Solving the Schrödinger equation of atoms and molecules: Chemical-formula theory,free-complement chemical-formula theory, and intermediate variational theoryHiroshi Nakatsuji, Hiroyuki Nakashima, and Yusaku I. Kurokawa

Citation: The Journal of Chemical Physics 149, 114105 (2018); doi: 10.1063/1.5040376View online: https://doi.org/10.1063/1.5040376View Table of Contents: http://aip.scitation.org/toc/jcp/149/11Published 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-formulatheory, and intermediate variational theory

Hiroshi Nakatsuji,a) Hiroyuki Nakashima, and Yusaku I. KurokawaQuantum 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 Schrodingerequation (SE) and Dirac equation (DE). The exact general theory for solving these fundamentalequations is therefore a key for formulating accurately predictive theory in chemical science. Thefree-complement (FC) theory for solving the SE of atoms and molecules proposed by one of theauthors is such a general theory. On the other hand, the working theory most widely used in chemistryis 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 chemistryever existed, we formulate our FC theory to have the structure reflecting the chemical formula. Ourbasic postulate is that as far as the SE is the principle of chemistry, its solutions for chemistry shouldhave the structure that can be related to the chemical formulas. So, in this paper, we first formulatea theory that designs the wave function to reflect the structure of the chemical formula. We callthis theory chemical formula theory (CFT). In the CFT, we place the valence ground and excitedstates of each atom at each position of the chemical formula of the molecule and let them interactusing their free valences to form the ground and excited states of the molecule. The principle thereis the variational principle so that the ground and excited states obtained satisfy the orthogonalityand Hamiltonian-orthogonality relations. Then, we formulate the exact FC theory starting from theinitial 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 theabove reason. The FC-CFT wave function is modified from that of CFT. Since this modification isdone 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 intermediatetheory between CFT and FC-CFT would also be useful. There, we use only integratable functionsand 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 westill need an exact level of solutions, we add the remaining functions of the FC-CFT and perform theexact 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, theinter-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 forexactly 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

Chemistry is governed by the principles of quantummechanics,1 the Schrodinger equation (SE),2 and the Diracequation (DE)3 expressed commonly by

a)Author to whom correspondence should be addressed: h.nakatsuji@qcri.or.jp

(H − E)ψ = 0, (1)

where identical elementary particles involved must satisfythe fermion or boson exchange requirements. Electrons mustsatisfy the antisymmetry relation expressed by

Pψ = (−)Pψ, (2)

where P is a permutation operator. The molecular Hamiltonianof 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)

H = −∑

i

12∆i −

∑A

12∆A −

∑i,A

ZA

riA+

∑i<j

1rij

+∑

A<B

ZAZB

RAB. (3)

The theory presented here is applicable to both non-Born-Oppenheimer (BO) and BO cases,4 and in the BO case, thelast term is dealt with as a constant. In this paper, we assumethe BO case.

If we can solve the SE exactly, elementary chemical phe-nomena should be predictable in high accuracy. This wasa dream of many theoretical chemists. Heitler and London5

applied quantum theory to the hydrogen molecule, and theirtheory became an origin of the valence bond (VB) theory.6

Hylleraas tried to solve the SE of the helium atom consider-ing the r12 coordinate explicitly as early as 1928, and highlyaccurate results were reported.7 His studies became the ori-gin of the so-called Hylleraas-CI.7–10 Calais and Lowdin11

reported a general method of calculating integrals includ-ing r12. One of the authors considered a method of solvingthe Schrodinger equation projected on the density matrixspace12 with some interesting applications.13,14 Explicitlycorrelated R12 and F12 methods have been actively devel-oped and applied to many systems with high accuracy.15–19

The explicitly correlated Gaussian method20,21 utilized themathematical merits of the special functions including theinter-electron coordinates rij explicitly and was used forthe systems of basic importance. The full-CI extrapolationmethod gave highly accurate energies and potential curves ofdiatomic molecules without using rij.22–24 Quantum MonteCarlo methods have a long history of calculating highly accu-rate wave functions including rij without using their difficultintegrals and have been applied to many different subjects ofchemistry.25–27

We studied the structure of the exact wave function ina series of papers28–37 and developed the wave function the-ories that have the exact structure, namely, the theories thatinclude the exact wave function within their variational space.Then, by applying the variational principle or the equivalent,we can get the exact wave function of the SE automatically.When the theory is given in an expanded form, we have toterminate at some order, which determines the accuracy ofthe result. When we solve the SE, we aim to obtain chemicalaccuracy, kcal/mol accuracy in the absolute value of the totalenergy.

