A broken-(spin) symmetry (BS) method is now widely used for systems that involve (quasi) degenerated frontier orbitals because of their lower cost of computation. The BS method splits up-spin and down-spin electrons into two different special orbitals, so that a singlet spin state of the degenerate system is expressed as a singlet biradical. In the BS solution, therefore, the spin symmetry is no longer retained. Due to such spin-symmetry breaking, the BS method often suffers from a serious problem called a spin contamination error, so that one must eliminate the error by some kind of projection method. An approximate spin projection (AP) method, which is one of the spin projection procedures, can eliminate the error from the BS solutions by assuming the Heisenberg model and can recover the spin symmetry. In this chapter, we illustrate a theoretical background of the BS and AP methods, followed by some examples of their applications, especially for calculations of the exchange interaction and for the geometry optimizations.
- quantum chemistry
- ab initio calculation
- orbital degeneracy
- electron correlation
- broken-(spin) symmetry (BS) method
- approximate spin projection (AP) method
- spin polarization
- spin contamination error
- effective exchange integral (Jab) values
For the past few decades, many reports about “polynuclear metal complexes” have been presented actively in the field of the coordination chemistry [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. Those systems usually have complicated electronic structures that are constructed by metal–metal (d-d) and metal–ligand (d-p) interactions. Those electronic structures caused by their unique molecular structures often bring many interesting and noble physical functionalities such as a magnetism [8, 9, 10, 11, 12, 13, 14, 15, 16, 17], a nonlinear optics , an electron conductivity , as well as their chemical functionalities, e.g., a catalyst and so on. For example, some three-dimensional (3D) metal complexes show interesting magnetic behaviors and are expected to be possible candidates for a single molecule magnet, a quantum dot, and so on [11, 12, 13, 14, 15, 16]. On the other hand, one-dimensional (1D) metal complexes are studied for the smallest electric wire, i.e., the nanowire [3, 4, 5, 6, 7, 17, 19]. In addition, it has been elucidated that the polynuclear metal complexes play an important role in the biosystems [20, 21, 22, 23, 24], e.g., Mn cluster [25, 26] in photosystem II and 4Fe-4S cluster [27, 28, 29, 30] in electron transfer proteins. In this way, the polynuclear metal complexes are widely noticed from a viewpoint of fundamental studies on their peculiar characters and of applications to materials. From those reasons, an elucidation of a relation among electronic structures, molecular structures, and physical properties is a quite important current subject.
Physical properties of molecules are sometimes discussed by using several parameters such as an exchange integrals (
2. Theoretical background of AP method
In this section, the theoretical background of the BS and AP methods for the biradical systems is explained with the simplest two-spin model (e.g., a dissociated H2) as illustrated in Figure 1(a).
2.1. Broken-symmetry (BS) solution and approximate spin projection (AP) methods for the (two-spin) biradical state
In the BS method, the spin-polarized orbitals are obtained from HOMO-LUMO mixing [55, 56]. For example, HOMO orbitals for up-spin () and down-spin () electrons of the simple H2 molecule are expressed as follows (Figure 1(b)):
where 0 ≤
where and express up- and down-spin electrons in orbital , respectively. If
We often regard such spin densities as an existence of localized spins. An interaction between localized spins can be expressed by using Heisenberg Hamiltonian:
where and are spin operators for spin sites a and b, respectively, and
Similarly, for triplet state
The energy difference between singlet () and triplet () states (S-T gap) within Heisenberg Hamiltonian should be equal to the S-T gap calculated by the difference in total energies of ab initio calculations (here we denote and for the BS singlet and triplet states, respectively). And if we can assume that spin densities of the BS singlet state on spin site
If the method is exact and the spin contamination error is not found in both singlet and triplet states (i.e., and ), the S-T gap between those states can be expressed as
The spin contamination in the triplet state is usually negligible (i.e., ), and one must consider the error only in the BS singlet state, so the S-T gap becomes
A second term in a right side of Eq. (10) indicates the spin contamination error in the S-T gap, and consequently, a second term in a denominator of Eq. (8) eliminates the spin contamination in the BS singlet solution. In this way, Eq. (8) gives approximately spin-projected (AP)
Eq. (11) is the so-called Yamaguchi equation to calculate
2.2. Approximate spin projection for BS energy and energy derivatives
Then, we explain about derivatives of this spin-projected energy (). In order to carry out the geometry optimization using the AP method, an energy gradient of is necessary. can be expanded by using Taylor expansion:
where and are the first and second derivatives (i.e., gradient and Hessian) of , respectively [62, 63, 64, 65]; and
where and are the first energy derivatives (energy gradients) of the BS and the HS states, respectively. As mentioned above, the spin contamination in the HS state is negligible, so that is usually a constant. Then can be written as
where and are the Hessians calculated by the BS and the HS states, respectively. And a second derivative of α can be expressed by
By using Eqs. (18) and (19), the spin-projected vibrational frequencies are also calculated. The AP optimization can be carried out based on Eq. (16) with obtained by numerical fitting or analytical ways.
