Solving the Schrödinger equation of atoms and molecules: Chemical-formula theory, free-complement chemical-formula theory, and intermediate variational theory Hiroshi Nakatsuji, Hiroyuki Nakashima, and Yusaku I. Kurokawa
Citation: The Journal of Chemical Physics 149, 114105 (2018); doi: 10.1063/1.5040376 View online: https://doi.org/10.1063/1.5040376 View Table of Contents: http://aip.scitation.org/toc/jcp/149/11 Published by the American Institute of Physics THE JOURNAL OF CHEMICAL PHYSICS 149, 114105 (2018)
Solving the Schrodinger¨ equation of atoms and molecules: Chemical-formula theory, free-complement chemical-formula theory, and intermediate variational theory Hiroshi Nakatsuji,a) Hiroyuki Nakashima, and Yusaku I. Kurokawa Quantum Chemistry Research Institute, Kyoto Technoscience Center 16, 14 Yoshida Kawaramachi, Sakyo-ku, Kyoto 606-8305, Japan (Received 16 May 2018; accepted 28 August 2018; published online 20 September 2018)
Chemistry is governed by the principle of quantum mechanics as expressed by the Schrodinger¨ equation (SE) and Dirac equation (DE). The exact general theory for solving these fundamental equations is therefore a key for formulating accurately predictive theory in chemical science. The free-complement (FC) theory for solving the SE of atoms and molecules proposed by one of the authors is such a general theory. On the other hand, the working theory most widely used in chemistry is the chemical formula that refers to the molecular structural formula and chemical reaction formula, collectively. There, the central concepts are the local atomic concept, transferability, and from-atoms- to-molecule concept. Since the chemical formula is the most successful working theory in chemistry ever existed, we formulate our FC theory to have the structure reflecting the chemical formula. Our basic postulate is that as far as the SE is the principle of chemistry, its solutions for chemistry should have the structure that can be related to the chemical formulas. So, in this paper, we first formulate a theory that designs the wave function to reflect the structure of the chemical formula. We call this theory chemical formula theory (CFT). In the CFT, we place the valence ground and excited states of each atom at each position of the chemical formula of the molecule and let them interact using their free valences to form the ground and excited states of the molecule. The principle there is the variational principle so that the ground and excited states obtained satisfy the orthogonality and Hamiltonian-orthogonality relations. Then, we formulate the exact FC theory starting from the initial functions produced by the CFT. This FC theory is referred to as free-complement chemical- formula theory (FC-CFT), which is expected to describe efficiently the solution of the SE by the above reason. The FC-CFT wave function is modified from that of CFT. Since this modification is done by the exact SE, its analysis may give some insights to chemists that assist their chemistry. Thus, this theory would be not only exact but also conceptually useful. Furthermore, the intermediate theory between CFT and FC-CFT would also be useful. There, we use only integratable functions and apply the variational principle so that we refer to this theory as FC-CFT-variational (FC-CFT- V). It is an advanced theory of CFT. Since the variational method is straightforward and powerful, we can do extensive chemical studies in a reasonable accuracy. After finishing such studies, if we still need an exact level of solutions, we add the remaining functions of the FC-CFT and perform the exact calculations. Furthermore, when we deal with large and even giant molecules, the inter-exchange (iExg) theory for the antisymmetry rule introduced previously leads to a large simplification. There, the inter-exchanges between distant electron pairs fade away so that only Coulombic interactions survive. Further in giant systems, even an electrostatic description becomes possible. Then, the FC-CFT for exactly solving the SE would behave essentially to order N for large and giant molecular systems, though the pre-factor should be very large and must be minimized. Published by AIP Publishing. https://doi.org/10.1063/1.5040376
I. INTRODUCTION (H − E)ψ = 0, (1)
Chemistry is governed by the principles of quantum where identical elementary particles involved must satisfy mechanics,1 the Schrodinger¨ equation (SE),2 and the Dirac the fermion or boson exchange requirements. Electrons must equation (DE)3 expressed commonly by satisfy the antisymmetry relation expressed by Pψ = (−)Pψ, (2) a)Author to whom correspondence should be addressed: h.nakatsuji@ where P is a permutation operator. The molecular Hamiltonian qcri.or.jp of the SE is given by
0021-9606/2018/149(11)/114105/14/$30.00 149, 114105-1 Published by AIP Publishing. 114105-2 Nakatsuji, Nakashima, and Kurokawa J. Chem. Phys. 149, 114105 (2018)
X 1 X 1 X ZA X 1 was introduced.34 Here, g is the scaling operator defined by H = − ∆i − ∆A − + i 2 A 2 i,A riA i
FIG. 1. Example of the chemical formula.
In this paper, we will propose a general electronic struc- atoms in molecules, while for bonds, some ambiguities and ture theory that covers from approximate to exact levels in a freedom are left, though they are also essentially local and unified way. We start from atoms, extend to molecules, and transferable. The basic implication of the chemical formula is want to further expand to large and giant molecules. For this that molecules are formed from sparsely arranged atoms kept purpose, the first step will be to build an electronic structure by bonds and that the electronic structure of each atomic region theory that reflects faithfully the chemical formulas. We named is essentially similar to that of the isolated atom. Hydrocar- this theory chemical formula theory (CFT). Then, we apply bons, for example, consist of carbons and hydrogens connected the FC theory to the wave functions produced by the CFT, by the C−−C and C−−H bonds, but for the C−−C bond, it may which leads to the exact wave function that has the structure of be single, double, triple, or even the median of them, and the the chemical formula. This theory is named free-complement nature of the CH bond is affected by them. chemical-formula theory (FC-CFT). We then investigate the The locality in the chemical formula originates from the possibility of the variational theory that lies in the interme- Coulombic potential included in the Hamiltonian given by diate of the CFT and the FC-CFT. Some applications of the Eq. (3). Near the nucleus, the electrons are captured in the theory presented in this paper will be given in Paper II39 and exponential orbitals around the nuclei, like an electron of the the separate papers. hydrogen atom captured in the orbital ψ = e−r. Therefore, the When systems become larger, some simplifications occur probability of finding an electron decays exponentially as r in their electronic structure. The largest reason lies in the increases. The inter-electron repulsion term, the fourth term fading away of the Fermion antisymmetry rule between dis- in the Hamiltonian, works to shield the effects of other nuclei tant electron pairs. The basis of this phenomenon has been so that the locality as expressed by “atoms in a molecule” is described before by the inter-exchange (iExg) theory.40 For realized. distant electron pairs, the exchange interactions fade away Chemists use chemical formulas in the molecular design and the Coulomb interactions survive, resulting large sim- and chemical synthesis, as if they had “seen” their molecules plifications in their electronic structures. In giant systems, and reactions, and with adding their inspirations and knowl- these Coulombic interactions are further simplified to the edges accumulated through chemical studies, they finally syn- electrostatic interactions even in the exact sense. The CFT thesize successfully complex molecules and materials. Chem- for large and giant systems must describe these changes ical formulas are the central tool for chemists to describe their properly. ideas, i.e., their chemistry. When one refers to the modern text