It is easy to prove that by performing the recursiveformula

ψn+1 = [1 + Cn(H − En)]ψn (4)

we can reach the exact solution of the SE, if the HamiltonianH does not include diverging potentials. For harmonic oscil-lators, for instance, we can obtain their exact solutions in highaccuracy. However, if the Hamiltonian includes the Coulom-bic potential like in Eq. (3), we have a divergence difficultyin solving Eq. (4) with the variational principle. To overcomethis difficulty, the inverse SE33 was first introduced and thenthe scaled SE defined by

g(H − E)ψ = 0 (5)

was introduced.34 Here, g is the scaling operator defined by

g =∑i,A

riA +∑i>j

rij, (6)

which is always positive and so the scaled SE is equivalent tothe original SE. The g operator was introduced to avoid thedivergence due to the collisions between i and A and betweeni and j. When the electron i belongs to a special atom A, asin the local VB form, it can collide only with A and with jbelonging to the same atom A so that the number of the termsin Eq. (6) reduces. This is a merit when a local descriptionis possible. With the MO formalism, an electron delocalizesall over the molecule so that this merit does not arise. Then,it is easy to prove that by performing the modified recursiveformula

ψn+1 = [1 + Cng(H − En)]ψn,

ψexact =∏

n

[1 + Cng(H − En)] · ψ0,(7)

starting from some initial function ψ0, one gets the exact solu-tion of the SE without meeting the divergence difficulty. Whenone performs several times the recursion formula (7) startingfrom an initial function ψ0, the right-hand side becomes a sumof the analytical functions including Cn. If we collect all inde-pendent and non-diverging analytical functions from them as{φI }, then the exact wave function is written as

ψexact =∑

I

cIφI . (8)

In this equation, all independent analytical functions {φI } aregiven the independent coefficients {cI }. So, Eq. (8) is moreefficient than Eq. (7) for calculating the solution of the SE.This theory was initially referred to as free ICI [iterative com-plement (configuration) interaction] theory. But later, since itis not an iterative theory, we renamed it to the free complement(FC) theory. The functions {φI } are referred to as complementfunctions (cf’s) since they are the elements of the completesolution of the SE. Thus, with the FC theory, the exact wavefunction is obtained by applying the FC theory to some initialfunction ψ0, which is written as

ψexact = FC(ψ0). (9)

An important feature of the FC theory is that the exact wavefunction produced by the FC theory has the mathematicalstructure of the initial function ψ0. Therefore, it is importantto start from the initial function that is useful from a chemicalpoint of view. We note here that an account of the FC theoryand an earlier review were published37 and that a similar for-mulation to solving the SE was also valid for solving the Diracequation.38

In chemistry, the most important working theory lies inmolecular structural formulas and chemical reaction formu-las, which are collectively referred to in this paper as thechemical formula. Figure 1 illustrates some examples. Theessence is the locality and transferability concept of atoms inmolecules that has a long history from Dalton. Because of itsbasic importance in chemistry and biochemistry, we must learnsome implications from the chemical formula.

114105-3 Nakatsuji, Nakashima, and Kurokawa J. Chem. Phys. 149, 114105 (2018)

FIG. 1. Example of the chemical formula.

In this paper, we will propose a general electronic struc-ture theory that covers from approximate to exact levels in aunified way. We start from atoms, extend to molecules, andwant to further expand to large and giant molecules. For thispurpose, the first step will be to build an electronic structuretheory that reflects faithfully the chemical formulas. We namedthis theory chemical formula theory (CFT). Then, we applythe FC theory to the wave functions produced by the CFT,which leads to the exact wave function that has the structure ofthe chemical formula. This theory is named free-complementchemical-formula theory (FC-CFT). We then investigate thepossibility of the variational theory that lies in the interme-diate of the CFT and the FC-CFT. Some applications of thetheory presented in this paper will be given in Paper II39 andthe separate papers.

