf x})$

{\tilde{g}}_{p q r s} (x)

at the reference geometry is also of particular interest. Evaluating the first-derivatives of one- and two-electron integrals at reference geometry x₀, we obtain^{44,49,51–54}

\begin{eqnarray}\tilde{h}_{pq}^{x} &=& h_{pq}^{x} - \frac{1}{2} \sum _{t} \big(h_{tq} S_{tp}^{x} + h_{pt} S_{tq}^{x} \big),\end{eqnarray}

\begin{matrix} {\tilde{h}}_{p q}^{x} & = & h_{p q}^{x} - \frac{1}{2} \sum_{t} (h_{t q} S_{t p}^{x} + h_{p t} S_{t q}^{x}), \end{matrix}

(52)

\begin{eqnarray}\tilde{g}_{pqrs}^{x} &=& g_{pqrs}^{x} \nonumber \\&& -\, \frac{1}{2} \sum _{t} \big(g_{tqrs} S_{tp}^{x} + g_{ptrs} S_{tq}^{x} + g_{pqts} S_{tr}^{x} + g_{pqrt} S_{ts}^{x}\big), \nonumber \\\end{eqnarray}

\begin{matrix} {\tilde{g}}_{p q r s}^{x} & = & g_{p q r s}^{x} \\ - \frac{1}{2} \sum_{t} (g_{t q r s} S_{t p}^{x} + g_{p t r s} S_{t q}^{x} + g_{p q t s} S_{t r}^{x} + g_{p q r t} S_{t s}^{x}), \end{matrix}

(53)

where

$S_{pq}^{x}$

S_{p q}^{x}

⁠,

$h_{pq}^{x}$

h_{p q}^{x}

⁠, and

$g_{pqrs}^{x}$

g_{p q r s}^{x}

are the skeleton (core) derivative integrals,⁵⁵ which are corresponding to the purely AO derivative contributions

\begin{eqnarray}S_{pq}^{x} &=& \sum _{\mu \nu } C_{\mu p} C_{\nu q} S_{\mu \nu }^{x},\end{eqnarray}

\begin{matrix} S_{p q}^{x} & = & \sum_{μ ν} C_{μ p} C_{ν q} S_{μ ν}^{x}, \end{matrix}

(54)

\begin{eqnarray}h_{pq}^{x} &=& \sum _{\mu \nu } C_{\mu p} C_{\nu q} h_{\mu \nu }^{x},\end{eqnarray}

\begin{matrix} h_{p q}^{x} & = & \sum_{μ ν} C_{μ p} C_{ν q} h_{μ ν}^{x}, \end{matrix}

(55)

\begin{eqnarray}g_{pqrs}^{x} &=& \sum _{\mu \nu \lambda \sigma } C_{\mu p} C_{\nu q} C_{\lambda r} C_{\sigma s} \langle \mu \nu ||\lambda \sigma \rangle ^{x}.\end{eqnarray}

\begin{matrix} g_{p q r s}^{x} & = & \sum_{μ ν λ σ} C_{μ p} C_{ν q} C_{λ r} C_{σ s} {⟨ μ ν ‖ λ σ ⟩}^{x} . \end{matrix}

(56)

C. OMP2.5 gradient

The gradient of the energy can be obtained from the first derivative of the Lagrangian in Eq. (8) as follows:^44,45,56,57

\begin{eqnarray}\frac{{\rm d}E}{{\rm d}x}\left|_{x=x_{0}} = \frac{\partial {\cal L}}{\partial x}\right|_{x=x_{0}} .\end{eqnarray}

\begin{matrix} \frac{d E}{d x} |_{x = x_{0}} = \frac{\partial L}{\partial x} |_{x = x_{0}} . \end{matrix}

(57)

Hence, we can write

\begin{eqnarray}\frac{{\rm d}E}{{\rm d}x} &=& \langle 0| \frac{\partial \hat{\!\tilde{H}}}{\partial x}\bigg |_{x=x_{0}} | 0 \rangle \nonumber \\&& + \, \langle 0| \bigg (\frac{\partial \hat{\!\tilde{W}}_{N}}{\partial x}\bigg |_{x=x_{0}} \hat{T}_{2}^{(1)}\bigg )_{c} |0\rangle \!+\! \langle 0| \bigg (\frac{\partial \hat{\!\tilde{W}}_{N}}{\partial x}\bigg |_{x=x_{0}} \hat{T}_{2}^{(2)}\bigg )_{c} |0\rangle \nonumber \\&& +\, \langle 0| \bigg\lbrace \hat{\Lambda }_{2}^{(1)} \bigg (\frac{\partial \hat{\!\tilde{f}_{N}^{d}}}{\partial x}\bigg |_{x=x_{0}}\hat{T}_{2}^{(1)} + \frac{\partial \hat{\!\tilde{W}}_{N}}{\partial x}\bigg |_{x=x_{0}} \bigg )_{c}\bigg\rbrace _{c} |0\rangle \nonumber \\&& +\, \frac{1}{2} \langle 0| \bigg\lbrace \hat{\Lambda }_{2}^{(1)} \bigg (\frac{\partial \hat{\!\tilde{f}_{N}^{d}}}{\partial x}\bigg |_{x=x_{0}}\hat{T}_{2}^{(2)} \!+\! \frac{\partial \hat{\!\tilde{W}}_{N}}{\partial x}\bigg |_{x=x_{0}}\hat{T}_{2}^{(1)} \bigg )_{c}\bigg\rbrace _{c} |0\rangle \nonumber \\&& + \, \frac{1}{2} \langle 0| \bigg\lbrace \hat{\Lambda }_{2}^{(2)} \bigg (\frac{\partial \hat{\!\tilde{f}_{N}^{d}}}{\partial x}\bigg |_{x=x_{0}}\hat{T}_{2}^{(1)} + \frac{\partial \hat{\!\tilde{W}}_{N}}{\partial x}\bigg |_{x=x_{0}} \bigg )_{c}\bigg\rbrace _{c} |0\rangle .\nonumber \\\end{eqnarray}

\begin{matrix} \frac{d E}{d x} & = & ⟨ 0 | \frac{\partial \hat{\tilde{H}}}{\partial x} |_{x = x_{0}} | 0 ⟩ \\ + ⟨ 0 | {(\frac{\partial {\hat{\tilde{W}}}_{N}}{\partial x}, |_{x = x_{0}}, {\hat{T}}_{2}^{(1)})}_{c} | 0 ⟩ + ⟨ 0 | {(\frac{\partial {\hat{\tilde{W}}}_{N}}{\partial x}, |_{x = x_{0}}, {\hat{T}}_{2}^{(2)})}_{c} | 0 ⟩ \\ + ⟨ 0 | {\{{\hat{Λ}}_{2}^{(1)}, {(\frac{\partial \hat{{\tilde{f}}_{N}^{d}}}{\partial x} |_{x = x_{0}} {\hat{T}}_{2}^{(1)} + \frac{\partial {\hat{\tilde{W}}}_{N}}{\partial x} |_{x = x_{0}})}_{c}\}}_{c} | 0 ⟩ \\ + \frac{1}{2} ⟨ 0 | {\{{\hat{Λ}}_{2}^{(1)}, {(\frac{\partial \hat{{\tilde{f}}_{N}^{d}}}{\partial x} |_{x = x_{0}} {\hat{T}}_{2}^{(2)} + \frac{\partial {\hat{\tilde{W}}}_{N}}{\partial x} |_{x = x_{0}} {\hat{T}}_{2}^{(1)})}_{c}\}}_{c} | 0 ⟩ \\ + \frac{1}{2} ⟨ 0 | {\{{\hat{Λ}}_{2}^{(2)}, {(\frac{\partial \hat{{\tilde{f}}_{N}^{d}}}{\partial x} |_{x = x_{0}} {\hat{T}}_{2}^{(1)} + \frac{\partial {\hat{\tilde{W}}}_{N}}{\partial x} |_{x = x_{0}})}_{c}\}}_{c} | 0 ⟩ . \end{matrix}