2.3. Relationship between the BS and projected wavefunctions
As well as a calculated energy and its derivatives, the BS wavefunction itself has also vital information. Here let us go back to Eq. (3). From the equation, an overlap between up-spin (so-called alpha) and down-spin (so-called beta) orbitals (
And because occupation number (
On the other hand, we can define projected wavefunction (PUHF) by eliminating triplet species from BS singlet wavefunction from Eq. (3) as follows:
Different from the
Those indices show how the BS and AP wavefunctions are connected. In addition, one can utilize the indices to estimate the contribution of double excitation for very large systems that CAS and MR methods cannot be applied.
Finally, a relationship between the BS wavefunction and values are briefly explained. The values of the BS singlet states do not show the exact value by the spin contamination error. value of the SA calculation is.
However, in the case of the BS singlet state of H2 molecule, it becomes
Here we explain another aspect of the spin projection method. As depicted in Figure 1(c), the BS wavefunction indicates only one spin-polarized configuration, e.g., BS1 in the figure. However, in order to obtain a pure singlet wavefunction, which satisfies the spin symmetry, the opposite spin-polarized state (BS2) must be included. The projection method can give a linear combination of the both BS states, and therefore it can give an appropriate quantum state for the singlet state.
3. Application of BS and AP methods to several biradical systems
3.1. Hydrogen molecule: comparison among SA, BS, and AP methods by simple biradical system
In this section, we briefly illustrate how the BS and AP methods approximate a dissociation of a hydrogen molecule. Figure 2(a) shows potential energy curves of Hartree-Fock and full CI methods. In the case of the spin-adapted (SA) HF, i.e., the spin-restricted (R) HF method, the curve does not converge to the dissociation limit. On the other hand, the BS HF, i.e., spin-unrestricted (U) HF calculation, successfully reproduces the dissociation limit of full CI method. This result indicates that the static correlation is included in the BS procedure. Around 1.2 Å, there is a bifurcation point between RHF and UHF methods. Within the closed shell (i.e., SA) region, where
In order to elucidate how the double-excitation state is included in the BS solution, the occupation numbers of the highest occupied natural orbital (HONO) are plotted along the H-H distance in Figure 2(b). The figure indicates that the occupation number is 2.0 in the closed shell region, while it suddenly decreases at the bifurcation point. And it finally closes to 1.0 at the dissociation limit. In Figure 2(c), calculated
Next, we illustrate results of calculated effective exchange integral (
3.2. Dichromium (II) complex: effectiveness of hybrid DFT method for calculation of
Next, the BS and AP methods are applied for Cr2(O2CCH3)4(OH2)2 (
As depicted in Figure 3(b), there are eight magnetic orbitals, i.e., bonding and antibonding σ, π //, π⊥, and δ orbitals that concern about the direct bond between Cr(II) ions. The NO analysis clarifies the nature of the Cr-Cr bond. If d-orbitals of two Cr(II) ions have sufficient overlap to form the stable covalent bond, the occupation numbers of each occupied orbital will be almost 2.0 (i.e.,
|Orbital||Occupation number (||Overlap (|
By substituting the obtained energies and values into Eq. (11),
|BH and HLYP||−520|
3.3. Singlet methylene molecule: Spin contamination error in optimized geometry by BS method and its elimination by AP method
Finally, we examine the spin contamination error in the optimized structure. Here we focus on a singlet methylene (CH2). As illustrated in Figure 4(a), the methylene molecule has two valence orbitals (