When systems become larger, some simplifications occurin their electronic structure. The largest reason lies in thefading away of the Fermion antisymmetry rule between dis-tant electron pairs. The basis of this phenomenon has beendescribed before by the inter-exchange (iExg) theory.40 Fordistant electron pairs, the exchange interactions fade awayand the Coulomb interactions survive, resulting large sim-plifications in their electronic structures. In giant systems,these Coulombic interactions are further simplified to theelectrostatic interactions even in the exact sense. The CFTfor large and giant systems must describe these changesproperly.

II. CHEMICAL FORMULA

Chemistry is the science dealing with atoms, molecules,and molecular systems including the biological ones. There,the most intuitive working theory is represented by molecularstructural formulas and chemical reaction formulas which arecollectively referred to as chemical formulas. Figure 1 showssome molecules and a reaction as examples of chemical formu-las. The structural information given by the X-ray spectroscopyfor biological systems is also included in the chemical formula.There, the basic concept is the locality and transferability of

atoms in molecules, while for bonds, some ambiguities andfreedom are left, though they are also essentially local andtransferable. The basic implication of the chemical formula isthat molecules are formed from sparsely arranged atoms keptby bonds and that the electronic structure of each atomic regionis essentially similar to that of the isolated atom. Hydrocar-bons, for example, consist of carbons and hydrogens connectedby the C−−C and C−−H bonds, but for the C−−C bond, it maybe single, double, triple, or even the median of them, and thenature of the CH bond is affected by them.

The locality in the chemical formula originates from theCoulombic potential included in the Hamiltonian given byEq. (3). Near the nucleus, the electrons are captured in theexponential orbitals around the nuclei, like an electron of thehydrogen atom captured in the orbital ψ = e−r . Therefore, theprobability of finding an electron decays exponentially as rincreases. The inter-electron repulsion term, the fourth termin the Hamiltonian, works to shield the effects of other nucleiso that the locality as expressed by “atoms in a molecule” isrealized.

Chemists use chemical formulas in the molecular designand chemical synthesis, as if they had “seen” their moleculesand reactions, and with adding their inspirations and knowl-edges accumulated through chemical studies, they finally syn-thesize successfully complex molecules and materials. Chem-ical formulas are the central tool for chemists to describe theirideas, i.e., their chemistry. When one refers to the modern textbooks of organic chemistry41 and chemical biology,42 one willfind that almost all basic and central concepts are written usingchemical formulas. Without chemical formulas, chemistry andbiochemistry have no way of expressing their sciences. Thus,chemical formulas have long been the best chemical theoryever existed and used by chemists as their conceptual basesof the chemistry. Chemical formulas represent the essence ofchemistry. Based on the ultimate importance of chemical for-mulas in the world of chemistry, this author postulates that asfar as the SE is the principle of chemistry, its solutions forchemistry should have the structure that can be related to thechemical formulas.

114105-4 Nakatsuji, Nakashima, and Kurokawa J. Chem. Phys. 149, 114105 (2018)

Since the FC theory produces, as Eq. (9) implies, the exactwave function that has the structure similar to the initial wavefunctionψ0, we want to formulate a general electronic structuretheory that simulates faithfully the structure of the chemicalformula and the origins thereof from a quantum chemical pointof view. We refer to this theory as chemical formula theory(CFT).

III. CHEMICAL FORMULA THEORY AS A GENERALELECTRONIC STRUCTURE THEORY OF ATOMSAND MOLECULES

Since locality and transferability are central to the chem-ical formula, we use the local atomic concept from the begin-ning of the CFT. Another central implication of the chemicalformula is that the molecule is formed from atoms whose three-dimensional arrangement is indicated in the formula. However,the chemical formula does not describe the chemical bonds andthe interactions involved there in detail. Therefore, they mustbe described by the quantum-chemical formulations of CFTand the variational principle. In the bond formation processfrom atoms to molecule, not only the ground state but also theexcited states of atoms are cooperatively transformed to theground and excited states of the molecules. For example, whenC−−C and C−−H bonds are formed from carbon and hydrogen,the ground state and many excited states of the carbon atomof 2s22p2 and 2s2p3 configurations participate. The participa-tion of the excited state from the 2s2p3 configuration is usuallymore important than that of the ground state of 2s22p2 config-uration. Therefore, CFT must describe the chemical changesfrom the ground and excited states of the constituent atomsto the ground and excited states of the molecule. This is animportant feature of the CFT. The detailed information ofthe atomic energy levels is given in Moore’s book, AtomicEnergy Levels, and NIST Atomic Spectra Database includingupdated data.43 They give basic information for building theCFT. For molecules, such levels of information are impos-sible because of the numerous numbers of molecules. Forthis reason, when we deal with large molecules, we still startfrom atoms and synthesize large molecules from individualatoms.

