Open shells implementing UHF

Another way of constructing wave functions for open-shell molecules is the restricted open shell Hartree-Fock method (ROHF). In this method, the paired electrons share the same spatial orbital thus, there is no spin contamination. The ROHF technique is more difficult to implement than UHF and may require slightly more CPU time to execute. ROHF is primarily used for cases where spin contamination is large using UHF. 

We have previously defined the one-electron spin-density matrix in the context of standard HF methodology (Eq. (6.9)), which includes semiempirical methods and both the UHF and ROHF implementations of Hartree-Fock for open-shell systems. In addition, it is well defined at the MP2, CISD, and DFT levels of theory, which permits straightforward computation of h.f.s. values at many levels of theory. Note that if the one-electron density matrix is not readily calculable, the finite-field methodology outlined in the last section allows evaluation of the Fermi contact integral by an appropriate perturbation of the quantum mechanical Hamiltonian. 

In this section, we briefly discuss some of the electronic structure methods which have been used in the calculations of the PE functions which are discussed in the following sections. There are variety of ab initio electronic structure methods which can be used for the calculation of the PE surface of the electronic ground state. Most widely used are Hartree-Fock (HF) based methods. In this approach, the electronic wavefunction of a closed-shell system is described by a determinant composed of restricted one-electron spin orbitals. The unrestricted HF (UHF) method can handle also open-shell electronic systems. The limitation of HF based methods is that they do not account for electron correlation effects. For the electronic ground state of closed-shell systems, electron correlation effects can be accounted for relatively easily by second-order Mpller-Plesset perturbation theory (MP2). In modern implementations of MP2, linear scaling with the size of the system has been achieved. It is thus possible to treat quite large molecules and clusters at this level of theory. 

The HF method treats electron-electron interactions at a mean field level, with the Hartree and exchange interactions exactly written. The method can be implemented either in its spin restricted form (RHF), for closed shell systems, or in the unrestricted form (UHF) for open-shell or strongly correlated systems. In the first case, the one-electron orbitals are identical for electrons of both spin directions, while UHF can account for a non-uniform spin density. The one electron orbitals, which are determined in the course of the self-consistent resolution of the HF equations, are expanded on an over-complete basis set of optimized variational functions. 

Gradients provide structures and their numerical second derivatives characterize critical points while providing vibrational frequencies. However, there are still great advantages if the second derivatives can be determined analytically, too. This unique development was accomplished by Gauss and Stanton [128], who implemented analytical Hessians for CC methods from CCSD to CCSD(T) and the full CCSDT. Open-shell (UHF) second derivatives for CC methods were presented by Szalay et al. [129]. Now this has been done automatically for even higher levels of theory with Kallay and Gauss automated approach [130]. 

Moreover, the Hartree-Fock Eqs. (50) implemented in Eqs. (51) and (52) are known as Roothaan equations [109] and constitute the basis for closed-shell (or restricted Hartree-Fock, RHF) molecular orbitals calculations. Their extension to spin effects provides the equations for the open shell (or unrestricted Hartree-Fock, UHF), which are also known as the Pople-Nesbet Unrestricted equations [118]. 

The main goal of the present work is to report the implementation of the explicitly-correlated coupled-cluster singles-and-doubles method (CCSD(F12)) in Turbomole. This tool is capable of very efficient calculating the CCSD energies at the basis set limit with relatively small orbital basis sets. The implementation works with RHF, UHF and ROHF reference wave functions, which means that it can treat both closed- and open-shell species. The formulation in terms of intermediate quantities and the application of density fitting techniques make this implementation quite unique. 

All explicitly correlated calculations were performed at the CCSD(F12) level of theory, as implemented in the TurbomOLE program [58, 69]. The Slater-type correlation factor was used with the exponent 7 = 1.0 aQ. It was approximated by a linear combination of six Gaussian functions with linear and nonlinear coefficients taken from Ref. [44]. The CCSD(F12) electronic energies were computed in an all-electron calculation with the d-aug-cc-pwCV5Z basis set [97]. For all cases we used full CCSD(F12) model (see Subsection 4.9 for the discussion about models implemented in Turbomole), the open-shell species were computed with a UHF reference wave function. The explicitly correlated contributions to the relative quantities are collected in Tables 10 and 11 under the label F12 . 

More physically correct than ROHF, and much easier to implement computationally, is another scheme called unrestricted Hartree-Fock (UHF), wherein the manifold of occupied MOs is not subdivided into closed and open shells. Instead, a standard HF program is used to carry out parallel sets of HF calculations on two different sets of MOs, one containing only the a and the other only the p electrons. The resulting pairs of UHF MOs for a and p electrons, which are identical in an ROHF calculation, have similar nodal properties, but they differ from each other in spatial detail. The restriction, Inherent in the ROHF scheme, that paired electrons of opposite spin occupy identical MOs is thus removed in UHF calculations. ... 

Electronic energies for a given molecular structure can be computed on the SCF, DFT, and MP2 level for closed shell states, and for open shell states within the UHF description. Open shell ROHF treatments are supported for the SCF method. All common DFT functionals are implemented, including the hybrid functionals which include exact exchange, e.g., modified BLYP (B3LYP). Treatments within the non-hybrid functionals can be efficiently earned out in the RI mode. Version 4.1 will support the efficient RI-MP2 technique. 

MINDO/3, MNDO, MNDO/C, AM1, and PM3 are implemented in VAMP. Default calculations employ the RHF formalism (half-electron for open-shell species), but UHF and annihilated UHF (AUHF) are also available. 

The main disadvantage of the MBPT/CCM approach, the restriction to a single reference function, is not essential to the theory but only to the current implementation (see references 30-32 for the multi-reference MBPT/CCM theory). To solve open-shell problems we normally use an unrestricted-Hartree-Fock (UHF) reference function. 

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