(58)

Then, we can express the gradient as follows:

\begin{eqnarray}\frac{{\rm d}E}{{\rm d}x} & = & \sum _{pq} \gamma _{pq} \tilde{h}_{pq}^{x} + \sum _{pqrs} \Gamma _{pqrs} \tilde{g}_{pqrs}^{x} .\end{eqnarray}

\begin{matrix} \frac{d E}{d x} & = & \sum_{p q} γ_{p q} {\tilde{h}}_{p q}^{x} + \sum_{p q r s} Γ_{p q r s} {\tilde{g}}_{p q r s}^{x} . \end{matrix}

(59)

When we insert the explicit equations for

$\tilde{h}_{pq}^{x}$

{\tilde{h}}_{p q}^{x}

and

$\tilde{g}_{pqrs}^{x}$

{\tilde{g}}_{p q r s}^{x}

into Eq. , the gradient expression can be cast into the following form:

\begin{eqnarray}\frac{{\rm d}E}{{\rm d}x} & = & \sum _{pq} \gamma _{pq} h_{pq}^{x} + \sum _{pqrs} \Gamma _{pqrs} g_{pqrs}^{x} - \sum _{pq} F_{pq} S_{pq}^{x}.\qquad\end{eqnarray}

\begin{matrix} \frac{d E}{d x} & = & \sum_{p q} γ_{p q} h_{p q}^{x} + \sum_{p q r s} Γ_{p q r s} g_{p q r s}^{x} - \sum_{p q} F_{p q} S_{p q}^{x} . \end{matrix}

(60)

Note the PDMs appear in Eq. (60) are the “unrelaxed” density matrices; because the orbitals are optimized, we do not need the additional orbital response contributions that would normally present. Similarly, we do not need to consider the orbital response contribution to the GFM, it is simply zero. The required density matrices are defined in Sec. II A 3.

The gradient is evaluated in the usual way by back-transforming the PDMs and GFM into the AO basis and contracting against the appropriate AO derivative integrals^58,59

\begin{eqnarray}F_{\mu \nu } &=& \sum _{pq} C_{\mu p} C_{\nu q} F_{pq},\end{eqnarray}

\begin{matrix} F_{μ ν} & = & \sum_{p q} C_{μ p} C_{ν q} F_{p q}, \end{matrix}

(61)

\begin{eqnarray}\gamma _{\mu \nu } &=& \sum _{pq} C_{\mu p} C_{\nu q} \gamma _{pq},\end{eqnarray}

\begin{matrix} γ_{μ ν} & = & \sum_{p q} C_{μ p} C_{ν q} γ_{p q}, \end{matrix}

(62)

\begin{eqnarray}\Gamma _{\mu \nu \lambda \sigma } &=& \sum _{pqrs} C_{\mu p} C_{\nu q} C_{\lambda r} C_{\sigma s} \Gamma _{pqrs},\end{eqnarray}

\begin{matrix} Γ_{μ ν λ σ} & = & \sum_{p q r s} C_{μ p} C_{ν q} C_{λ r} C_{σ s} Γ_{p q r s}, \end{matrix}

(63)

where F_μν, γ_μν, and Γ_μνλσ are the AO basis GFM, one- and two-particle density matrices, respectively. Hence, the final gradient equation in the AO basis can be written as

\begin{eqnarray}\frac{{\rm d}E}{{\rm d}x} &=& \sum _{\mu \nu } \gamma _{\mu \nu } h_{\mu \nu }^{x} + \sum _{\mu \nu \lambda \sigma } \Gamma _{\mu \nu \lambda \sigma } \langle \mu \nu ||\lambda \sigma \rangle ^{x} - \sum _{\mu \nu } F_{\mu \nu } S_{\mu \nu }^{x}. \nonumber \\\end{eqnarray}

\begin{matrix} \frac{d E}{d x} & = & \sum_{μ ν} γ_{μ ν} h_{μ ν}^{x} + \sum_{μ ν λ σ} Γ_{μ ν λ σ} {⟨ μ ν ‖ λ σ ⟩}^{x} - \sum_{μ ν} F_{μ ν} S_{μ ν}^{x} . \end{matrix}

(64)

D. MP2.5 gradient

For the MP2.5 wave function, we can write the following Lagrangian:

\begin{eqnarray}{\cal L} &=& \langle 0| \hat{H} | 0 \rangle \nonumber \\&& + \, \langle 0| \big (\hat{W}_{N}\hat{T}_{2}^{(1)}\big )_{c} |0\rangle + \frac{1}{2} \langle 0| \big (\hat{W}_{N}\hat{T}_{2}^{(2)}\big )_{c} |0\rangle \nonumber \\&& + \, \langle 0| \big\lbrace \hat{\Lambda }_{2}^{(1)} \ \big (\hat{f}_{N}^{d}\hat{T}_{2}^{(1)} + \hat{W}_{N} \big )_{c}\big\rbrace _{c} |0\rangle \nonumber \\&& + \, \frac{1}{2} \langle 0| \big\lbrace \hat{\Lambda }_{2}^{(1)} \ \big (\hat{f}_{N}^{d}\hat{T}_{2}^{(2)} + \hat{W}_{N}\hat{T}_{2}^{(1)} \big )_{c}\big\rbrace _{c} |0\rangle \nonumber \\&& + \, \frac{1}{2} \langle 0| \big\lbrace \hat{\Lambda }_{2}^{(2)} \ \big (\hat{f}_{N}^{d}\hat{T}_{2}^{(1)} + \hat{W}_{N} \big )_{c}\big\rbrace _{c} |0\rangle \nonumber \\&& + \, \langle 0|[\hat{H},\hat{Z}] |0 \rangle ,\end{eqnarray}

\begin{matrix} L & = & ⟨ 0 | \hat{H} | 0 ⟩ \\ + ⟨ 0 | {({\hat{W}}_{N}, {\hat{T}}_{2}^{(1)})}_{c} | 0 ⟩ + \frac{1}{2} ⟨ 0 | {({\hat{W}}_{N}, {\hat{T}}_{2}^{(2)})}_{c} | 0 ⟩ \\ + ⟨ 0 | {\{{\hat{Λ}}_{2}^{(1)}, {({\hat{f}}_{N}^{d} {\hat{T}}_{2}^{(1)} + {\hat{W}}_{N})}_{c}\}}_{c} | 0 ⟩ \\ + \frac{1}{2} ⟨ 0 | {\{{\hat{Λ}}_{2}^{(1)}, {({\hat{f}}_{N}^{d} {\hat{T}}_{2}^{(2)} + {\hat{W}}_{N} {\hat{T}}_{2}^{(1)})}_{c}\}}_{c} | 0 ⟩ \\ + \frac{1}{2} ⟨ 0 | {\{{\hat{Λ}}_{2}^{(2)}, {({\hat{f}}_{N}^{d} {\hat{T}}_{2}^{(1)} + {\hat{W}}_{N})}_{c}\}}_{c} | 0 ⟩ \\ + ⟨ 0 | [\hat{H}, \hat{Z}] | 0 ⟩, \end{matrix}

(65)

where the operator

$\hat{Z}$

\hat{Z}

is defined as

\begin{eqnarray}\hat{Z} &=& \sum _{ai} z_{ai} (\hat{a}^{\dagger } \hat{i} - \hat{i}^{\dagger } \hat{a}) ,\end{eqnarray}

\begin{matrix} \hat{Z} & = & \sum_{a i} z_{a i} ({\hat{a}}^{†} \hat{i} - {\hat{i}}^{†} \hat{a}), \end{matrix}

(66)

and let us recall that the Hartree–Fock (HF) MO gradient is

\begin{eqnarray}w_{ai}^{HF} &=& \langle 0| (\hat{a}^{\dagger } \hat{i} - \hat{i}^{\dagger } \hat{a}) |0 \rangle \ = \ 2f_{ai} .\end{eqnarray}

\begin{matrix} w_{a i}^{H F} & = & ⟨ 0 | ({\hat{a}}^{†} \hat{i} - {\hat{i}}^{†} \hat{a}) | 0 ⟩ = 2 f_{a i} . \end{matrix}

(67)

Differentiating the MP2.5 Lagrangian in Eq. (65) with respect to orbital rotation parameters we can obtain the following linear equation, which defines the Z-vector:

\begin{eqnarray}{\bf A z + w} &=& 0 ,\end{eqnarray}

\begin{matrix} A z + w & = & 0, \end{matrix}

(68)

where A is the HF MO Hessian and w is the MP2.5 MO gradient.