Let us consider some organic molecules. The ordinaryσ-bonds that constitute their molecular framework would bedescribed well with the local bonds represented by (ab+ba)αβand (a + b)2αβ, which are VB and MO type, respectively. Forthe π-bonds, the VB and MO descriptions are different. TheVB description is essentially local, like Kekule and Dewarstructures of the benzene molecule, but the MO descriptionis delocalized over 6pπ orbitals. In the present theory, wechoose the physically and chemically best description, or wecan choose even both, if it is effective, since our theory isbased on the non-orthogonal cf’s. We further want to extendour theory to large and giant molecular systems. An exam-ple of a large system is the fullerene shown in Fig. 1. For theσ-framework, the local bond description as given above wouldbe appropriate. For the π-framework, again the MO and VBdescriptions exist, but the MO description may be easier touse. This 60 π description is still local, like 6 π of ben-zene. An interesting example of the giant system is a carbon

nanotube. For itsσ-framework, it is a strong material, and fromits π-framework, high electric conductivity along the tube isrealized. The description of the σ-framework would be thesame as above, but for the π-framework, a delocalized picturemay become necessary. Thus, in the CFT, the descriptions ofthe bonds and the inter-atomic interactions are flexible. Forlarge and giant systems, the CFT can utilize the simplicitiesshown in Sec. VI like the electrostatic model which can beexact, though it is used frequently in molecular dynamics (MD)simulations.44,45 Finally, we note that in the FC calculations,the local description is computationally more feasible than thedelocalized one.

If we define the exact wave function of a molecule in itsIth electronic state as ψI , then the ground and excited states ofthis molecule satisfy the following important relation:

〈ψI |ψJ〉 = δIJ ,

〈ψI |H |ψJ〉 = EIδIJ ,(10)

where the Hamiltonian is given by Eq. (3) and EI is the energyof the Ith electronic state of this molecule. Since we use thevariational principle in CFT, the relation similar to Eq. (10)holds also for the CFT-level of the results. Since a motivationof CFT is to construct intuitively useful initial functions for theFC theory, it needs not to be highly accurate. Rather, it mustbe conceptually simple and useful. Accuracy can be improvedlater by applying the FC theory.

A. CFT for atoms

First, we consider the CFT for atoms. This phrase maysound strange because an atom is simply a dot in the chemicalformula. In CFT, however, to this dot, all the electronic statesof the atom must be associated. When we refer to the AtomicEnergy Levels of Moore,43 we find a detailed list of energylevels for each atom of the periodic table. Behind a “dot” ofthe chemical formula, plenty of atomic electronic structures arehidden. We want to describe these atomic electronic structureswith CFT. Furthermore, for atoms, the explanation of CFT issimpler than that for molecules.

If we define the exact wave function of the electronic stateI of each atom A byψA

I , then they must satisfy the orthogonalityand Hamiltonian-orthogonality relations as given by Eq. (10)with adding the superscript A on ψI , EI , and H, where HA isthe atomic Hamiltonian of atom A given by

HA = −∑

i

12∆i −

∑i,A

ZA

riA+

∑i<j

1rij

, (11)

where the nucleus A is fixed at origin, which is a dot A of thechemical formula.

When only the ground state is of interest, one may limitthe calculations to the ground state alone. Then, the ground-state wave function ψA

G′

may be expanded as ψAG′=

∑i ciϕ

Ai,G

(here, the prime indicates that it is tentative). The first com-plement function (cf) ϕA

0,G is the best possible approximation

to ψAG, and other terms ϕA

i,G (i , 0) are added to improve it.(We use the term “complement function,” which was usedin the FC theory, again here since the role of this functionis essentially the same as before. It is abbreviated as cf, butnote that it is different from CF.) In CFT, however, we wantto describe both ground and excited states at the same time.