books of organic chemistry41 and chemical biology,42 one will find that almost all basic and central concepts are written using II. CHEMICAL FORMULA chemical formulas. Without chemical formulas, chemistry and Chemistry is the science dealing with atoms, molecules, biochemistry have no way of expressing their sciences. Thus, and molecular systems including the biological ones. There, chemical formulas have long been the best chemical theory the most intuitive working theory is represented by molecular ever existed and used by chemists as their conceptual bases structural formulas and chemical reaction formulas which are of the chemistry. Chemical formulas represent the essence of collectively referred to as chemical formulas. Figure1 shows chemistry. Based on the ultimate importance of chemical for- some molecules and a reaction as examples of chemical formu- mulas in the world of chemistry, this author postulates that as las. The structural information given by the X-ray spectroscopy far as the SE is the principle of chemistry, its solutions for for biological systems is also included in the chemical formula. chemistry should have the structure that can be related to the There, the basic concept is the locality and transferability of chemical formulas. 114105-4 Nakatsuji, Nakashima, and Kurokawa J. Chem. Phys. 149, 114105 (2018)
Since the FC theory produces, as Eq. (9) implies, the exact nanotube. For its σ-framework, it is a strong material, and from wave function that has the structure similar to the initial wave its π-framework, high electric conductivity along the tube is function ψ0, we want to formulate a general electronic structure realized. The description of the σ-framework would be the theory that simulates faithfully the structure of the chemical same as above, but for the π-framework, a delocalized picture formula and the origins thereof from a quantum chemical point may become necessary. Thus, in the CFT, the descriptions of of view. We refer to this theory as chemical formula theory the bonds and the inter-atomic interactions are flexible. For (CFT). large and giant systems, the CFT can utilize the simplicities shown in Sec.VI like the electrostatic model which can be exact, though it is used frequently in molecular dynamics (MD) III. CHEMICAL FORMULA THEORY AS A GENERAL simulations.44,45 Finally, we note that in the FC calculations, ELECTRONIC STRUCTURE THEORY OF ATOMS the local description is computationally more feasible than the AND MOLECULES delocalized one. Since locality and transferability are central to the chem- If we define the exact wave function of a molecule in its ical formula, we use the local atomic concept from the begin- Ith electronic state as ψI , then the ground and excited states of ning of the CFT. Another central implication of the chemical this molecule satisfy the following important relation: formula is that the molecule is formed from atoms whose three- hψI | ψJ i = δIJ , dimensional arrangement is indicated in the formula. However, (10) hψ |H|ψ i = E δ , the chemical formula does not describe the chemical bonds and I J I IJ the interactions involved there in detail. Therefore, they must where the Hamiltonian is given by Eq. (3) and EI is the energy be described by the quantum-chemical formulations of CFT of the Ith electronic state of this molecule. Since we use the and the variational principle. In the bond formation process variational principle in CFT, the relation similar to Eq. (10) from atoms to molecule, not only the ground state but also the holds also for the CFT-level of the results. Since a motivation excited states of atoms are cooperatively transformed to the of CFT is to construct intuitively useful initial functions for the ground and excited states of the molecules. For example, when FC theory, it needs not to be highly accurate. Rather, it must C−−C and C−−H bonds are formed from carbon and hydrogen, be conceptually simple and useful. Accuracy can be improved the ground state and many excited states of the carbon atom later by applying the FC theory. of 2s22p2 and 2s2p3 configurations participate. The participa- A. CFT for atoms tion of the excited state from the 2s2p3 configuration is usually more important than that of the ground state of 2s22p2 config- First, we consider the CFT for atoms. This phrase may uration. Therefore, CFT must describe the chemical changes sound strange because an atom is simply a dot in the chemical from the ground and excited states of the constituent atoms formula. In CFT, however, to this dot, all the electronic states to the ground and excited states of the molecule. This is an of the atom must be associated. When we refer to the Atomic important feature of the CFT. The detailed information of Energy Levels of Moore,43 we find a detailed list of energy the atomic energy levels is given in Moore’s book, Atomic levels for each atom of the periodic table. Behind a “dot” of Energy Levels, and NIST Atomic Spectra Database including the chemical formula, plenty of atomic electronic structures are updated data.43 They give basic information for building the hidden. We want to describe these atomic electronic structures CFT. For molecules, such levels of information are impos- with CFT. Furthermore, for atoms, the explanation of CFT is sible because of the numerous numbers of molecules. For simpler than that for molecules. this reason, when we deal with large molecules, we still start If we define the exact wave function of the electronic state from atoms and synthesize large molecules from individual A I of each atom A by ψI , then they must satisfy the orthogonality atoms. and Hamiltonian-orthogonality relations as given by Eq. (10) A Let us consider some organic molecules. The ordinary with adding the superscript A on ψI , EI , and H, where H is σ-bonds that constitute their molecular framework would be the atomic Hamiltonian of atom A given by described well with the local bonds represented by (ab+ba)α β X 1 X Z X 1 2 A A and (a + b) α β, which are VB and MO type, respectively. For H = − ∆i − + , (11) i 2 i,A r i Equation (10) implies that when direct interactions between number M is here 5. The atomic orbitals, ns and 2p, are given ground and excited states are included, both ground and excited in a minimal Slater form by states are improved efficiently. Suppose that there are M such excited states of the same symmetry within some energy range 1s = exp(−α1sr), A and that we have an approximate cf φ0,m for each state of 2s = (b0 + r) exp(−α2sr), m = 0...M. Then, we make these approximate ground and 2p = x exp(−α r), 2p = y exp(−α r), excited states interact with each other as x 2p y 2p 2pz = z exp(−α2pr), M A,CFT X 2 (17) I A 3s = (b0 + b1r + r ) exp(−α3sr), ψI = cmφ0,m (I = 0, ··· , M) (12) m=0 2 3 4s = (b0 + b1r + b2r + r ) exp(−α4sr), 2 3 4 and obtain, by applying the variational principle, the best 5s = (b0 + b1r + b2r + b3r + r ) exp(−α5sr), possible expression of each state. The resultant states satisfy 2 3 4 5 6s = (b0 + b1r + b2r + b3r + b4r + r ) exp(−α6sr). D A,CFT A,CFT E ψI ψJ = δIJ , (13) The coefficients b ’s and the exponents α’s may be found in D A,CFT A A,CFT E A,CFT i ψ H ψ = E δIJ , I J I the literature or may be optimized by doing, for example, the atomic Hartree-Fock calculations. which is an approximate relation corresponding to the exact Finally, our CFT wave function corresponding to Eq. (12) relation given by Eq. (10). is given by The expression given by Eq. (12) corresponds to the “min- A imal” approximation for each state since only one cf φ was Be,CFT Be Be Be Be 0,m ψ = c0φ + c1φ + c2φ + c3φ chosen for each electronic state m. Then, the first index of m 0,G 0,2p2 0,3s 0,4s Be Be the subscript is 0. Since the minimal approximation is practi- + c4φ0,5s + c5φ0,6s. (18) cally important, we recommend optimizing variationally not only the coefficients cm but also the non-linear parameters By applying the variational principle to this wave function, we A obtain 6 solutions that satisfy Eq. (13). They correspond to the like orbital exponents included in the cf φ0,m. If this mini- mal approximation is insufficient, we may introduce double, approximations