** Analytic Gradients**

**Analytic gradiens for HF-SCF**

`P. Pulay, Ab initio calculation of force constants and equilibrium geometries in polyatomic molecules, Mol. Phys. 17, 197 (1969)`

**Analytic gradients for MBPT/CC methods**

review articles

`R.J. Bartlett, J.F. Stanton, and J.D. Watts, in Advances in Molecular Vibrations and Collision Dynamics}, Vol. 1, Ed.: J.Bowman (JAI Press, 1991) p. 139`

`J. Gauss and D. Cremer, Analytical Energy Gradients in Møller-Plesset Perturbation and Quadratic Configuration Interaction Methods: Theory and Application, Adv. Quant. Chem. 23, 205-299 (1992)`

`P. Pulay, in Modern Electronic Structure Theory Ed.: D.R. Yarkony (World Scientific, Singapore, 1995), p. 1191 `

`T. Helgaker, Gradient Theory, in Encyclopedia of Computational Chemistry, Eds.: P.v.R. Schleyer et al. (Wiley, 1998) p. 1157 `

`J.F. Stanton and J. Gauss, Analytic second derivatives in high-order many-body perturbation and coupled-cluster theories: Computational considerations and applications, Int. Rev. Phys. Chem. 19, 61-95 (2000)`

MBPT(2)/MP2 gradients

`J.A. Pople, R.Krishnan, H.B. Schlegel, and J.S. Binkley, Derivative studies in hartree‐fock and møller‐plesset theories, Int. J. Quant. Chem. Symp. 13, 225-241 (1979) `

general MBPT/CC gradient theory

`E.A. Salter, G.W. Trucks, and R.J. Bartlett, Analytic energy derivatives in many‐body methods. I. First derivatives, J. Chem. Phys. 90, 1752 (1989)`

MBPT(3)/MP3 and SDQ-MBPT(4)/SDQ-MP4 gradients

`J. Gauss and D. Cremer, Implementation of analytical energy gradients at third- and fourth-order Møller-Plesset perturbation theory, Chem. Phys. Lett. 138, 131-140 (1987)`

MBPT(4)/MP4 gradients

`J. Gauss and D. Cremer, Analytical differentiation of the energy contribution due to triple excitations in fourth-order Møller-Plesset perturbation theory, Chem. Phys. Lett. 153, 303-308 (1988)`

ROHF-MBPT(2)/ROHF-MP2 gradients

`W.J. Lauderdale, J.F. Stanton, J. Gauss, J.D. Watts, and R.J. Bartlett, Restricted open‐shell Hartree–Fock‐based many‐body perturbation theory: Theory and application of energy and gradient calculations, J. Chem. Phys. 97, 6606 (1992)`

ROHF-MBPT(3)/ROHF-MP3 gradients

`J. Gauss, unpublished`

CCSD gradients

`A.C. Scheiner, G.E. Scuseria, J.E. Rice, T.J. Lee, and H.F. Schaefer III, Analytic evaluation of energy gradients for the single and double excitation coupled cluster (CCSD) wave function: Theory and application, J. Chem. Phys. 87, 5361 (1987) (closed-shell CCSD)`

`J. Gauss, J.F. Stanton, and R.J. Bartlett, Coupled‐cluster open‐shell analytic gradients: Implementation of the direct product decomposition approach in energy gradient calculations, J. Chem. Phys. 95, 2623 (1991) (UHF-CCSD)`

`J. Gauss, W.J. Lauderdale, J.F. Stanton, J.D. Watts, and R.J. Bartlett, Analytic energy gradients for open-shell coupled-cluster singles and doubles (CCSD) calculations using restricted open-shell Hartree—Fock (ROHF) reference functions, Chem. Phys. Lett. 182, 207-215 (1991) (ROHF-CCSD)`

`J. Gauss, J.F. Stanton, and R.J. Bartlett, Analytic evaluation of energy gradients at the coupled‐cluster singles and doubles level using quasi‐restricted Hartree–Fock open‐shell reference functions, J. Chem. Phys. 95, 2639 (1991) (QRHF-CCSD)`

QCISD gradients

`J. Gauss and D. Cremer, Analytical evaluation of energy gradients in quadratic configuration interaction theory, Chem. Phys. Lett. 150, 280-286 (1988)`

CCSD(T) gradients

` G.E. Scuseria, Analytic evaluation of energy gradients for the singles and doubles coupled cluster method including perturbative triple excitations: Theory and applications to FOOF and Cr2, J. Chem. Phys. 94, 442 (1991) (closed-shell CCSD(T))`

`T.J. Lee and A.P. Rendell, Analytic gradients for coupled‐cluster energies that include noniterative connected triple excitations: Application to cis‐ and trans‐HONO, J. Chem. Phys. 94, 6229 (1991) (closed-shell CCSD(T) with perturbed canonical orbitals)`