114105-5 Nakatsuji, Nakashima, and Kurokawa J. Chem. Phys. 149, 114105 (2018)

Equation (10) implies that when direct interactions betweenground and excited states are included, both ground and excitedstates are improved efficiently. Suppose that there are M suchexcited states of the same symmetry within some energy rangeand that we have an approximate cf φA

0,m for each state ofm = 0. . .M. Then, we make these approximate ground andexcited states interact with each other as

ψA,CFTI =

M∑m=0

cImφ

A0,m (I = 0, · · · , M) (12)

and obtain, by applying the variational principle, the bestpossible expression of each state. The resultant states satisfy⟨

ψA,CFTI

���ψA,CFTJ

⟩= δIJ ,⟨

ψA,CFTI

���HA���ψ

A,CFTJ

⟩= EA,CFT

I δIJ ,(13)

which is an approximate relation corresponding to the exactrelation given by Eq. (10).

The expression given by Eq. (12) corresponds to the “min-imal” approximation for each state since only one cf φA

0,m waschosen for each electronic state m. Then, the first index ofthe subscript is 0. Since the minimal approximation is practi-cally important, we recommend optimizing variationally notonly the coefficients cm but also the non-linear parameterslike orbital exponents included in the cf φA

0,m. If this mini-mal approximation is insufficient, we may introduce double,triple, . . . approximations. But, the accuracy can be improvedlater by using the FC theory, which will be more efficient andsystematic. The FC theory produces from φA

0,m (used as an

initial function) the complete cf’s {φAi,m} (i = 0, . . ., Lm) for

each mth electronic state of the atom A, as will be explainedin Sec. IV A.

We next explain how to construct the function φA0,m. Sup-

pose we are dealing with the beryllium atom, A = Be. It hasfour electrons, and the ground state is composed of two 1s andtwo 2s electrons as

φBe0,G = A[1s(1)1s(2)α(1)β(2) · 2s(3)2s(4)α(3)β(4)], (14)

where we use G for the ground state. Referring to NIST AtomicSpectra Database, we found that the next higher valence-like1S state over the ground state is the 2p2 state lying as high asat 76 190 cm−1. This state is written similarly as

φBe0,2p2 = A

[1s(1)1s(2)αβ ·

(2px(3)2px(4) + 2py(3)2py(4)

+ 2pz(3)2pz(4))αβ

], (15)

where the electron assignment on the spin function was abbre-viated. Within this energy range, there are many Rydbergexcited 1S states at 54 677(3s), 65 245(4s), 69 322(5s),71 321(6s), etc., in cm−1. They are 2s → ns (n = 3, 4, 5, 6,. . .) states and are written as

φBe0,2s→ns = A[1s(1)1s(2)αβ · (2s(3)ns(4) + ns(3)2s(4))αβ].

(16)

We truncate ns to 6s because higher Rydberg states scarcelyinteract with the valence electronic states. Therefore, the

number M is here 5. The atomic orbitals, ns and 2p, are givenin a minimal Slater form by

1s = exp(−α1sr),

2s = (b0 + r) exp(−α2sr),

2px = x exp(−α2pr), 2py = y exp(−α2pr),

2pz = z exp(−α2pr),

3s = (b0 + b1r + r2) exp(−α3sr),

4s = (b0 + b1r + b2r2 + r3) exp(−α4sr),

5s = (b0 + b1r + b2r2 + b3r3 + r4) exp(−α5sr),

6s = (b0 + b1r + b2r2 + b3r3 + b4r4 + r5) exp(−α6sr).

(17)

The coefficients bi’s and the exponents α’s may be found inthe literature or may be optimized by doing, for example, theatomic Hartree-Fock calculations.