to the ground state, 3s to 6s Rydberg states, triple, ... approximations. But, the accuracy can be improved and 2p2 valence excited states in energy order. A merit of CFT later by using the FC theory, which will be more efficient and is that it gives the wave functions not only for the ground A state but also for the excited states included in the energy systematic. The FC theory produces from φ0,m (used as an A range of the calculations. We can calculate therefrom the initial function) the complete cf’s {φi,m} (i = 0, ..., Lm) for each mth electronic state of the atom A, as will be explained valence and Rydberg excitation energies and their wave func- in Sec.IVA. tions. From them, we can calculate various properties of these A states. We next explain how to construct the function φ0,m. Sup- pose we are dealing with the beryllium atom, A = Be. It has We have three notes here. First, in the above cf’s, the inner four electrons, and the ground state is composed of two 1s and core was written in a doubly occupied closed-shell form. Here, two 2s electrons as instead, we can introduce in-out correlation form, replacing 1s1s α β by (1s1s0+1s01s) α β, where 1s and 1s0 have differ- Be φ0,G = A[1s(1)1s(2)α(1)β(2) · 2s(3)2s(4)α(3)β(4)], (14) ent exponents. The in-out form takes the in-out correlation effect into account and so can be advantageous. Second, in where we use G for the ground state. Referring to NIST Atomic the above formulation, the Slater functions in Eq. (18) were Spectra Database, we found that the next higher valence-like minimal. Though the minimal approximation is a rather poor 1S state over the ground state is the 2p2 state lying as high as approximation in the modern quantum chemistry, the expe- at 76 190 cm−1. This state is written similarly as riences from the applications of the FC theory show that the minimal Slater approximation is acceptable at least for φBe A α β · 0,2p2 = 1s(1)1s(2) 2px(3)2px(4) + 2py(3)2py(4) small organic molecules: we could calculate chemical accu- racy results with this approximation. Third, in the above for- + 2pz(3)2pz(4) α β , (15) mulation for atoms, the number of the atomic excited states where the electron assignment on the spin function was abbre- having the same symmetry as the ground state, M, is usually viated. Within this energy range, there are many Rydberg small. A reason of including the 1S(2p2) state lying as high as excited 1S states at 54 677(3s), 65 245(4s), 69 322(5s), 76 190 cm−1 was that it is the lowest valence 1S excited state. 71 321(6s), etc., in cm−1. They are 2s → ns (n = 3, 4, 5, 6, However, for molecules, this situation changes much. ...) states and are written as Furthermore, a note is on the cationic and anionic states of atoms. As will become clear, for constructing the CFT of Be φ0,2s→ns = A[1s(1)1s(2)α β · (2s(3)ns(4) + ns(3)2s(4))α β]. molecules, we need not only the neutral atomic states but also (16) cationic and anionic states. For the cationic states, we have detailed energy levels in Moore’s book,43 but for the anionic We truncate ns to 6s because higher Rydberg states scarcely states, only a few (stable) levels exist for atoms, which are interact with the valence electronic states. Therefore, the available through the Internet. 114105-6 Nakatsuji, Nakashima, and Kurokawa J. Chem. Phys. 149, 114105 (2018) B. CFT for molecules from CFT for atoms Eq. (10). We understand from this equation that the stabiliza- tion energy relative to the atomic energy is distributed among For studying the molecular electronic structure with CFT, the bond energies and that the electronic excitations occur with we need to know all lower electronic states of different sym- the weakening of the bonds involved. These concepts are useful metries for all atoms involved in the molecule under study. In for studying photo-chemical reactions. molecules, the symmetry becomes lower so that the mixing Now let us apply the above procedures to the hydro- of different atomic states is facilitated. Here again, Moore’s carbon molecule C H as an example. First, we put carbon and NIST tables of atomic energy level are very helpful for n m and hydrogen atoms at the positions of the chemical formula. planning the CFT studies. From the transferability, all n carbons are the same and the m Let us now construct electronic structures of molecules hydrogens are the same. Second, we prepare the cf’s of lower with CFT. The procedure may be summarized as follows. electronic states of the constituent atoms and investigate them. 1. Put all constituent atoms at their positions of the chemical The hydrogen atom has a very discrete electronic structure. formula. The ground state is the valence 1s state, and the degenerate 2s A −1 2. Examine the atomic cf’s φ0,m to the energy level of inter- and 2p states are too high (82 259 cm ) and diffuse to form est, EEL, for all constituent atoms and prepare them. The chemical bonds with other atoms. So, we need to consider non-linear parameters in the atomic cf’s are assumed to solely the 1s state for hydrogen atoms. On the other hand, be already optimized. The energy level, EEL, is important the carbon atom has many valence atomic states. They are, since it decides the accuracy and the necessary amounts from lower with the energy in cm−1 in parentheses, the 3P(0), of computations. Moore’s book43 is very useful here as 1D(10193), and 1S(21648) states of the 2s22p2 configuration will be seen below for carbons. and the 5S(33735), 3D(64090), and 3P(75256) states of the 3. This step is the “from atomic states to molecular states” 2s2p3 configuration. There are no other valence states below step, which is central in the CFT. We produce molecu- the H(2s) state at 82 259 cm−1.43 The Rydberg states start from lar cf’s from atomic cf’s by making bonds with nearby 60 333(3P) to 61 982(1P) cm−1, which are 3s states, and the atoms using free electrons of the ground & excited states 3p Rydberg states exist from 68 856 to 73 976 cm−1.43 For the of the constituent atoms. As will be given explicitly in the reactions and properties near the ground state of carbons, the examples below, the two-center σ bond is expressed with 3P, 1D, and 1S states of the 2s22p2 configuration and the lowest the covalent term and the cationic-anionic ionic term. 5S state of the 2s2p3 configuration are important because they The local MO form is also useful near the equilibrium are relatively low (<34 000 cm−1) in energy. Therefore, for distance. When the system includes the π-framework, ordinary chemistry of the ground state alone, the EEL defined like benzene, the valence bond “resonance” form and/or above may be around 34 000 cm−1. The lowest 3P(2s22p2) state the delocalized MO form are useful. The former has a has two free valences and can form two bonds with the neigh- local structure, but the latter does not. The paired elec- boring atoms, while the 5S(2s2p3) state has four free valences trons in the valence orbital of atoms may become a lone and can form four bonds. Therefore, the 5S(2s2p3) state can get pair in a molecule. Different combinations of free valence very large stabilization by forming four bonds with the neigh- electrons give different sets of bonds and lone pairs that boring atoms, the total stabilization energy exceeding its high correspond to different sets of molecular cf’s. We des- energy level (33 735 cm−1) relative to the ground 3P(2s22p2) Mol 5 3 ignate the cf’s of the molecule as φ0,I . The suffix I state. Therefore, this S(2s2p ) state plays very important roles represents a bonding form used in this cf, and the number in carbon chemistry, which are sometimes more important than of them is defined as M, similarly to the atomic case. The the roles of the ground 3P state of the s2p2 configuration. The pre-suffix 0 represents that it is the initial function of the existence of these many interesting electronic states of carbons FC theory. This pre-suffix will be used later in the FC may be the origin of the variety of carbon chemistry and further theory. the origin of life produced mainly by organic molecules and { Mol} 4. By a linear combination of all molecular cf’s φ0,I systems. prepared above, the CFT wave