Both 1A1 and 3B1 methylene molecules have bent structures, but the experimental data indicates a large structural difference between them. For example, as summarized in Table 3, experimental HCH angles (θHCH) of 1A1 and 3B1 states are 102.4° and 134.0°, respectively [66, 67]. There have also been many reports of the SA results as summarized in Ref. . On the other hand, the BS method is a convenient substitute for CI and CAS method, so here we examined the optimized geometry of the 1A1 methylene by SA and BS methods. In order to elucidate a dependency of the spin contamination error on the calculation methods, HF, configuration interaction method with all double substitutions (CID), coupled-cluster method with double substitutions (CCD), several levels of Møller-Plesset energy correction methods (MP2, MP3, and MP4(SDQ)), and a hybrid DFT (B3LYP) method are also examined. In the case of 1A1 state, all SA results are in good agreement with the experimental values; however, it is reported that energy gap between the singlet and triplet (S-T gap) value is too much underestimated . On the other hand, all BS results overestimate the HCH angle. The difference in HCH angle between the BS values and experimental one is about 10–20°. The HCH angles of UCI and UCC methods are especially larger than MP and DFT methods, indicating that the post-HF methods even require some correction for such systems if the BS procedure is utilized. Therefore it is difficult to use the BS solution for 1A1 state without some corrections. On the other hand, by applying the AP method to the BS solution, the error is drastically improved, and the optimized structural parameters became in good agreement with experimental ones. The difference in the optimized
By calculating Hessian, one can also obtain frequencies of the normal modes. In Table 4, the calculated frequencies of the normal mode singlet methylene are summarized. The significant difference between the BS and AP methods can be found in a bending mode. The BS result underestimates the binding mode frequency by the contamination of the triplet state. On the other hand, the AP result gives close to the experimental result of 1A1 species. In this way, the AP method is also effective for the normal mode analysis as well as the geometry optimization.
In this chapter, we explain how the BS method breaks the spin symmetry and AP method recover it. In addition, we also demonstrate how those methods work the biradical systems. The theoretical studies of the large biradical and polyradical systems such as polynuclear metal complexes have been fairly realized by the BS HDFT methods in this decade. The BS method is quite powerful for the large degenerate systems, but one must be careful about the spin contamination error. Therefore the AP method would be important for those studies. For example, it is suggested that the spin contamination error misleads a reaction path that involves biradical transition states (TS) or intermediate state (IM) . In addition, in the case of the more larger systems, e.g., metalloproteins, some kind of semiempirical approach combined with the AP hybrid DFT method by ONIOM method will be effective . By using the method, the mechanisms of the long-distance electron transfers and so on will be elucidated. In such cases, one also must be careful about the parameter of the semiempirical approach to fit the spin-polarized systems. Recently, some improvements for PM6 method have been proposed [75, 76]. Because the PM6 calculation can be utilized for the outer region in ONIOM approach, therefore the AP method is also the effective method for the larger systems. In addition, the BS wavefunction can be applied for other molecular properties by combining with other theoretical procedures. For example, it was reported that the electron conductivity of spin-polarized systems could be simulated by using the BS wavefunction together with elastic Green’s function method , and some applications for one-dimensional complexes have reported [78, 79]. The results indicate that the BS wavefunctions can be applied for calculations of the physical properties of the strong electron correlation systems as well as their electronic structures. The spin-projected wavefunctions seem to be effective for such simulations of the physical properties. From those points of view, the BS and AP methods have a great potential to clarify chemical and physical phenomena that are still open questions.