Finally, our CFT wave function corresponding to Eq. (12)is given by

ψBe,CFTm = c0φ

Be0,G + c1φ

Be0,2p2 + c2φ

Be0,3s + c3φ

Be0,4s

+ c4φBe0,5s + c5φ

Be0,6s. (18)

By applying the variational principle to this wave function, weobtain 6 solutions that satisfy Eq. (13). They correspond to theapproximations to the ground state, 3s to 6s Rydberg states,and 2p2 valence excited states in energy order. A merit of CFTis that it gives the wave functions not only for the groundstate but also for the excited states included in the energyrange of the calculations. We can calculate therefrom thevalence and Rydberg excitation energies and their wave func-tions. From them, we can calculate various properties of thesestates.

We have three notes here. First, in the above cf’s, the innercore was written in a doubly occupied closed-shell form. Here,instead, we can introduce in-out correlation form, replacing1s1s αβ by (1s1s′+1s′1s) αβ, where 1s and 1s′ have differ-ent exponents. The in-out form takes the in-out correlationeffect into account and so can be advantageous. Second, inthe above formulation, the Slater functions in Eq. (18) wereminimal. Though the minimal approximation is a rather poorapproximation in the modern quantum chemistry, the expe-riences from the applications of the FC theory show thatthe minimal Slater approximation is acceptable at least forsmall organic molecules: we could calculate chemical accu-racy results with this approximation. Third, in the above for-mulation for atoms, the number of the atomic excited stateshaving the same symmetry as the ground state, M, is usuallysmall. A reason of including the 1S(2p2) state lying as high as76 190 cm−1 was that it is the lowest valence 1S excited state.However, for molecules, this situation changes much.

Furthermore, a note is on the cationic and anionic statesof atoms. As will become clear, for constructing the CFT ofmolecules, we need not only the neutral atomic states but alsocationic and anionic states. For the cationic states, we havedetailed energy levels in Moore’s book,43 but for the anionicstates, only a few (stable) levels exist for atoms, which areavailable through the Internet.

114105-6 Nakatsuji, Nakashima, and Kurokawa J. Chem. Phys. 149, 114105 (2018)

B. CFT for molecules from CFT for atoms

For studying the molecular electronic structure with CFT,we need to know all lower electronic states of different sym-metries for all atoms involved in the molecule under study. Inmolecules, the symmetry becomes lower so that the mixingof different atomic states is facilitated. Here again, Moore’sand NIST tables of atomic energy level are very helpful forplanning the CFT studies.

Let us now construct electronic structures of moleculeswith CFT. The procedure may be summarized as follows.

1. Put all constituent atoms at their positions of the chemicalformula.

2. Examine the atomic cf’s φA0,m to the energy level of inter-

est, EEL, for all constituent atoms and prepare them. Thenon-linear parameters in the atomic cf’s are assumed tobe already optimized. The energy level, EEL, is importantsince it decides the accuracy and the necessary amountsof computations. Moore’s book43 is very useful here aswill be seen below for carbons.

3. This step is the “from atomic states to molecular states”step, which is central in the CFT. We produce molecu-lar cf’s from atomic cf’s by making bonds with nearbyatoms using free electrons of the ground & excited statesof the constituent atoms. As will be given explicitly in theexamples below, the two-center σ bond is expressed withthe covalent term and the cationic-anionic ionic term.The local MO form is also useful near the equilibriumdistance. When the system includes the π-framework,like benzene, the valence bond “resonance” form and/orthe delocalized MO form are useful. The former has alocal structure, but the latter does not. The paired elec-trons in the valence orbital of atoms may become a lonepair in a molecule. Different combinations of free valenceelectrons give different sets of bonds and lone pairs thatcorrespond to different sets of molecular cf’s. We des-ignate the cf’s of the molecule as φMol

0,I . The suffix Irepresents a bonding form used in this cf, and the numberof them is defined as M, similarly to the atomic case. Thepre-suffix 0 represents that it is the initial function of theFC theory. This pre-suffix will be used later in the FCtheory.

4. By a linear combination of all molecular cf’s {φMol0,I }

prepared above, the CFT wave function of the targetmolecule Mol is described as

ψMol,CFTK =

M∑I

cKI φ

Mol0,I . (19)

5. Apply the variational principle to Eq. (19) and obtainCFT wave functions and energies of all the ground andexcited states of the target molecule. After applying thevariational principle, the suffix K denotes the groundand excited states of the molecule. They satisfy veryimportant orthogonality and Hamiltonian-orthogonalityrelations like Eq. (10).