function of the target For the next step, we prepare the cf’s for the 1s state of molecule Mol is described as hydrogen as φH = 1sHα (20) XM 0,1s Mol,CFT K Mol 3 2 2 5 3 ψK = cI φ0,I . (19) and for the P(2s 2p ) and S(2s2p ) states of carbon as I C C 2 · C 2 · C C φ0,3P(s2p2) = A[(1s ) α β (2s ) α β 2px 2py αα] (21) 5. Apply the variational principle to Eq. (19) and obtain and CFT wave functions and energies of all the ground and C C 2 · C C C C excited states of the target molecule. After applying the φ0,5S(sp3) = A[(1s ) α β 2s 2px 2py 2pz αααα], (22) variational principle, the suffix K denotes the ground respectively. Here, we omitted the electron numbers for and excited states of the molecule. They satisfy very brevity. For molecular calculations, the atomic state cf important orthogonality and Hamiltonian-orthogonality of the 5S(2s2p3) state is often rewritten in the hybrid relations like Eq. (10). C C C C C C C C C C C C forms σ1 σ2 2py 2pz , σ1 σ2 σ3 pz , and σ1 σ2 σ3 σ4 , depend- From the above procedure, some important features of ing on the sp, sp2, and sp3 hybrids, respectively. They CFT become evident. In CFT, both ground and excited states are not strictly the atomic states, but useful for molecular adjust themselves through their interactions, as expressed by calculations. 114105-7 Nakatsuji, Nakashima, and Kurokawa J. Chem. Phys. 149, 114105 (2018) Now, in the third stage, we prepare molecular cf’s from role for the higher potential curves of the CH molecule around atomic cf’s. This is the most important “from-atoms-to- this energy level. molecule” step of CFT. In this step, we make bonds using The C2 molecule is an interesting example. It is a closed- free valence orbitals of atomic cf’s. For the H2 molecule, the shell molecule, and its bonding is still a subject of discus- cf of CFT is written as sions.24,46–48 Important contributions to the ground state come C C C C 5 3 from the σ1 σ2 2py 2pz expression of the S(2s2p ) state and 3 2 2 H2 H1 H2 H2 H1 from the P(2s 2p ) state given by Eq. (21). Before giving their φ0,G1 = A[(1s (1)1s (2) + 1s (1)1s (2))α(1)β(2)] expressions, let us define the spin part of the cf’s used in this H1 H2 = A[(1s (1)1s (2))(α(1)β(2) + β(1)α(2))], (23) paper by α βα βα β ··· α βααα ··· α, (27) which is obtained by making a singlet coupling from the atomic cf φH given by Eq. (20). This is the simplest Heitler-London as far as this spin function is possible for the chemistry 0,1s C C C C function for the hydrogen molecule. Another different form of under consideration. From the σ1 σ2 2py 2pz expression of 5 3 bond formation is possible, which is a local MO form given the S(2s2p ) state of the two carbons, C1 and C2, we obtain by quadruple bonds CC C1 2 C2 2 C1 C2 C2 C1 φ0,G = A (1s ) (1s ) (σ1 σ1 + σ1 σ1 ) φH2 = A[(1sH1 + 1sH2 )(1)(1sH1 + 1sH2 )(2)α(1)β(2)]. (24) 0,G2 × C1 C2 C2 C1 · C1 C2 C2 C1 (σ2 σ2 + σ2 σ2 ) (py py + py py ) C1 C2 C2 C1 This MO form has a well-known deficiency that it breaks off × (pz pz + pz pz ) , (28) at the dissociation limit. So, we use this local MO form only where we have omitted the spin function of the form given by when we study the H2 molecule near its equilibrium geom- Eq. (27) and the electron numbers. We assumed the C2 axis etry. There, we often use both cf’s of the covalent VB and to be on x. From the 3P(2s22p2) state of the two carbons, we MO forms as our cf’s of CFT: this corresponds to taking obtain double π bonds two-electron full CI into account. The so-called ionic con- CC C1 2 C2 2 C1 2 C2 2 C1 C2 C2 C1 tribution is included in the cf given by Eq. (24). We usually φ0,G1 = A (1s ) (1s ) (2s ) (2s ) (py py + py py ) use the covalent VB and MO forms simultaneously for the × (pC1 pC2 + pC2 pC1 ) (29) local bond near the equilibrium distance, when we want to z z z z calculate the exact solution of the SE in chemical accuracy. and double σ and π bonds We may consider the ionic terms separately. They are given CC C1 2 C2 2 C1 2 C2 2 by φ0,G2 = A (1s ) (1s ) (2s ) (2s ) C1 C2 C2 C1 C1 C2 C2 C1 × (px px + px px )(py py + py py ) H2 H1 H1 φ = A[1s (1)1s (2)α(1)β(2)], C1 C2 C2 C1 C1 C2 C2 C1 0,I1 (25) + (px px + px px )(pz pz + pz pz ) . (30) φH2 = A[1sH2 (1)1sH2 (2)α(1)β(2)]. 0,I2 Equations (28)–(30) represent the most important cf’s of the ground state of the C2 molecule. The strength of the C−−C For the ground state of the CH molecule, whose symmetry bond of C depends on the extent of the mixing of these two 2 3 2 2 2 is Π, the main cf is prepared from the ground P(2s 2p ) state kinds of cf’s. At infinite separation, the contribution of Eq. (28) of carbon φC given by Eq. (21) and the function φH 0,3P(s2p2) 0,1s becomes zero since the ground state of the carbon atom is the given by Eq. (20) as 3P(2s22p2) state. The contribution of the cf given by Eq. (28) increases as the two carbons come closer. In the above, we CH C 2 C 2 omitted the cf’s representing the contributions of the ionic φ0,G = A (1s ) α β · (2s ) α β terms, but now, it will be easily written down. When one studies · C H H C · C (2px 1s + 1s 2px )α β 2py α (26) only near the equilibrium geometry, the local bond forms are useful for both σ and π bonds. The detailed study will be by forming the CH σ bond with the singlet coupling of the reported in a separate paper. C H 2px and 1s electrons. For the CH bond, the ionic terms The above examples were given only for some of the C 2 H 2 (2px ) α β and (1s ) α β are also considered. Instead, the local valence atomic states. There are many other states particularly C H 2 MO bond form (2px + 1s ) α β is convenient when we study for the states including carbon. The Rydberg atomic states lie only near the equilibrium distance. They give different inde- at rather high energy levels, and their orbital exponents are pendent cf’s. The 5S(2s2p3) state does not contribute to this diffuse in comparison with the valence states. Therefore, their ground state by spin-symmetry, but the 3P(75256) state of the contributions to the lower electronic states are limited. 2s2p3 configuration contributes, but slightly because this state Next is the fourth stage. We describe our CFT wave func- is much higher in energy than the 3P(2s22p2) state. For gen- tion by a linear combination of the molecular cf’s, as given eral hydrocarbons, there are no symmetry restrictions. There- by Eq. (19). Note that the CFT is not a theory for the ground fore, both atomic states, 3P(2s22p2) and 5S(2s2p3), given by state alone, but it describes also the excited states of the target Eqs. (21) and (22), respectively, play important roles, which molecule. Usually, as the energy becomes higher, the accuracy will be discussed in the succeeding papers. The Rydberg decreases because we limit the energy range of the atomic cf’s 3s state of carbon 3P(2s22p3s) at 60 333 cm−1 shows some to be within some range. 114105-8 Nakatsuji, Nakashima, and Kurokawa J. Chem. Phys. 149, 114105 (2018) The final fifth stage is the harvesting stage: the variational Eq. (19) for molecules. We use them as the initial function determination of the ground and excited states. The ground of the FC theory. However, since the FC theory is linear on A state is the most tightly bound state in which the stabilization the cf’s, the operation is done to the elementary cf’s {φ0,m} energy relative to a sum of the atomic separate states is used Mol of Eq. (12) for atoms and {φ0,I } of Eq. (19) for molecules. to strengthen the chemical bonds within the molecule. The In other words, the operation of the FC theory is done to higher excited states consist of the chemical bonds whose sum these elementary cf’s, not to the optimized CFT wave func- of the bond energies is less than that of the ground state: some tion, because their coefficients are reoptimized again by the FC bonds are weaken in some way, which might be interested in theory. the reactions of excited states. The CFT is a general electronic structure theory for atoms A. Atoms and molecules that try to construct the electronic structures of For atoms, the cf’s of the FC theory are produced using the target system as naturally as possible like in the chemical the recursion formula given by Eq. (7) with the scaling