The gradient of energy can be obtained from the first derivative of the Lagrangian in Eq. (65) as follows:^44,45,56,57

\begin{eqnarray}\frac{{\rm d}E}{{\rm d}x}\left |_{x=x_{0}} = \frac{\partial {\cal L}}{\partial x}\right |_{x=x_{0}} .\end{eqnarray}

\begin{matrix} \frac{d E}{d x} |_{x = x_{0}} = \frac{\partial L}{\partial x} |_{x = x_{0}} . \end{matrix}

(69)

Hence, we can write

\begin{eqnarray}\frac{{\rm d}E}{{\rm d}x} &=& \langle 0| \frac{\partial \hat{\!\tilde{H}}}{\partial x}\bigg |_{x=x_{0}} | 0 \rangle \nonumber \\&& + \langle 0| \bigg (\frac{\partial \hat{\!\tilde{W}}_{N}}{\partial x}\bigg |_{x=x_{0}} \hat{T}_{2}^{(1)}\bigg )_{c} |0\rangle \!+\! \langle 0| \bigg (\frac{\partial \hat{\!\tilde{W}}_{N}}{\partial x}\bigg |_{x=x_{0}} \hat{T}_{2}^{(2)}\bigg )_{c} |0\rangle \nonumber \\&& + \langle 0| \left\lbrace \hat{\Lambda }_{2}^{(1)} \ \bigg (\frac{\partial \hat{\!\tilde{f}_{N}^{d}}}{\partial x}\bigg |_{x=x_{0}}\hat{T}_{2}^{(1)} + \frac{\partial \hat{\!\tilde{W}}_{N}}{\partial x}\bigg |_{x=x_{0}} \bigg )_{c}\right\rbrace _{c} |0\rangle \nonumber \\&& + \frac{1}{2} \langle 0| \left\lbrace \hat{\Lambda }_{2}^{(1)} \ \bigg (\frac{\partial \hat{\!\tilde{f}_{N}^{d}}}{\partial x}\bigg |_{x=x_{0}}\hat{T}_{2}^{(2)} + \frac{\partial \hat{\!\tilde{W}}_{N}}{\partial x}\bigg |_{x=x_{0}}\hat{T}_{2}^{(1)} \bigg )_{c}\right\rbrace _{c} |0\rangle \nonumber \\&& + \frac{1}{2} \langle 0| \left\lbrace \hat{\Lambda }_{2}^{(2)} \ \bigg (\frac{\partial \hat{\!\tilde{f}_{N}^{d}}}{\partial x}\bigg |_{x=x_{0}}\hat{T}_{2}^{(1)} \!+\! \frac{\partial \hat{\!\tilde{W}}_{N}}{\partial x}\bigg |_{x=x_{0}} \bigg )_{c}\right\rbrace _{c} |0\rangle \nonumber \\&& + \, \langle 0|\bigg[\frac{\partial \hat{\!\tilde{H}}}{\partial x}\bigg |_{x=x_{0}},\hat{Z}\bigg] |0 \rangle .\!\!\!\!\!\!\end{eqnarray}

\begin{matrix} \frac{d E}{d x} & = & ⟨ 0 | \frac{\partial \hat{\tilde{H}}}{\partial x} |_{x = x_{0}} | 0 ⟩ \\ + ⟨ 0 | {(\frac{\partial {\hat{\tilde{W}}}_{N}}{\partial x}, |_{x = x_{0}}, {\hat{T}}_{2}^{(1)})}_{c} | 0 ⟩ + ⟨ 0 | {(\frac{\partial {\hat{\tilde{W}}}_{N}}{\partial x}, |_{x = x_{0}}, {\hat{T}}_{2}^{(2)})}_{c} | 0 ⟩ \\ + ⟨ 0 | {\{{\hat{Λ}}_{2}^{(1)}, {(\frac{\partial \hat{{\tilde{f}}_{N}^{d}}}{\partial x} |_{x = x_{0}} {\hat{T}}_{2}^{(1)} + \frac{\partial {\hat{\tilde{W}}}_{N}}{\partial x} |_{x = x_{0}})}_{c}\}}_{c} | 0 ⟩ \\ + \frac{1}{2} ⟨ 0 | {\{{\hat{Λ}}_{2}^{(1)}, {(\frac{\partial \hat{{\tilde{f}}_{N}^{d}}}{\partial x} |_{x = x_{0}} {\hat{T}}_{2}^{(2)} + \frac{\partial {\hat{\tilde{W}}}_{N}}{\partial x} |_{x = x_{0}} {\hat{T}}_{2}^{(1)})}_{c}\}}_{c} | 0 ⟩ \\ + \frac{1}{2} ⟨ 0 | {\{{\hat{Λ}}_{2}^{(2)}, {(\frac{\partial \hat{{\tilde{f}}_{N}^{d}}}{\partial x} |_{x = x_{0}} {\hat{T}}_{2}^{(1)} + \frac{\partial {\hat{\tilde{W}}}_{N}}{\partial x} |_{x = x_{0}})}_{c}\}}_{c} | 0 ⟩ \\ + ⟨ 0 | [\frac{\partial \hat{\tilde{H}}}{\partial x} |_{x = x_{0}}, \hat{Z}] | 0 ⟩ . \end{matrix}

(70)

Now, we can express the gradient as follows:

\begin{eqnarray}\frac{{\rm d}E}{{\rm d}x} & = & \sum _{pq} \gamma _{pq} \tilde{h}_{pq}^{x} + \sum _{pqrs} \Gamma _{pqrs} \tilde{g}_{pqrs}^{x} + 2\sum _{ai} z_{ai} \tilde{f}_{ai}^{x} .\qquad\end{eqnarray}

\begin{matrix} \frac{d E}{d x} & = & \sum_{p q} γ_{p q} {\tilde{h}}_{p q}^{x} + \sum_{p q r s} Γ_{p q r s} {\tilde{g}}_{p q r s}^{x} + 2 \sum_{a i} z_{a i} {\tilde{f}}_{a i}^{x} . \end{matrix}

(71)

The gradient of the Fock matrix,

$\tilde{f}_{ai}^{x}$

{\tilde{f}}_{a i}^{x}

⁠, can be written as

\begin{eqnarray}\tilde{f}_{ai}^{x} &=& f_{ai}^{x} - \frac{1}{2} \big ( \varepsilon _{i} S_{ia}^{x} + \varepsilon _{a} S_{ai}^{x} \big ) - \frac{1}{2} \sum _{m,t} (g_{atim} + g_{amit} ) S_{tm}^{x},\nonumber\\\end{eqnarray}

\begin{matrix} {\tilde{f}}_{a i}^{x} & = & f_{a i}^{x} - \frac{1}{2} (ɛ_{i} S_{i a}^{x} + ɛ_{a} S_{a i}^{x}) - \frac{1}{2} \sum_{m, t} (g_{a t i m} + g_{a m i t}) S_{t m}^{x}, \end{matrix}