From the above procedure, some important features ofCFT become evident. In CFT, both ground and excited statesadjust themselves through their interactions, as expressed by

Eq. (10). We understand from this equation that the stabiliza-tion energy relative to the atomic energy is distributed amongthe bond energies and that the electronic excitations occur withthe weakening of the bonds involved. These concepts are usefulfor studying photo-chemical reactions.

Now let us apply the above procedures to the hydro-carbon molecule CnHm as an example. First, we put carbonand hydrogen atoms at the positions of the chemical formula.From the transferability, all n carbons are the same and the mhydrogens are the same. Second, we prepare the cf’s of lowerelectronic states of the constituent atoms and investigate them.The hydrogen atom has a very discrete electronic structure.The ground state is the valence 1s state, and the degenerate 2sand 2p states are too high (82 259 cm−1) and diffuse to formchemical bonds with other atoms. So, we need to considersolely the 1s state for hydrogen atoms. On the other hand,the carbon atom has many valence atomic states. They are,from lower with the energy in cm−1 in parentheses, the 3P(0),1D(10193), and 1S(21648) states of the 2s22p2 configurationand the 5S(33735), 3D(64090), and 3P(75256) states of the2s2p3 configuration. There are no other valence states belowthe H(2s) state at 82 259 cm−1.43 The Rydberg states start from60 333(3P) to 61 982(1P) cm−1, which are 3s states, and the3p Rydberg states exist from 68 856 to 73 976 cm−1.43 For thereactions and properties near the ground state of carbons, the3P, 1D, and 1S states of the 2s22p2 configuration and the lowest5S state of the 2s2p3 configuration are important because theyare relatively low (<34 000 cm−1) in energy. Therefore, forordinary chemistry of the ground state alone, the EEL definedabove may be around 34 000 cm−1. The lowest 3P(2s22p2) statehas two free valences and can form two bonds with the neigh-boring atoms, while the 5S(2s2p3) state has four free valencesand can form four bonds. Therefore, the 5S(2s2p3) state can getvery large stabilization by forming four bonds with the neigh-boring atoms, the total stabilization energy exceeding its highenergy level (33 735 cm−1) relative to the ground 3P(2s22p2)state. Therefore, this 5S(2s2p3) state plays very important rolesin carbon chemistry, which are sometimes more important thanthe roles of the ground 3P state of the s2p2 configuration. Theexistence of these many interesting electronic states of carbonsmay be the origin of the variety of carbon chemistry and furtherthe origin of life produced mainly by organic molecules andsystems.

For the next step, we prepare the cf’s for the 1s state ofhydrogen as

φH0,1s = 1sHα (20)

and for the 3P(2s22p2) and 5S(2s2p3) states of carbon as

φC0,3P(s2p2) = A[(1sC)2αβ · (2sC)2αβ · 2pC

x 2pCy αα] (21)

and

φC0,5S(sp3)

= A[(1sC)2αβ · 2sC2pCx 2pC

y 2pCz αααα], (22)

respectively. Here, we omitted the electron numbers forbrevity. For molecular calculations, the atomic state cfof the 5S(2s2p3) state is often rewritten in the hybridforms σC

1 σC2 2pC

y 2pCz , σC

1 σC2 σ

C3 pC

z , and σC1 σ

C2 σ

C3 σ

C4 , depend-

ing on the sp, sp2, and sp3 hybrids, respectively. Theyare not strictly the atomic states, but useful for molecularcalculations.

114105-7 Nakatsuji, Nakashima, and Kurokawa J. Chem. Phys. 149, 114105 (2018)

Now, in the third stage, we prepare molecular cf’s fromatomic cf’s. This is the most important “from-atoms-to-molecule” step of CFT. In this step, we make bonds usingfree valence orbitals of atomic cf’s. For the H2 molecule, thecf of CFT is written as

φH20,G1 = A[(1sH1 (1)1sH2 (2) + 1sH2 (1)1sH1 (2))α(1)β(2)]

= A[(1sH1 (1)1sH2 (2))(α(1)β(2) + β(1)α(2))], (23)

which is obtained by making a singlet coupling from the atomiccf φH

0,1s given by Eq. (20). This is the simplest Heitler-Londonfunction for the hydrogen molecule. Another different form ofbond formation is possible, which is a local MO form givenby