`J.D. Watts, J. Gauss, and R.J. Bartlett, Open-shell analytical energy gradients for triple excitation many-body, coupled-cluster methods: MBPT(4), CCSD+T(CCSD), CCSD(T),and QCISD(T), Chem. Phys. Lett. 200, 1-7 (1992) (UHF-CCSD(T))`

`J.D. Watts, J. Gauss, and R.J. Bartlett, Coupled‐cluster methods with noniterative triple excitations for restricted open‐shell Hartree–Fock and other general single determinant reference functions. Energies and analytical gradients, J. Chem. Phys. 98, 8718 (1993) (ROHF-CCSD(T))`

QCISD(T) gradients

`J. Gauss and D. Cremer, Analytical differentiation of the energy contribution due to triple excitations in quadratic configuration interaction theory, Chem. Phys. Lett. 163, 549-554 (1989)`

CCSDT-n (n=1-3) and CC3 gradients

`J. Gauss and J.F. Stanton, Analytic first and second derivatives for the CCSDT-n (n=1–3) models: a first step towards the efficient calculation of CCSDT properties, Phys. Chem. Chem. Phys. 2, 2047-2060 (2000) (closed-shell)`

CCSDT gradients

`J. Gauss and J.F. Stanton, Analytic gradients for the coupled-cluster singles, doubles, and triples (CCSDT) model, J. Chem. Phys. 116, 1773 (2002)(closed-shell CCSDT)`

`M. Kállay, J. Gauss, and P.G. Szalay, Analytic first derivatives for general coupled-cluster and configuration interaction models, J. Chem. Phys. 119, 2991 (2003)(open-shell CCSDT)`

CCSDT(Q), CCSDTQ-1, and CC4 gradients

`D.A. Matthews, Analytic gradients of approximate coupled cluster methods with quadruple excitations J. Chem. Theor. Comp. 16, 6195-6206 (2020)`

gradients for general CC methods (closed and open-shell)

`M. Kállay, J. Gauss, and P.G. Szalay, Analytic first derivatives for general coupled-cluster and configuration interaction models, J. Chem. Phys. 119, 2991 (2003)`

gradients for two-component CCSD and CCSD(T) including spin-orbit coupling

`F. Wang and J. Gauss, Analytic energy gradients in closed-shell coupled-cluster theory with spin-orbit coupling , J. Chem. Phys. 129, 174110 (2008)`

gradients for Mukherjee's Multireference CC method (Mk-MRCC)

`E. Prochnow, F.A. Evangelista, H.F. Schaefer III, W.D. Allen, and J. Gauss, Analytic gradients for the state-specific multireference coupled cluster singles and doubles model, J. Chem. Phys. 131, 064109 (2009)`

gradients for Mukherjee's Multireference CC method (Mk-MRCC) using TCSCF orbitals

`T.-C. Jagau, E. Prochnow, F.A. Evangelista, and J. Gauss, Analytic gradients for Mukherjee’s multireference coupled-cluster method using two-configurational self-consistent-field orbitals, J. Chem. Phys. 132, 144110 (2010)`

gradients for Mukherjee's Multireference CC method with triples (Mk-MRCCSDT)

`E. Prochnow and J. Gauss, unpublished (2011)`

Z-vector method

`N.C. Handy and H.F. Schaefer III, On the evaluation of analytic energy derivatives for correlated wave functions, J. Chem. Phys. 51, 5031 (1984)`

Dalgarno-Stewart interchange theorem

`A. Dalgarno and A.L. Stewart, A perturbation calculation of properties of the helium iso-electronic sequence, Proc. Roy. Soc. A 247, 245 (1958)`

perturbed-canonical orbitals

`T.J. Lee and A.P. Rendell, Analytic gradients for coupled‐cluster energies that include noniterative connected triple excitations: Application to cis‐ and trans‐HONO, J. Chem. Phys. 94, 6229 (1991)`

**Analytic gradients for TC-SCF**

`J.D. Goddard, N.C. Handy, and H.F. Schaefer III, Gradient techniques for open‐shell restricted Hartree–Fock and multiconfiguration self‐consistent‐field methods , J. Chem. Phys. 71, 1525 (1979)`

**Analytic first derivatives for the unitary group based spin-adapted COS-CC method**

CCSD

`D. Datta and J. Gauss, Analytic first derivatives for a spin-adapted open-shell coupled cluster theory: Evaluation of first-order electrical properties, J. Chem. Phys. 141, 104102 (2014); implementation in CFour`

CCSD(T)