(72)

where ɛ_i and ɛ_a are the occupied and virtual orbital energies, respectively. When we insert the explicit equations for

$\tilde{h}_{pq}^{x}$

{\tilde{h}}_{p q}^{x}

⁠,

$\tilde{g}_{pqrs}^{x}$

{\tilde{g}}_{p q r s}^{x}

⁠, and

$\tilde{f}_{ai}^{x}$

{\tilde{f}}_{a i}^{x}

into Eq. , the first derivative equation can be cast into the following form:

\begin{eqnarray}\frac{{\rm d}E}{{\rm d}x} & = & \sum _{pq} \gamma _{pq}^{eff} h_{pq}^{x} + \sum _{pqrs} \Gamma _{pqrs}^{eff} g_{pqrs}^{x} - \sum _{pq} F_{pq}^{eff} S_{pq}^{x} ,\nonumber\\\end{eqnarray}

\begin{matrix} \frac{d E}{d x} & = & \sum_{p q} γ_{p q}^{e f f} h_{p q}^{x} + \sum_{p q r s} Γ_{p q r s}^{e f f} g_{p q r s}^{x} - \sum_{p q} F_{p q}^{e f f} S_{p q}^{x}, \end{matrix}

(73)

where

$\gamma _{pq}^{eff}$

γ_{p q}^{e f f}

⁠,

$\Gamma _{pqrs}^{eff}$

Γ_{p q r s}^{e f f}

⁠, and

$F_{pq}^{eff}$

F_{p q}^{e f f}

are effective PDMs and the GFM,^{8,16,19,22,23} which include the Z-vector contributions. The Z-vector contributions to PDMs can be written as

\begin{eqnarray}\gamma _{ai}^{eff} & = & \gamma _{ia}^{eff} \ = \ z_{ai} ,\end{eqnarray}

\begin{matrix} γ_{a i}^{e f f} & = & γ_{i a}^{e f f} = z_{a i}, \end{matrix}

(74)

\begin{eqnarray}\Gamma _{amin}^{eff} & = & \Gamma _{inam}^{eff} \ = \ z_{ai} \delta _{mn} .\end{eqnarray}

\begin{matrix} Γ_{a m i n}^{e f f} & = & Γ_{i n a m}^{e f f} = z_{a i} δ_{m n} . \end{matrix}

(75)

Similarly, the orbital relaxation contributions to the GFM can be expressed as follows:^13,60

\begin{eqnarray}F_{ia}^{eff} & = & F_{ia} + z_{ai} \varepsilon _{i} ,\end{eqnarray}

\begin{matrix} F_{i a}^{e f f} & = & F_{i a} + z_{a i} ɛ_{i}, \end{matrix}

(76)

\begin{eqnarray}F_{ai}^{eff} & = & F_{ai} + z_{ai} \varepsilon _{a} + \sum _{em} z_{em} (g_{eami} + g_{eima}) ,\end{eqnarray}

\begin{matrix} F_{a i}^{e f f} & = & F_{a i} + z_{a i} ɛ_{a} + \sum_{e m} z_{e m} (g_{e a m i} + g_{e i m a}), \end{matrix}

(77)

\begin{eqnarray}F_{ij}^{eff} & = & F_{ij} + \sum _{em} z_{em} (g_{eimj} + g_{ejmi}) .\end{eqnarray}

\begin{matrix} F_{i j}^{e f f} & = & F_{i j} + \sum_{e m} z_{e m} (g_{e i m j} + g_{e j m i}) . \end{matrix}

(78)

The gradient is evaluated in the usual way by back-transforming the relaxed PDMs and the relaxed GFM into the AO basis and contracting against the appropriate AO derivative integrals^58,59

\begin{eqnarray}F_{\mu \nu } &=& \sum _{pq} C_{\mu p} C_{\nu q} F_{pq}^{eff},\end{eqnarray}

\begin{matrix} F_{μ ν} & = & \sum_{p q} C_{μ p} C_{ν q} F_{p q}^{e f f}, \end{matrix}

(79)

\begin{eqnarray}\gamma _{\mu \nu } &=& \sum _{pq} C_{\mu p} C_{\nu q} \gamma _{pq}^{eff},\end{eqnarray}

\begin{matrix} γ_{μ ν} & = & \sum_{p q} C_{μ p} C_{ν q} γ_{p q}^{e f f}, \end{matrix}

(80)

\begin{eqnarray}\Gamma _{\mu \nu \lambda \sigma } &=& \sum _{pqrs} C_{\mu p} C_{\nu q} C_{\lambda r} C_{\sigma s} \Gamma _{pqrs}^{eff},\end{eqnarray}

\begin{matrix} Γ_{μ ν λ σ} & = & \sum_{p q r s} C_{μ p} C_{ν q} C_{λ r} C_{σ s} Γ_{p q r s}^{e f f}, \end{matrix}

(81)

where F_μν, γ_μν, and Γ_μνλσ are the AO basis relaxed GFM, one- and two-particle density matrices, respectively. Hence, the final gradient equation in the AO basis can be written as

\begin{eqnarray}\frac{{\rm d}E}{{\rm d}x} &=& \sum _{\mu \nu } \gamma _{\mu \nu } h_{\mu \nu }^{x} + \sum _{\mu \nu \lambda \sigma } \Gamma _{\mu \nu \lambda \sigma } \langle \mu \nu ||\lambda \sigma \rangle ^{x} - \sum _{\mu \nu } F_{\mu \nu } S_{\mu \nu }^{x}.\nonumber\\\end{eqnarray}

\begin{matrix} \frac{d E}{d x} & = & \sum_{μ ν} γ_{μ ν} h_{μ ν}^{x} + \sum_{μ ν λ σ} Γ_{μ ν λ σ} {⟨ μ ν ‖ λ σ ⟩}^{x} - \sum_{μ ν} F_{μ ν} S_{μ ν}^{x} . \end{matrix}

(82)

III. RESULTS AND DISCUSSION

Results from the OMP2.5 method were obtained for geometries of closed- and open-shell molecules, energies of hydrogen transfer reactions, and noncovalent interaction energies for comparison with those from the canonical MP2, MP2.5, coupled-cluster singles and doubles (CCSD), and coupled-cluster singles and doubles with perturbative triples [CCSD(T)] methods. The MP2, CCSD, and CCSD(T)^1,61–63 energies are obtained from the PSI4³⁹ and MOLPRO⁶⁴ programs, while the MP2.5 and OMP2.5 energies are obtained from our PSI4 code.³⁹ For the comparison of geometries, Dunning's correlation-consistent polarized core-valence quadruple-ζ basis set (cc-pCVQZ)^65,66 was employed. The CCSD(T)/cc-pCVQZ level of theory was shown to be quite accurate for equilibrium geometries of molecules without a strong multi-reference character.^67–71 Hence, MP2, MP2.5, OMP2.5, and CCSD results are compared with those of CCSD(T)/cc-pCVQZ. Optimized geometries of the molecules considered in hydrogen-transfer reactions were taken from previous studies,^{11,12,72–76} while those for open-shell noncovalent interactions (the O23 database) were taken from our recent study.¹⁵ All electrons were correlated in all computations.