func- formulas. Atomic and molecular CFT describes both ground tion given by Eq. (6). More simply, they can be obtained by and excited states lying in the energy range of interest. Molec- multiplying the elements of the g operator to φ . Then, the cf’s ular CFT wave functions are produced by making chemical 0 {φA } of atoms are written as bonds from the ground and excited states of the constituent l,m atoms. For the reacting system, the states of interest along {φA } = φA , r φA , r φA , r r φA , r r φA , lm,m 0,m i 0,m ij 0,m j i 0,m k ij 0,m the reaction coordinate are calculated referring to the chemi- A cal reaction formula. These bond formation steps are probably rklrij φ0,m, ... , (32) the most important steps in CFT, and we use both VB and where i, j, ... denote the electrons and ri is actually riA with MO knowledge. The locality concept of VB is common to A denoting the nucleus. We define Lm to be the number of the CFT and useful for reducing the number of the cf’s. For most A cf’s produced from the initial cf φ0,m and lm to be the index σ-framework, this would be valid. At the same time, if the running from 0 (initial cf) to Lm. With these definitions, the delocalized nature is the essence of the bond, we must use it FC wave function for the Ith state of the atom A is written as since by doing so the convergence to the exact solution of the XM XLm SE will be accelerated. These CFT wave functions are used to ψA,FC-CFT = cI φA (I, m = 0, ... , M). (33) study molecular structural properties and the potential energy I lm,m lm,m m lm surfaces along the reaction. The central concept of CFT is A the locality concept of the electronic structure, its transferabil- The number of the cf’s included in this expression, NCF, ity, and from-atoms-to-molecule concept. The central principle becomes M of CFT is the variational principle so that the resultant wave A X N = Lm. (34) functions satisfy the expressions, as given by Eq. (10). As in CF m=0 the chemical formula, the locality concept for bonds has still Thus, the SE of the atom A is solved for all the electronic states. some ambiguity, which might be considered as an aspect of In comparison with the CFT equation given by Eq. (12), the the diversity which is also an essential nature of chemistry and FC theory produced the cf’s denoted by the suffix l , and with biology. m these functions, each atomic state becomes exact, interacting A motivation of CFT was to construct the wave function with other atomic states. Then, the results satisfy theory that mimics the chemical formula. Then, by applying D A,FC-CFT A,FC-CFTE the FC theory to the initial functions obtained with the CFT, ψ ψ = δIJ , I J (35) we will be able to get the exact solution of the SE easier than D A,FC-CFT A,FC-CFTE A,FC-CFT ψ H ψ = E δIJ , starting from other initial functions. In the following section, I J I we apply the FC theory to the initial functions obtained by which is essentially the same quality equation, as given by the CFT. This level of theory is referred to as FC-CFT (free- Eq. (10). complement chemical-formula theory). The above formulation shows the motivation of the CFT for the use as the initial functions of the FC theory. We com- IV. FREE COMPLEMENT THEORY APPLIED pare Eqs. (12) and (13) for the CFT with Eqs. (33) and (35) TO THE CFT WAVE FUCTION: FC-CFT for the FC-CFT. In both formulations, we use each atomic state as a basis for improving the set of the atomic states For the importance of the chemical formula in chemistry (I,m = 0, ..., M). This originates from our experience of and biochemistry, the CFT would be a useful initial theory for studying the ground and excited states with the SAC-CI solving the SE. For this purpose, we use the CFT as (symmetry-adapted-cluster configuration-interaction) formal- 49,50 ψFC-CFT = FC(ψCFT), (31) ism. Rather than using some arbitrary basis function set, exact 0 it would be more efficient if we use both ground and excited where we use the FC theory as if it is an operator trans- states of the system as their interacting basis. Then, each state forming the CFT function to the exact wave function. In becomes better with the interactions among them and finally this section, we describe in some detail about this operation. becomes exact, as the FC-CFT result. Looking at the chem- One important note is given initially. The CFT wave func- ical formula, we imagined the formulation to be as close as tion is obtained by applying the variational principle to the possible as the nature does and formulated the CFT and FC- cf’s of CFT, as explained with Eq. (12) for atoms and with CFT. The CFT was an initial stage of this formulation and the 114105-9 Nakatsuji, Nakashima, and Kurokawa J. Chem. Phys. 149, 114105 (2018) FC-CFT is final: there are also interesting possibilities in the The principal part of the initial function for the ground state intermediate stage, as will be explained later. may be written as For obtaining the chemical accuracy, it is necessary to Acetylene ˆ C1 2 C2 2 2 2 3 φ0,G = A (1s ) (1s ) (σ(h1C1)) (σ(C1C2)) consider to the second order, but we usually include ri and 3 2 2 2 rij, but neglect the second-order terms of rij when elec- × (π1(C1C2)) (π2(C1C2)) (σ(C2h2)) . (37) trons i and j belong to different shells. The term rklrij is included only when all four electrons are different. However, The first two are inner 1s cores, and the followings are the σ and σ π depending on each atom, we modify this rule based on our π local bonds. Each of the local and bonds in Eq. (37) may experiences. include covalent, ionic, and local MO forms, giving indepen- dent cf’s. Next, we apply the g function given by Eq. (36). For B. Molecules chemical accuracy, we produce the molecular cf’s mostly to order two, though some third-order terms are added when they The locality of the electronic structures of the CFT brings are important. In addition to this standard general rule, we use much simplicity to the calculations of the exact solution of the following rules to reduce the computational labors. Most the SE of molecules. First, we show the simplicity produced of them are the results of the compromise between accuracy to the g-operator of the FC theory. As shown by Eq. (5), and costs. the g-operator was introduced to avoid the collisions between 1. When we study molecules in their equilibrium structure, charged particles in a molecule that causes the divergence dif- the local bonds in the initial functions like the one given ficulty of the integrals in the variational solution of the SE. It by Eq. (37) may be written in the local MO form: the was originally defined by Eq. (6). However, for the locality, the reason is that when we use the sampling method of the g-operator is much simplified. The electron belonging to atom local SE method,51,52 the local MO form is twice faster A, i , has almost no chance to collide with the other nuclei A than the covalent form. Then, each local bond is replaced, B. Likewise, i and i do not have a chance to collide with A B one by one order, with the covalent form and also with each other. Taking these facts into account, the g-operator to the ionic terms, if necessary. be applied to the molecular initial functions of CFT is proposed 2. When the functions r , x , and r are applied to the as iA iA ij X X electrons belonging to the local MO bond, the two-center g = (riAA + xiAA + yiAA + ziAA) + riAjA local MO is changed to the covalent VB form to make it A,iA A,iA>jA match with the local treatment. 3. For the 2s orbital, we add exp(−α r) at the first order A-B,bondX A-B,bondX 2S when we use a nodeless 2s orbital in the initial function. + riAB + riAjB , (36) A,iA,B iA,jB 4. For the second order function riAjA rkAlA , we include only when all four suffixes are different. The function r r where the first two terms are the ordinary terms and the last two iAjA jAkA may also be included, when it is thought not to be terms are sometimes added. Because molecules are asymmet- negligible. ric, we added x , et cetera in addition to the spherical term iAA 5. For the two center terms for the bond between atoms A r . They work to polarize the orbital. The last two terms are iAA and B, like r and r , originating from the last two exceptional since they are two-center r and r terms for iAB iAjB iAB iAjB terms of the g function given by Eq. (36), only the first the A-B bond. They might be unnecessary because they are order term is considered when A is the hydrogen atom. essentially proportional to the AB bond distance and therefore 6. Similarly, only up to the first order is considered for the might cause a redundancy. So, we consider them only when the inter-pair terms: between 1s and 2s pair, between 1s and hydrogen atom is involved in the bond since the bond distance bond pair, etc. is usually small. 