`D. Datta and J. Gauss, Accurate Prediction of Hyperfine Coupling Tensors for Main Group Elements Using a Unitary Group Based Rigorously Spin-Adapted Coupled-Cluster Theory, J. Chem. Theory Comput. 15, 1572-1592 (2019); implementation in CFour`

**Analytic second derivatives**

HF-SCF (RHF and UHF)

`J.A. Pople, R. Krishnan, H.B. Schlegel, and J.S. Binkley, Derivative studies in hartree‐fock and møller‐plesset theories, Int. J. Quant. Chem. Symp. 13, 225-241 (1979)`

ROHF-SCF

`P. Saxe, Y. Yamaguchi, and H.F. Schaefer III, Analytic second derivatives in restricted Hartree–Fock theory. A method for high‐spin open‐shell molecular wave functions, J. Chem. Phys. 77, 5647 (1982)`

TCSCF

`Y. Yamaguchi, Y. Osamura, H.F. Schaefer III, Analytic energy second derivatives for two-configuration self-consistent-field wave functions. Application to twisted ethylene and to the trimethylene diradical, J. Am. Chem. Soc. 105, 7506-7511 (1983)`

MBPT(2)/MP2

`N.C. Handy, R.D. Amos, J.F. Gaw, J.E. Rice, E. Simadiras, T.J. Lee, R.J. Harrison, W.D. Laidig, G.B. Fitzgerald, and R.J. Bartlett,`

`in `*Geometrical Derivatives of Energy Surfaces and Molecular Properties*,

`Eds.: P. Jørgensen and J. Simons (Reidel, Dordrecht, 1986), p. 179. (closed-shell)`

`N.C. Handy, R.D. Amos, J.F. Gaw, J.E. Rice, and E.D. Simadiras, The elimination of singularities in derivative calculations, Chem. Phys. Letters 120, 151-158 (1985) (closed-shell)`

`R.J. Harrison, G.B. Fitzgerald, W.D. Laidig, and R.J. Bartlett, Analytic MBPT(2) second derivatives, Chem. Phys. Letters 124, 291-294 (1986) (closed-shell)`

`J.F. Stanton, J. Gauss, and R.J. Bartlett, Analytic evaluation of second derivatives using second-order many-body perturbation theory and unrestricted Hartree—Fock reference functions, Chem. Phys. Letters 195, 194-199 (1992) (UHF-MBPT(2)/UHF-MP2)`

`J. Gauss, J.F. Stanton, and R.J. Bartlett, Analytic ROHF–MBPT(2) second derivatives, J. Chem. Phys. 97, 7825 (1992) (ROHF-MBPT(2)/ROHF-MP2)`

MBPT(3)/MP3 and MBPT(4)/MP4

CCSD

`H. Koch, H.J.AA. Jensen, T. Helgaker, P. Jørgensen, G.E. Scuseria, and H.F. Schaefer III, Coupled cluster energy derivatives. Analytic Hessian for the closed‐shell coupled cluster singles and doubles wave function: Theory and applications, J. Chem. Phys. 92, 4924 (1990) (closed-shell, RHF)`

`P.G. Szalay, J. Gauss, and J.F. Stanton, Analytic UHF-CCSD(T) second derivatives: implementation and application to the calculation of the vibration-rotation interaction constants of NCO and NCS, Theor. Chem. Acc. 100, 5-11 (1998) (open-shell, UHF)`

CCSD(T)

`J. Gauss and J.F. Stanton, Analytic CCSD(T) second derivatives, Chem. Phys. Letters 276, 70-77 (1997) (closed-shell, RHF)`

`P.G. Szalay, J. Gauss, and J.F. Stanton, Analytic UHF-CCSD(T) second derivatives: implementation and application to the calculation of the vibration-rotation interaction constants of NCO and NCS, Theor. Chim. Acta 100, 5-11 (1998) (open-shell, UHF)`

CCSDT-n, CC3

`J. Gauss and J.F. Stanton, Analytic first and second derivatives for the CCSDT-n (n=1–3) models: a first step towards the efficient calculation of CCSDT properties, Phys. Chem. Chem. Phys. 2, 2047-2060 (2000) (closed-shell)`

CCSDT

`J. Gauss, Analytic second derivatives for the full coupled-cluster singles, doubles, and triples model: Nuclear magnetic shielding constants for BH, HF, CO, N2, N2O, and O3, J. Chem. Phys. 116, 4773 (2002) (closed-shell)`

general CC

`M. Kállay and J. Gauss, Analytic second derivatives for general coupled-cluster and configuration-interaction models, J. Chem. Phys. 120, 6841 (2004) (RHF and UHF)`

two-component CCSD and CCSD(T) including spin-orbit coupling

`F. Wang and J. Gauss, Analytic second derivatives in closed-shell coupled-cluster theory with spin-orbit coupling, J. Chem. Phys. 131, 164113 (2009)`