For hydrogen transfer reactions and noncovalent interactions, single-point energies were computed at optimized geometries, and the total energies were extrapolated to complete basis set (CBS) limits.^77,78 The two-point extrapolation approach of Halkier et al.⁷⁹ was employed for this purpose

\begin{eqnarray}E_{X}^{HF} &=& E_{CBS}^{HF} + A e^{-\alpha X},\end{eqnarray}

\begin{matrix} E_{X}^{H F} & = & E_{C B S}^{H F} + A e^{- α X}, \end{matrix}

(83)

\begin{eqnarray}E_{X}^{corr} &=& E_{CBS}^{corr} + B X^{-3},\end{eqnarray}

\begin{matrix} E_{X}^{c o r r} & = & E_{C B S}^{c o r r} + B X^{- 3}, \end{matrix}

(84)

where

$E_{CBS}^{HF}$

E_{C B S}^{H F}

and

$E_{CBS}^{corr}$

E_{C B S}^{c o r r}

are Hartree–Fock and correlation energies at CBS limit, respectively, A and B are the fitting parameters, and X is the cardinal number of the (aug-)cc-pVXZ basis set. The exponent α was chosen 1.63 as suggested.⁷⁹ In the two-point extrapolation procedure, the cc-pVDZ and cc-pVTZ basis sets were employed for hydrogen transfer reactions. For closed-shell noncovalent interactions, the aug-cc-pVDZ and aug-cc-pVTZ basis sets were used, whereas for open-shell noncovalent interactions the aug-cc-pVTZ and aug-cc-pVQZ basis sets were employed. All intermolecular interaction energies are counterpoise corrected.⁸⁰

A. Geometries

As the first step of the assessment for geometries, we consider a set of bond lengths for closed-shell molecules introduced by Helgaker et al.⁶⁷ Table S1 of the supplementary material⁸¹ presents bond lengths of closed-shell molecules in order of increasing experimental value. Errors in bond lengths of closed-shell molecules for the MP2, MP2.5, OMP2.5, CCSD, and CCSD(T) methods with respect to experiment are presented graphically in Figure 1, while mean absolute errors (MAEs) are depicted in Figure 2. MAEs are 0.0062 (MP2), 0.0074 (MP2.5), 0.0046 (OMP2.5), 0.0066 (CCSD), and 0.0018 [CCSD(T)] Å. The CCSD(T) method provides the lowest error compared to experiment, while MP2.5 yields the largest error. The OMP2.5 method significantly improves upon MP2, MP2.5, and CCSD by 25%, 38%, and 31%, respectively.

FIG. 1.

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Errors in bond lengths of closed-shell molecules from Helgaker et al.⁶⁷ for the MP2, MP2.5, OMP2.5, CCSD, and CCSD(T) methods with respect to experiment (the cc-pCVQZ basis set was employed).

FIG. 2.

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Mean absolute errors in bond lengths of closed-shell molecules from Helgaker et al.⁶⁷ for the MP2, MP2.5, OMP2.5, CCSD, and CCSD(T) methods with respect to experiment (the cc-pCVQZ basis set was employed).

Now, we consider a set of bond lengths for open-shell molecules discussed by Byrd, Sherrill, and Head-Gordon⁸² as the second step of our assessment. Table S2 of the supplementary material⁸¹ presents bond lengths of open-shell molecules in order of increasing experimental value. Errors in bond lengths of open-shell molecules for the MP2, MP2.5, OMP2.5, CCSD, and CCSD(T) methods with respect to experiment are presented graphically in Figure 3, while the MAE values are depicted in Figure 4. The MAE values are 0.0153 (MP2), 0.0154 (MP2.5), 0.0087 (OMP2.5), 0.0121 (CCSD), and 0.0071 [CCSD(T)] Å. The CCSD(T) method again provides the lowest MAE value, while MP2.5 again yields the largest one. The performance of OMP2.5 is not as good as that of CCSD(T), but it still substantially improves upon MP2, MP2.5, and CCSD by 43%, 43%, and 28%, respectively. Hence, with the same computational cost one may approach CCSD(T) quality for open-shell bond lengths preferring the OMP2.5 method over CCSD.

FIG. 3.

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Errors in bond lengths of open-shell molecules from Byrd et al.⁸² for the MP2, MP2.5, OMP2.5, CCSD, and CCSD(T) methods with respect to experiment (the cc-pCVQZ basis set was employed).

FIG. 4.

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Mean absolute errors in bond lengths of open-shell molecules from Byrd et al.⁸² for the MP2, MP2.5, OMP2.5, CCSD, and CCSD(T) methods with respect to experiment (the cc-pCVQZ basis set was employed).

B. Hydrogen transfer reactions

It was shown that the standard methods, such as MP2 and CEPA(0), dramatically fail for several hydrogen transfer reactions between free radicals.^12,13,83 In our recent studies, we demonstrated that the OMP2 and OCEPA(0) methods exhibit substantially better performance than their standard counterparts [MP2 and CEPA(0)], providing 5- and 6-fold lower MAEs than those of MP2 and CEPA(0).^12,13 Hence, we consider the same test set¹² for comparison of MP2.5/OMP2.5.

For hydrogen transfer reactions, reaction energies (in kcal mol⁻¹) from the MP2, MP2.5, CCSD, and CCSD(T) methods at the CBS limit are reported in Table I. Errors with respect to CCSD(T) are presented graphically in Figure 5, while the MAE values are depicted in Figure 6. The MAE values are 14.6 (MP2),⁸⁴ 11.8 (MP2.5), 1.1 (OMP2.5), and 0.5 (CCSD) kcal mol⁻¹, indicating a reduction in MP2.5 errors by more than a factor of 10 when optimized orbitals are used, and comparing to MP2 there is a more than 12-fold decrease in errors. The CCSD method reduces the MAE by an additional 0.6 kcal mol⁻¹ compared to OMP2.5.

Table I.

Reaction energies (in kcal mol⁻¹) of hydrogen transfer reactions from MP2, MP2.5, OMP2.5, CCSD, and CCSD(T) at the CBS limit.

	Reaction	MP2	MP2.5	OMP2.5	CCSD	CCSD(T)
1	CH₃ + H₂ → CH₄ + H	−7.6	−5.9	−5.8	−2.5	−3.5
2	C₂H + H₂ → C₂H₂ + H	−50.9	−47.2	−35.2	−31.4	−31.9
3	C₂H₃ + H₂ → C₂H₄ + H	−20.0	−17.7	−11.7	−8.7	−9.3
4	C(CH₃)₃ + H₂ → HC(CH₃)₃ + H	−1.2	0.8	1.6	4.8	3.8
5	C₆H₅ + H₂ → C₆H₆ + H	−44.7	−37.6	−13.5	−10.6	−11.1
6	C₂H + C₂H₄ → C₂H₂ + C₂H₃	−30.9	−29.5	−23.4	−22.8	−22.6
7	C(CH₃)₃ + C₂H₄ → HC(CH₃)₃ + C₂H₃	18.8	18.6	13.4	13.4	13.1
8	C₆H₅ + C₂H₄ → C₆H₆ + C₂H₃	−24.6	−19.8	−1.7	−2.0	−1.7
9	C₂H + HC(CH₃)₃ → C₂H₂ + C(CH₃)₃	−49.7	−48.0	−36.8	−36.2	−35.7
10	C₆H₅ + HC(CH₃)₃ → C₆H₆ + C(CH₃)₃	−43.4	−38.4	−15.1	−15.4	−14.9
11	C₂H + C₆H₆ → C₂H₂ + C₆H₅	−6.2	−9.6	−21.7	−20.8	−20.8
12	C₂H + CH₄ → C₂H₂ + CH₃	−43.3	−41.3	−29.4	−29.0	−28.4
13	C₂H₃ + CH₄ → C₂H₄ + CH₃	−12.4	−11.9	−5.9	−6.2	−5.8
14	C(CH₃)₃ + CH₄ → HC(CH₃)₃ + CH₃	6.4	6.7	7.4	7.2	7.3
15	C₆H₅ + CH₄ → C₆H₆ + CH₃	−37.1	−31.7	−7.6	−8.2	−7.6