7. The diagonal terms are considered up to the third order, However, when we use MO theory, the above simplifica- 2 3 2 3 like ri A, r , r and ri j , r , r , when they are tions are not realized: since all electrons are delocalized over all A iAA iAA A A iAjA iAjA thought to be important. atoms in a molecule, we cannot assign an electron i to an atom A so that the number of the cf’s increases. In Sec. III, we have When we have several initial cf’s, they are grouped in order, considered the cases where the delocalized MO description according to the order of their “parent” initial function. For might be better or easier than the local VB-type description. the first-order cf’s, the g function is applied only to the first However, such cases occur usually for the system which has order. All cf’s generated with these rules are again local a large π-electron framework. Most σ-electron framework is functions. still described by the local two electron bonds. We should The cf’s thus produced by the FC theory for the molecule, choose the description as local as possible when we prepare Mol, may be written in a general form as the initial functions of the FC theory. lI ,I Mol m lI ,I lI ,I lI ,I φ (1, 2, ... , N) = A[{rij ij }· u1 (1)u2 (2) ··· uN (N)] For molecules, the method of constructing the molecular lI ,I { Mol} cf’s φ0,I of the CFT was explained in Sec. III B, and the (38) examples were given for some diatomic molecules. Here we use another example of the acetylene molecule, which is a including the initial functions, where the ith electron function ulI ,I (i) is obtained from the initial functions by multiplying the linear closed shell molecule i functions like riA, xiA, and/or rij. In Eq. (38), the suffix I of lI H − C ≡ C − H. is due to the fact that the number of the cf’s produced from the 114105-10 Nakatsuji, Nakashima, and Kurokawa J. Chem. Phys. 149, 114105 (2018) initial function φ0,I , which is defined as LI , is generally dif- and information at the CFT level. If we need only the ground ferent for each initial function I. Note that when φlI ,I includes state, we may apply the FC theory only to those cf’s that are lI ,I lI ,I important in the ground-state wave function at the CFT level the two-electron terms rij, the functions ui (i) and uj (j) are connected by rij, where usually the electrons i and j belong to to reduce the labor of the calculations. the same atom because only in such a case the electrons i and Like the initial functions, the higher-order cf’s of the FC- lI I CFT also have local and transferable structures. Therefore, j can collide. mij is 0, 1, 2, or 3 for the ij pair of the cf lI I. With these cf’s, the FC wave function for the Kth state of the final FC wave functions as the solutions of the SE have the molecule is written as the local and transferable structures. This is an important fea- ture of the present theory to obtain not only accurate but also XM XLI ψMol,FC-CFT = cK φMol (K, I = 0, ... , M). (39) understandable results. Then, the following quantities K lI ,I lI ,I I lI exact CFT ψFC-CFT − ψ0 , (42) Thus, the SE of the molecule is solved for the ground ∆Γexact = Γexact (1, 2, ··· , N) − Γ (1, 2, ··· , N), (43) and excited states. The number of the cf’s included in this FC-CFT FC-CFT CFT expression, NMol, is given by and cf exact exact − ∆ρFC-CFT = ρFC-CFT(1) ρCFT (1) (44) M X are of special interest for chemists, where Γ and ρ are the NMol = L . (40) cf I N-electron density matrix and electron density, respectively. I=0 These difference quantities are the modifications that the SE In comparison with the CFT equation given by Eq. (19), the has given over the CFT quantities. The FC-CF theory is based FC theory produced the functions denoted by the suffix lI , and on the non-orthogonal set of functions so that the formulations with these additional cf’s, the ground and lower excited states of Lowdin¨ 53 would be useful for analyzing the quantities given become exact, interacting with the other states at the same by Eqs. (43) and (44). Since the SE is the governing principle time. Then, the results satisfy of chemistry and biology, these quantities might give what we D Mol,FC-CFT Mol,FC-CFTE wanted to know. We want to learn from these quantities what ψK ψL = δIJ , D E (41) are necessary to make quantum chemical calculations useful in Mol,FC-CFT Mol,FC-CFT Mol,FC-CFT ψK H ψL = EK δKL, the research of chemistry. The exact concept obtained with the FC-CFT would be useful since it is reliable. In the chemical which is important showing the quality of the ground and formula, the descriptions of bonds and interactions are rather excited states calculated. vague in comparison with the atomic concepts. Therefore, we Very often, the calculations are limited only to the ground want to learn from the results of the present theory about their state of a molecule. In such a case, the calculation can be exact nature and concepts for the system of interest or under simplified. The number of the cf’s given by Eq. (40) can be the course of chemical reaction. reduced. There are two choices. One is to limit the calculations to the ground state from the beginning. Then, we calculate only the ground state, starting from the most probable initial func- V. POSSIBILITY OF THE INTERMEDIATE THEORY: FC-CFT-V tion. Usually, we know from the experience that what atomic state is most important for molecular formation. For example, The cf’s {φI } produced by the FC-CF theory include for carbon, we use the sp3 configuration rather than the s2p2 both integratable and non-integratable functions. We group configuration, though the former is the excited state, when them into those integratable, which is called the i-set, and the symmetry of the molecule permits this choice. Then, we those non-integratable, which is called the n-set. The i-set plus construct the initial wave functions using the idea of the CFT n-set forms the complete set. The n-set cf’s are characterized and apply the FC theory to this initial function. Even in this by the existence of the odd-powers of rij terms that makes them case, we can get the exact wave function and the energy with non-integratable. However, these cf’s characterized by the rij the FC-CFT because the FC theory leads to the exact wave terms may not be so important for chemical researches, though function from any initial function, if it has an overlap with the they work to lower the energy. For example, both MO and VB exact wave function, though we may need a larger number of theories do not include rij terms, but they are very important cf’s than the other cases we explain below. This is a simple for studying chemistry. way of performing the FC-CFT: the locality concept and other This suggests a possibility of the intermediate theory. We implications of the chemical formula are included in the ini- consider only the i-set cf’s at some order of the FC theory and tial wave function. Actually, for oxygen, for example, only the apply the variational principle to them. We refer to this theory s2p4 configuration lies below 73 000 cm−1 so that this choice as FC-CFT variational (FC-CFT-V), where “V” indicates vari- and the other choices may be the same. ational. The accuracy of the FC-CFT-V can be improved by The second choice is, for the carbon case, to choose the raising the order of the FC theory and by expanding the realm atomic states arising from both ground s2p2 and excited sp3 of the integral method. Because of its variational nature, we configurations for making bonds with other atoms. Then, we can study