	Reaction	MP2	MP2.5	OMP2.5	CCSD	CCSD(T)
1	CH₃ + H₂ → CH₄ + H	−7.6	−5.9	−5.8	−2.5	−3.5
2	C₂H + H₂ → C₂H₂ + H	−50.9	−47.2	−35.2	−31.4	−31.9
3	C₂H₃ + H₂ → C₂H₄ + H	−20.0	−17.7	−11.7	−8.7	−9.3
4	C(CH₃)₃ + H₂ → HC(CH₃)₃ + H	−1.2	0.8	1.6	4.8	3.8
5	C₆H₅ + H₂ → C₆H₆ + H	−44.7	−37.6	−13.5	−10.6	−11.1
6	C₂H + C₂H₄ → C₂H₂ + C₂H₃	−30.9	−29.5	−23.4	−22.8	−22.6
7	C(CH₃)₃ + C₂H₄ → HC(CH₃)₃ + C₂H₃	18.8	18.6	13.4	13.4	13.1
8	C₆H₅ + C₂H₄ → C₆H₆ + C₂H₃	−24.6	−19.8	−1.7	−2.0	−1.7
9	C₂H + HC(CH₃)₃ → C₂H₂ + C(CH₃)₃	−49.7	−48.0	−36.8	−36.2	−35.7
10	C₆H₅ + HC(CH₃)₃ → C₆H₆ + C(CH₃)₃	−43.4	−38.4	−15.1	−15.4	−14.9
11	C₂H + C₆H₆ → C₂H₂ + C₆H₅	−6.2	−9.6	−21.7	−20.8	−20.8
12	C₂H + CH₄ → C₂H₂ + CH₃	−43.3	−41.3	−29.4	−29.0	−28.4
13	C₂H₃ + CH₄ → C₂H₄ + CH₃	−12.4	−11.9	−5.9	−6.2	−5.8
14	C(CH₃)₃ + CH₄ → HC(CH₃)₃ + CH₃	6.4	6.7	7.4	7.2	7.3
15	C₆H₅ + CH₄ → C₆H₆ + CH₃	−37.1	−31.7	−7.6	−8.2	−7.6

FIG. 5.

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Errors in hydrogen transfer reaction energies (Table I) for the MP2, MP2.5, OMP2.5, and CCSD methods with respect to CCSD(T), all in the CBS limit.

FIG. 6.

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Mean absolute errors in hydrogen transfer reaction energies (Table I) for the MP2, MP2.5, OMP2.5, and CCSD methods with respect to CCSD(T), all in the CBS limit.

C. Noncovalent interactions

Now, we turn our attention to noncovalent interactions, and consider the closed-shell complexes at first. For this purpose, we considered the A24 database of Rezác and Hobza.⁸⁵ For the A24 database, noncovalent interaction energies (in kcal mol⁻¹) from the MP2, MP2.5, and CCSD methods at the CBS limit are reported in Table II. Errors with respect to reference energies (Table II) are presented graphically in Figure 7, while the MAE values are depicted in Figure 8. The MAE values are 0.11 (MP2), 0.06 (MP2.5), 0.04 (OMP2.5), and 0.26 (CCSD) kcal mol⁻¹. The OMP2.5 method improves upon MP2.5 only slightly, indicating that the orbital relaxation effects might be ignored in the case of closed-shell complexes. It is surprising that the CCSD method yields a larger MAE than MP2. However, this result is consistent with the recent study of Copan, Sokolov, and Schaefer,⁸⁶ where the MAE of CCSD was 0.25 kcal mol⁻¹ with the aug-cc-pVTZ basis set.

Table II.

Closed-shell noncovalent interaction energies (in kcal mol⁻¹) from MP2, MP2.5, OMP2.5, and CCSD at the CBS limit, and the mean absolute errors (MAE) with respect to reference energies.^a

	Complex	Interaction type	MP2	MP2.5	OMP2.5	CCSD	Ref.^a
1	Water⋯ammonia (C_s)	H-bond	−6.53	−6.46	−6.55	−6.14	−6.52
2	Water dimer (C_s)	H-bond	−4.89	−4.89	−4.97	−4.71	−5.01
3	HCN dimer (C_∞v)	H-bond	−4.89	−4.80	−4.82	−4.63	−4.75
4	HF dimer (C_s)	H-bond	−4.33	−4.39	−4.45	−4.30	−4.57
5	Ammonia dimer (C_2h)	H-bond	−3.14	−3.10	−3.16	−2.89	−3.16
6	Methane⋯HF (C_3v)	Mixed	−1.68	−1.66	−1.73	−1.52	−1.68
7	Ammonia⋯methane (C_3v)	Mixed	−0.71	−0.72	−0.75	−0.64	−0.78
8	Methane⋯water (C_s)	Mixed	−0.62	−0.63	−0.65	−0.56	−0.67
9	Formaldehyde dimer (C_s)	Mixed	−4.46	−4.30	−4.61	−4.01	−4.47
10	Ethene⋯water (C_s)	Mixed	−2.78	−2.68	−2.68	−2.30	−2.58
11	Ethene⋯formaldehyde (C_s)	Mixed	−1.69	−1.64	−1.67	−1.36	−1.63
12	Ethyne dimer (C_2v)	Mixed	−1.67	−1.62	−1.59	−1.36	−1.54
13	Ethene⋯ammonia (C_s)	Mixed	−1.52	−1.46	−1.44	−1.17	−1.39
14	Ethene dimer (C_2v)	Mixed	−1.28	−1.18	−1.17	−0.78	−1.11
15	Methane⋯ethene (C_s)	Mixed	−0.56	−0.54	−0.52	−0.37	−0.51
16	Borane⋯methane (C_s)	DD^b	−1.48	−1.45	−1.54	−1.16	−1.52
17	Methane⋯ethane (C_s)	DD	−0.81	−0.78	−0.82	−0.62	−0.84
18	Methane⋯ethane (C₃)	DD	−0.55	−0.55	−0.59	−0.45	−0.62
19	Methane dimer (D_3d)	DD	−0.49	−0.49	−0.51	−0.39	−0.54
20	Methane⋯Ar (C_3v)	DD	−0.41	−0.37	−0.38	−0.28	−0.41
21	Ethene⋯Ar (C_2v)	DD	−0.43	−0.38	−0.36	−0.23	−0.37
22	Ethene⋯ethyne (C_2v)	DD	0.47	0.73	0.80	1.23	0.78
23	Ethene dimer (D_2h)	DD	0.68	0.87	0.93	1.39	0.90
24	Ethyne dimer (D_2h)	DD	0.70	1.03	1.11	1.50	1.08
		MAE	0.11	0.06	0.04	0.26