extensively the chemistry of the system of interest. construct all cf’s at the CFT level and apply the FC theory to Furthermore, when we start from the CFT, we generate the cf’s them to proceed the FC-CFT. The diagonalization at the CFT not only for the ground states but also for the excited states. level is optional, if only the FC-CFT is the purpose, though we These ground and excited states satisfy the orthogonality and recommend doing so, because we can get some understanding Hamiltonian orthogonality. This relation is important for the 114105-11 Nakatsuji, Nakashima, and Kurokawa J. Chem. Phys. 149, 114105 (2018) (k) study of the ground and excited states. The quantities defined where EAB is the k-electron exchange operator between A and by Eqs. (42)–(44) are useful to investigate the chemistry under B, K is the maximum number of electrons belonging to either A study. or B, and AA and AB are the antisymmetrizers for the electrons The FC-CFT-V is an advanced theory of the CFT. Since belonging to the atoms A and B, respectively. Note that each the FC theory always improves the theory, FC-CFT-V is electron belongs to the orbital of an exponential form that always more accurate than the CFT itself. When integrations belongs to either A or B. When we apply the one-electron (1) are possible for all cf’s, the FC-CFT-V is just FC-CFT and exchange operator EAB to the molecular wave function ψAB, (1) highly accurate results are obtained. Such cases were already the magnitude of E ψAB becomes exponentially smaller than 54–56 AB reported for small systems like the He atom and H2 the magnitude of ψAB itself since by the exchange, the electron- 57 + molecule. For H2 , highly accurate BO and non-BO results nucleus distances riA and rjB become longer by about the bond 58,59 were reported. In particular, highly accurate analytical distance R. Then, as the distance R increases, the result of the + potential curves of H2 were derived from the non-BO wave exchange becomes smaller exponentially and finally becomes 59 function. negligibly small, when the following Eq. (46) is satisfied. Such Then, we propose the two-step approach. After studying a situation can be measured by using the overlap T ab between the basic chemistry with the FC-CFT-V theory, we study only the squares of orbitals χa and χb to which the electron pair the most important subject by considering the effects of the belongs, n-set cf’s by performing FC-CFT. Then, we can extend the D 2 2E accuracy to the exact level and we can predict and perform Tab = χa χb ≤ λ. (46) chemistry with confidence. For this purpose, we may use the 2 local Schrodinger¨ equation (LSE) method.51,52 It is interesting We use T ab instead of Sab to avoid the case of Sab = 0 by sym- metry. For the pair of valence electrons of carbon atoms, λ is to formulate the LSE method to be useful at this stage of the −6 about 10 and the distance Rλ corresponds to the four bonds of FC theory. (k) hydrocarbons. For the k electron exchange EAB, this decrease (1) is k times faster than that of EAB. In large molecules, there are (n) VI. POSSIBILITIES OF THE CFT AND FC-CFT many atomic pairs for which EABψ is negligibly small: then FOR LARGE AND GIANT MOLECULES all terms of k ≥ n are negligible. Furthermore, the operation counts for A A also reduce from N ! × N ! to N ! + N ! When we deal with large and giant molecules, the local A B A B A B and further to N 3 + N 3 with the use of the Nk theory. Thus, nature of the electronic structure of the CFT and the FC-CFT A B the labor of antisymmetrizations can be reduced much with a leads to much simplifications. This arises from the nature of the combinatorial use of the iExg and Nk theories. antisymmetry requirement for the electron pairs in molecules. Now let us consider what the iExg theory implies when The inter-exchange (iExg) theory40 tells us that as the distance we apply the FC theory to large molecules and even to giant of the electron pair increases, the antisymmetry rule between molecular systems. them fades away. As the system becomes larger, the number of the distant electron pairs in the molecule increases rapidly so that much simplifications result since antisymmetrizations B. Simplification in large molecules are usually very time-consuming processes. Here, we discuss As molecules become larger, the number of electrons this important effect in large and giant molecules. increases, but at the same time, the number of distant atomic pairs in the molecule also increases. With the use of the expres- sion of the cf given by Eq. (38), the matrix elements between A. iExg theory for antisymmetrization different cf’s are written as Electrons are fermions, and therefore the electronic wave N functions must satisfy the antisymmetry requirement, as X D 1 X Z E hφ | H| φ i = u I (1) − ∆ − A u J (1) D(I | J ) expressed by Eq. (2). We have proposed an efficient anti- I J i 2 1 r j i j i,j=1 A A1 symmetry theory, called Nk theory,60 which is useful for N correlated theory including rij explicitly. This algorithm is X D I I 1 J J E + ui (1)uj (2) uk (1)ul (2) determinant-based, and its order is roughly N3-4. Therefore, for " r i With this equation, let us consider the behaviors of the distance Rλ related to the parameter λ given by Eq. (46). To integrals when the orbitals involved are so distant that the eliminate the overlapping effect between A and D, the AD exchange of the electrons between these orbitals is negligi- distance necessary for Eq. (46) is roughly 2Rλ. For a carbon ble as the iExg theory implies. For the one-electron part, when pair, it is about 8 bond separation, ∼24 a.u. (12.7 Å). I J the orbitals ui and uj are far apart, the integral is negligibly small since each orbital is local and centered on a nucleus C. Further simplification in giant molecules and decays exponentially from the center of the orbital, while When we move to still larger systems, like giant molecular the kinetic and nuclear attraction operators have little effect systems, further simplification occurs. There, the distance of on these behaviors of the integrals. For the two-electron part, the AD pair can be much larger than the above case. When the when all four orbitals, uI , uI , uJ , and uJ , are far apart to i j k l AD distance is so large that the widths of the orbitals a and each other, the integral is negligible. When three of them are d are negligible, then the width of the orbital d looks like a far apart, again the integral is negligible. When two of them, point when it is seen from the orbital a. In such a situation, the (i, k) and (j, l), in parentheses are close, but (i, l) and (j, k) electron density distribution ρA looks like an electronic charge are far distant, the first term remains but the second term is A I J q on atom A, where negligible. This case occurs when ui and uk belong to one atom and uI and uJ belong to other different atom and the two j l qA = ρA(r )dr (50) atoms are far apart in the molecule. Similarly, when one is far 1A 1A apart though three are close, again both integrals are negli- gible. Both integrals survive only when all orbitals are close and Eq. (48) is transformed to to each other. Similar behaviors are also seen for the overlap δAδD matrix elements involved in the co-factor matrices D(I J ) and h ψ|H|ψi = , (51) i j A,D R D(I I |J J ). The above observations clearly imply that much AD i j k l simplification occurs when the molecule is very large: we can where δA is the gross charge of atom A, δA = ZA − qA. expect that roughly order-N behavior of the computational Such simple electrostatic forms have been introduced in labor would result. many molecular dynamics (MD) simulation programs,44,45 We apply the above argument to the interaction of the though their gross charges are empirical and/or approximate. two atoms A and D that lie far apart from each other in a They were also used in the quantum mechanical models of large molecule. We assume that both A and D have only one giant molecular systems like the elongation method for poly- electron, and the distance between A and D is so far apart mers,61 the divide and conquer method,62,63 and the fragment that the electron exchange between A and