	Complex	Interaction type	MP2	MP2.5	OMP2.5	CCSD	Ref.^a
1	Water⋯ammonia (C_s)	H-bond	−6.53	−6.46	−6.55	−6.14	−6.52
2	Water dimer (C_s)	H-bond	−4.89	−4.89	−4.97	−4.71	−5.01
3	HCN dimer (C_∞v)	H-bond	−4.89	−4.80	−4.82	−4.63	−4.75
4	HF dimer (C_s)	H-bond	−4.33	−4.39	−4.45	−4.30	−4.57
5	Ammonia dimer (C_2h)	H-bond	−3.14	−3.10	−3.16	−2.89	−3.16
6	Methane⋯HF (C_3v)	Mixed	−1.68	−1.66	−1.73	−1.52	−1.68
7	Ammonia⋯methane (C_3v)	Mixed	−0.71	−0.72	−0.75	−0.64	−0.78
8	Methane⋯water (C_s)	Mixed	−0.62	−0.63	−0.65	−0.56	−0.67
9	Formaldehyde dimer (C_s)	Mixed	−4.46	−4.30	−4.61	−4.01	−4.47
10	Ethene⋯water (C_s)	Mixed	−2.78	−2.68	−2.68	−2.30	−2.58
11	Ethene⋯formaldehyde (C_s)	Mixed	−1.69	−1.64	−1.67	−1.36	−1.63
12	Ethyne dimer (C_2v)	Mixed	−1.67	−1.62	−1.59	−1.36	−1.54
13	Ethene⋯ammonia (C_s)	Mixed	−1.52	−1.46	−1.44	−1.17	−1.39
14	Ethene dimer (C_2v)	Mixed	−1.28	−1.18	−1.17	−0.78	−1.11
15	Methane⋯ethene (C_s)	Mixed	−0.56	−0.54	−0.52	−0.37	−0.51
16	Borane⋯methane (C_s)	DD^b	−1.48	−1.45	−1.54	−1.16	−1.52
17	Methane⋯ethane (C_s)	DD	−0.81	−0.78	−0.82	−0.62	−0.84
18	Methane⋯ethane (C₃)	DD	−0.55	−0.55	−0.59	−0.45	−0.62
19	Methane dimer (D_3d)	DD	−0.49	−0.49	−0.51	−0.39	−0.54
20	Methane⋯Ar (C_3v)	DD	−0.41	−0.37	−0.38	−0.28	−0.41
21	Ethene⋯Ar (C_2v)	DD	−0.43	−0.38	−0.36	−0.23	−0.37
22	Ethene⋯ethyne (C_2v)	DD	0.47	0.73	0.80	1.23	0.78
23	Ethene dimer (D_2h)	DD	0.68	0.87	0.93	1.39	0.90
24	Ethyne dimer (D_2h)	DD	0.70	1.03	1.11	1.50	1.08
		MAE	0.11	0.06	0.04	0.26

^a

At the CCSD(T)/CBS + ΔE_cc + ΔE_rel + ΔCCSDT(Q) level, where ΔE_cc and ΔE_rel are the core correlation and the relativity corrections, respectively.⁸⁵

^b

Dispersion dominated.

FIG. 7.

View large Download slide

Errors in closed-shell noncovalent interaction energies (Table II) for the MP2, MP2.5, OMP2.5, and CCSD methods (all in the CBS limit) with respect to reference energies.⁸⁵

FIG. 8.

View large Download slide

Mean absolute errors in closed-shell noncovalent interaction energies (Table II) for the MP2, MP2.5, OMP2.5, and CCSD methods (all in the CBS limit) with respect to reference energies.⁸⁵

Finally, we assess the performance of OMP2.5 for open-shell noncovalent interactions. For this purpose, we considered a set of open-shell complexes (the O23 database) introduced recently.^15,16 For the O23 database, noncovalent interaction energies (in kcal mol⁻¹) from the MP2, MP2.5, CCSD, and CCSD(T) methods at the CBS limit are reported in Table III. Errors with respect to CCSD(T) are presented graphically in Figure 9, while the MAE values are depicted in Figure 10. The MAE values are 0.68 (MP2), 0.60 (MP2.5), 0.12 (OMP2.5), and 0.38 (CCSD) kcal mol⁻¹, indicating a reduction in MP2.5 errors by a factor of 5 when optimized orbitals are employed, and comparing to CCSD there is a more than 3-fold decrease in errors. Further, for a few complexes the standard methods fail significantly, while the OMP2.5 method provides quite reasonable results. For example, for the HF⋯ CO⁺ complex, MP2, MP2.5, and CCSD yield errors of −5.45, −4.09, and 1.55 kcal mol⁻¹, respectively, whereas the OMP2.5 error is 0.19 kcal mol⁻¹. Similarly, for H₂O⋯ F, the MP2, MP2.5, and CCSD methods yield errors of 3.55, 3.81, and 1.59 kcal mol⁻¹, respectively, while the OMP2.5 error is only 0.08 kcal mol⁻¹. Overall, it appears that the OMP2.5 method can be reliably used for open-shell noncovalent interactions when the more sophisticated CCSD(T) approach is too computationally costly.

Table III.

Open-shell noncovalent interaction energies (in kcal mol⁻¹) from MP2, MP2.5, OMP2.5, CCSD, and CCSD(T) at the CBS limit, and the mean absolute errors (MAE) with respect to CCSD(T).

	Complex	MP2^a	MP2.5	OMP2.5	CCSD^a	CCSD(T)^a
1	H₂O⋯ ${\rm NH_3^{+}}$ ${NH}_{3}^{+}$	−17.40	−17.32	−18.32	−17.68	−18.40
2	HOH⋯CH₃	−1.67	−1.63	−1.72	−1.49	−1.75
3	NH⋯NH^b	−1.04	−1.04	−1.03	−1.01	−1.02
4	Li⋯Li^c	0.04	−0.16	−0.19	−0.94	−0.97
5	H₂O⋯ ${\rm HNH_2^{+}}$ ${HNH}_{2}^{+}$	−25.58	−25.66	−25.62	−25.09	−25.41
6	H₂⋯Li	−0.02	−0.02	−0.02	−0.02	−0.02
7	H₂O⋯F	−0.16	0.10	−3.64	−2.12	−3.71
8	FH⋯BH₂	−4.11	−4.07	−4.17	−3.95	−4.22
9	He⋯Li	0.00	0.00	0.00	0.00	0.00
10	H₂O⋯HO₂	−2.10	−2.10	−2.21	−2.06	−2.24
11	H₂O⋯Al	−7.12	−7.18	−7.70	−6.84	−7.75
12	Ar⋯NO	−0.33	−0.31	−0.54	−0.24	−0.34
13	Ar⋯OH	−0.16	−0.15	−0.15	−0.14	−0.16
14	FH⋯OH	−6.02	−6.02	−6.12	−5.84	−6.10
15	He⋯OH	−0.02	−0.02	−0.03	−0.03	−0.05
16	H₂O⋯Be⁺	−63.95	−64.81	−64.72	−65.42	−65.22
17	HF⋯CO⁺	−35.82	−34.46	−30.19	−28.82	−30.37
18	H₂O⋯Cl	−2.95	−2.74	−3.69	−2.66	−3.58
19	H₂O⋯Br	−3.11	−2.83	−3.46	−2.64	−3.48
20	H₂O⋯Li	−11.64	−11.96	−12.36	−12.46	−12.63
21	FH⋯NH₂	−10.43	−10.35	−10.41	−10.00	−10.33
22	NC⋯Ne	−0.06	−0.06	−0.05	−0.06	−0.07
23	He⋯NH^c	−0.02	−0.03	−0.03	−0.03	−0.04
	MAE	0.68	0.60	0.12	0.38

	Complex	MP2^a	MP2.5	OMP2.5	CCSD^a	CCSD(T)^a
1	H₂O⋯ ${\rm NH_3^{+}}$ ${NH}_{3}^{+}$	−17.40	−17.32	−18.32	−17.68	−18.40
2	HOH⋯CH₃	−1.67	−1.63	−1.72	−1.49	−1.75
3	NH⋯NH^b	−1.04	−1.04	−1.03	−1.01	−1.02
4	Li⋯Li^c	0.04	−0.16	−0.19	−0.94	−0.97
5	H₂O⋯ ${\rm HNH_2^{+}}$ ${HNH}_{2}^{+}$	−25.58	−25.66	−25.62	−25.09	−25.41
6	H₂⋯Li	−0.02	−0.02	−0.02	−0.02	−0.02
7	H₂O⋯F	−0.16	0.10	−3.64	−2.12	−3.71
8	FH⋯BH₂	−4.11	−4.07	−4.17	−3.95	−4.22
9	He⋯Li	0.00	0.00	0.00	0.00	0.00
10	H₂O⋯HO₂	−2.10	−2.10	−2.21	−2.06	−2.24
11	H₂O⋯Al	−7.12	−7.18	−7.70	−6.84	−7.75
12	Ar⋯NO	−0.33	−0.31	−0.54	−0.24	−0.34
13	Ar⋯OH	−0.16	−0.15	−0.15	−0.14	−0.16
14	FH⋯OH	−6.02	−6.02	−6.12	−5.84	−6.10
15	He⋯OH	−0.02	−0.02	−0.03	−0.03	−0.05
16	H₂O⋯Be⁺	−63.95	−64.81	−64.72	−65.42	−65.22
17	HF⋯CO⁺	−35.82	−34.46	−30.19	−28.82	−30.37
18	H₂O⋯Cl	−2.95	−2.74	−3.69	−2.66	−3.58
19	H₂O⋯Br	−3.11	−2.83	−3.46	−2.64	−3.48
20	H₂O⋯Li	−11.64	−11.96	−12.36	−12.46	−12.63
21	FH⋯NH₂	−10.43	−10.35	−10.41	−10.00	−10.33
22	NC⋯Ne	−0.06	−0.06	−0.05	−0.06	−0.07
23	He⋯NH^c	−0.02	−0.03	−0.03	−0.03	−0.04
	MAE	0.68	0.60	0.12	0.38