D does not occur in molecular orbital method.64 It is remarkable that this expres- chemical accuracy. Then, the direct interaction term between sion is valid even in the exact theory for solving the SE, which A and D is written as is the present FC theory, when it is combined with the iExg ZAZD ZD theory. Thus, the exact FC theory transforms its theoretical h ψ|H|ψi = − ρA(r )dr structure automatically to a lower-order N type theory for large A,D R r 1A 1A AD 1AD molecules and even to the theory of giant molecular systems A Z D as the system size increases further. This transformation of − ρ (r1D )dr1D r1DA the theoretical structure of the FC theory is ideal as a chem- 1 ical theory. Intuitively, this is a natural transformation of the + ρA(r ) ρD(r )dr dr , (48) 1A 2D 1A 2D chemical theory that reflects the chemical formulas central to r1A2D chemistry. The basis is the locality and the transferability of where r represents the coordinate of an electron belong- 1A chemistry combined with the iExg theory. ing to atom A and ρA(r ) is the electron density distribution 1A We note that the quantum-mechanical theories for giant associated with atom A, molecular systems, like the elongation method, divide-and- conquer method, and fragment MO method, cited above61–64 ρA(r ) = ψ(r r ··· r ) ψ(r r ··· r ) dr ··· dr . 1A 1A 2B NG 1A 2B NG 2B NG are not the theories one obtains when the MO theory is (49) applied to giant molecular systems, but these theories are the original products of the authors invented based on the MO The first term of Eq. (48) represents the nuclear repulsion, theory. Furthermore, the correlation theories over Hartree- the second and third terms represent the attractions between Fock are difficult to apply to giant molecular systems because the electron density of one atom with the nucleus of another integral transformations necessary for the MO-based corre- atom, and the last term represents the repulsion between the lation theories are too time-consuming for giant molecular electron density of the atoms A and D. Since the electron systems. exchange between A and D is negligible, only the classical The CFT for large and giant molecular systems must Coulomb interactions remain, and they sum up to the above describe the above simplifications well. Though this is the simple formula. subject of our separate paper, the FC-CFT based on such con- We note that r1A in Eq. (48) denotes the coordinate of the cepts must be designed to work to order N as the system size electron that belongs to the atom A, but due to the overlap increases to giant systems. Fine coding will be necessary to and antisymmetry principle, the electron delocalizes out of fully utilize the simplifications in large and giant molecular the atom A, but the delocalization limit is within the limiting systems. 114105-13 Nakatsuji, Nakashima, and Kurokawa J. Chem. Phys. 149, 114105 (2018) VII. CONCLUDING REMARKS theory and is an advanced variational theory of the CFT. For the efficiency of the variational method, we can study chem- We aimed in this paper to combine two major principles of istry extensively. Afterwards, if it is necessary to go to the exact chemical science, Schrodinger¨ equation and chemical formu- SE level, we add the non-integratable cf’s, perform FC-CFT las, to produce useful predictive theory in chemistry. Chemical for both ground and excited states, if necessary, and obtain the formulas are used daily by chemists and biochemists as work- exact-level of the results. The chemical formula concepts are ing concepts and the sources of their ideas. Therefore, if the valid in all levels of the theory. wave function of the exact theory can have the structure of The examinations of the locality-based concepts for the the chemical formula, it must be very useful for chemists. The possibility toward very large and even giant molecular systems FC theory has a merit that it produces the exact wave function gave some light for the future. Because of the locality-based that has the structure assumed to the initial function used at structure of the present theory, the inter-exchange (iExg) the- the beginning. Therefore, in this paper, we first proposed the ory of antisymmetrization introduced previously could pro- CFT that gives the wave function whose structure reflects that duce large simplifications in the treatment of large systems. of the chemical formula. Then, starting from CFT, we can pro- There, the exchanges between distant electron pairs fade away duce, by FC-CFT, the exact wave function that has the structure and Coulombic interactions become dominant. For still larger similar to the chemical formula by using the FC theory. The dif- giant systems, even the electrostatic descriptions become pos- ference of the results of the CFT and the FC-CFT is due to the sible. Then, we have to develop the CFT for such systems con- exact SE itself. So, by analyzing the difference, the chemists sidering the simplifications investigated in this paper. The code may get deep understanding that assists their researches. Thus, of the FC-CFT for exactly solving the SE must be designed the CFT is designed from the beginning to be used by the FC to work to essentially order N for large and giant molecular theory as its initial functions so that it affects the descriptions systems. of the exact wave function. In applications of the present theory, the automated intel- In CFT, our basic idea is that the molecular electronic ligence (AI) technology would be useful to simplify the inputs, states are produced from atomic electronic structures. The to connect different interfaces of the FC theory, and to clar- detailed knowledge of the atomic electronic structures is sum- ify the implications of the calculated results, given in chemical marized in Moore’s book of Atomic Energy Levels,43 which formula language, obtained by the various levels of the present provides us useful basis. The basic ideas on the natures of the theories. Furthermore, the feedbacks from the experimental- interactions between atoms are included in chemical formulas. ists about the merits and the faults of our theory will be useful It is not decisive, but suggestive and must be described well by for further development of our methodology. Such collabora- the methods of quantum chemistry. We use all real electronic tions may be a key for the future progress of the theory and states, ground and excited states, and ionic and anion states algorithms. of atoms as if they are the basis functions for constructing the molecular ground and excited states. Atoms react with the ACKNOWLEDGMENTS nearby ones using free valences with forming bonds for stabi- lization. The governing principle is the variational principle. This paper is a tribute to the late Professor R. G. Parr We use both VB and MO concepts, but the basic and dom- who has encouraged much one of the authors from the inant local structures in chemical formulas are close to the beginning of the studies. The authors thank Research Center VB concept. The delocalized π-framework may be described for Computational Science, Institute for Molecular Science, well with the MO concept. The resultant electronic states pro- Okazaki, for their encouragement to our research project from duced by CFT satisfy the orthogonality and the Hamiltonian computational sides. We also thank Mr. Nobuo Kawakami orthogonality between the different states. for his continuous support to the activities of QCRI. This This result of CFT is elevated to the exact level by the work was supported by the JSPS KAKENHI, Grant Nos. FC theory, which is referred to as FC-CFT. When we study 16H02257, 17H06233, 16H00943, 17H04867, 17KT0103, both ground and excited states of the molecule with CFT, all and 16K17864. of them are made exact by the FC theory. The CFT is designed 1 to be used by the FC theory as its initial function so that this P. A. M. Dirac, Proc. R. Soc. London, Ser. A 123, 714 (1929). 2E. Schrodinger,¨ Ann. Phys. 384, 361 (1926); 384, 489 (1926); 384, 734 connection is smooth and natural: the important relation, the (1926); 385, 437 (1926); 386, 109 (1926); Phys. 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