^a

From Soydaş and Bozkaya.¹⁵ All systems are in doublet states unless otherwise noted.

^b

The lowest quintet state of the dimer is considered, the lowest singlet and triplet states require multireference wave functions.⁸⁹

^c

The lowest triplet state of the dimer is considered.

FIG. 9.

View large Download slide

Errors in open-shell noncovalent interaction energies (Table III) for the MP2, MP2.5, OMP2.5, and CCSD methods with respect to CCSD(T), all in the CBS limit.

FIG. 10.

View large Download slide

Mean absolute errors in open-shell noncovalent interaction energies (Table III) for the MP2, MP2.5, OMP2.5, and CCSD methods with respect to CCSD(T), all in the CBS limit.

IV. CONCLUSIONS

OMP2.5 and its analytic energy gradients have been presented. Results from the OMP2.5 method have been obtained for geometries, hydrogen transfer reaction energies, and noncovalent interaction energies for comparison with those from the standard MP2, MP2.5, CCSD, and CCSD(T) methods.

For the variational optimization of the molecular orbitals for OMP2.5, a Lagrangian-based approach has been employed similar to the previously reported orbital-optimized methods.^8,9,13 For the MO optimization procedure, an orbital DIIS algorithm has been implemented, as described in our recent study.¹³ Both the OMP2.5 and CCSD methods scale formally as O(N⁶). However, the OMP2.5 t₂-amplitude equations are simpler than those of CCSD. For example, the W_pqrs intermediates,⁴⁰ which scale as O(N⁶) for CCSD, are equal to the corresponding two-electron integrals for OMP2.5. Hence, in each OMP2.5 iteration we avoid a cost of 3*O(N⁶) for the W_pqrs intermediates. Additionally, because

$\lambda _{ab}^{ij(1)} = t_{ij}^{ab(1)}$

λ_{a b}^{i j (1)} = t_{i j}^{a b (1)}

and

$\lambda _{ab}^{ij(2)} = t_{ij}^{ab(2)}$

λ_{a b}^{i j (2)} = t_{i j}^{a b (2)}

for MP2.5, orbital optimization does not require any extra solution of λ₂-amplitude equations to obtain the OMP2.5 energies. Hence, the cost of the OMP2.5 method is no more expensive than that of CCSD, but in practice it is several times faster than CCSD in energy computations. Further, in CCSD analytic gradient computations, one needs to solve the λ₂-amplitude equations, but for OMP2.5 one does not. Moreover, the OOVV block TPDM (Γ_ijab) scales as O(N⁶) for CCSD,^87,88 while it has no extra cost [Eq. ] for OMP2.5. Finally, for analytic gradient computations (and hence for geometry optimizations) the OMP2.5 method is as much as twice as fast as CCSD. Combining this feature with the demonstrated accuracy of OMP2.5, one may prefer OMP2.5 over CCSD for a number of purposes.

For bond lengths of closed- and open-shell molecules with a cc-pCVQZ basis, the OMP2.5 method provides lower mean absolute errors (0.0046, 0.0087 Å) than MP2.5 (0.0074, 0.0154 Å) and CCSD (0.0066, 0.0121 Å), but the MAE of OMP2.5 is still higher than that of CCSD(T) (0.0018, 0.0071 Å), as expected. Hence, with a substantially lower cost one may approach CCSD(T) quality for bond lengths of open-shell molecules with the OMP2.5 method. For hydrogen transfer reactions between free radicals in the CBS limit, the OMP2.5 method (MAE = 1.1 kcal mol⁻¹) exhibits a considerably better performance than MP2 (14.6 kcal mol⁻¹) and MP2.5 (11.8 kcal mol⁻¹), indicating a 10-fold reduction in MAE compared to canonical MP2.5 and a more than 12-fold reduction compared to MP2. The CCSD method further improves upon OMP2.5 and provides a MAE of 0.5 kcal mol⁻¹.

OMP2.5 maintains the very good performance of MP2.5 for closed-shell noncovalent interaction energies, but it provides a substantial improvement for open-shell systems. For the O23 test set, the OMP2.5 (MAE = 0.12 kcal mol⁻¹) method outperforms the MP2 (0.68 kcal mol⁻¹), MP2.5 (0.60 kcal mol⁻¹), and CCSD (0.38 kcal mol⁻¹) methods and provides quite accurate results. MP2.5 errors decrease by a factor of 5 when optimized orbitals are used, and comparing to CCSD there is a more than 3-fold reduction in errors. Hence, it appears that the OMP2.5 method is a good choice for the study of open-shell noncovalent interactions considering the computational cost and accuracy balance. Overall, the results presented in this study demonstrate that the OMP2.5 method is very beneficial for chemical systems with challenging electronic structures, such as open-shell noncovalent interaction complexes and free radicals. The inexpensive analytic gradients of OMP2.5 compared to CCSD or CCSD(T) make it promising for high-accuracy studies.

ACKNOWLEDGMENTS

This research was supported by the U.S. National Science Foundation (NSF) (Grant Nos. CHE-1300497 and ACI-1147843) and the Scientific and Technological Research Council of Turkey (TÜBİTAK-113Z203).

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2014

AIP Publishing LLC

Orbital-optimized MP2.5 and its analytic gradients: Approaching CCSD(T) quality for noncovalent interactions

I. INTRODUCTION

II. THEORETICAL APPROACH

A. OMP2.5 energy

1. The MP2.5-Λ energy functional and amplitude equations

2. The parametrization of the OMP2.5 wave function

3. Response density matrices

4. Generalized-Fock and orbital gradient

5. The orbital optimization procedure

B. Geometry dependent transformations

C. OMP2.5 gradient

D. MP2.5 gradient

III. RESULTS AND DISCUSSION

A. Geometries

B. Hydrogen transfer reactions

C. Noncovalent interactions

IV. CONCLUSIONS

ACKNOWLEDGMENTS

REFERENCES

Supplementary Material

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Orbital-optimized MP2.5 and its analytic gradients: Approaching CCSD(T) quality for noncovalent interactions

I. INTRODUCTION

II. THEORETICAL APPROACH

A. OMP2.5 energy

1. The MP2.5-Λ energy functional and amplitude equations

2. The parametrization of the OMP2.5 wave function

3. Response density matrices

4. Generalized-Fock and orbital gradient

5. The orbital optimization procedure

B. Geometry dependent transformations

C. OMP2.5 gradient

D. MP2.5 gradient

III. RESULTS AND DISCUSSION

A. Geometries

B. Hydrogen transfer reactions

C. Noncovalent interactions

IV. CONCLUSIONS

ACKNOWLEDGMENTS

REFERENCES

Supplementary Material

Citing articles via

Submit your article

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Resources

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