φH20,G2 = A[(1sH1 + 1sH2 )(1)(1sH1 + 1sH2 )(2)α(1)β(2)]. (24)

This MO form has a well-known deficiency that it breaks offat the dissociation limit. So, we use this local MO form onlywhen we study the H2 molecule near its equilibrium geom-etry. There, we often use both cf’s of the covalent VB andMO forms as our cf’s of CFT: this corresponds to takingtwo-electron full CI into account. The so-called ionic con-tribution is included in the cf given by Eq. (24). We usuallyuse the covalent VB and MO forms simultaneously for thelocal bond near the equilibrium distance, when we want tocalculate the exact solution of the SE in chemical accuracy.We may consider the ionic terms separately. They are givenby

φH20,I1 = A[1sH1 (1)1sH1 (2)α(1)β(2)],

φH20,I2 = A[1sH2 (1)1sH2 (2)α(1)β(2)].

(25)

For the ground state of the CH molecule, whose symmetryis 2Π, the main cf is prepared from the ground 3P(2s22p2) stateof carbon φC

0,3P(s2p2)given by Eq. (21) and the function φH

0,1s

given by Eq. (20) as

φCH0,G = A

[(1sC)2αβ · (2sC)2αβ

· (2pCx 1sH + 1sH2pC

x )αβ · 2pCy α

](26)

by forming the CH σ bond with the singlet coupling of the2pC

x and 1sH electrons. For the CH bond, the ionic terms(2pC

x )2αβ and (1sH)2αβ are also considered. Instead, the localMO bond form (2pC

x + 1sH)2αβ is convenient when we studyonly near the equilibrium distance. They give different inde-pendent cf’s. The 5S(2s2p3) state does not contribute to thisground state by spin-symmetry, but the 3P(75256) state of the2s2p3 configuration contributes, but slightly because this stateis much higher in energy than the 3P(2s22p2) state. For gen-eral hydrocarbons, there are no symmetry restrictions. There-fore, both atomic states, 3P(2s22p2) and 5S(2s2p3), given byEqs. (21) and (22), respectively, play important roles, whichwill be discussed in the succeeding papers. The Rydberg3s state of carbon 3P(2s22p3s) at 60 333 cm−1 shows some

role for the higher potential curves of the CH molecule aroundthis energy level.

The C2 molecule is an interesting example. It is a closed-shell molecule, and its bonding is still a subject of discus-sions.24,46–48 Important contributions to the ground state comefrom the σC

1 σC2 2pC

y 2pCz expression of the 5S(2s2p3) state and

from the 3P(2s22p2) state given by Eq. (21). Before giving theirexpressions, let us define the spin part of the cf’s used in thispaper by

αβαβαβ · · · αβααα · · · α, (27)

as far as this spin function is possible for the chemistryunder consideration. From the σC

1 σC2 2pC

y 2pCz expression of

the 5S(2s2p3) state of the two carbons, C1 and C2, we obtainquadruple bonds

φCC0,G = A

{(1sC1 )2(1sC2 )2(σC1

1 σC21 + σC2

1 σC11 )

× (σC12 σC2

2 + σC22 σC1

2 ) · (pC1y pC2

y + pC2y pC1

y )

× (pC1z pC2

z + pC2z pC1

z )}, (28)

where we have omitted the spin function of the form given byEq. (27) and the electron numbers. We assumed the C2 axisto be on x. From the 3P(2s22p2) state of the two carbons, weobtain double π bonds

φCC0,G1 = A

{(1sC1 )2(1sC2 )2(2sC1 )2(2sC2 )2(pC1

y pC2y + pC2

y pC1y )

× (pC1z pC2

z + pC2z pC1

z )}

(29)

and double σ and π bonds

φCC0,G2 = A

{(1sC1 )2(1sC2 )2(2sC1 )2(2sC2 )2

×[(pC1

x pC2x + pC2

x pC1x )(pC1

y pC2y + pC2

y pC1y )

+ (pC1x pC2

x + pC2x pC1

x )(pC1z pC2

z + pC2z pC1

z )] }

. (30)

Equations (28)–(30) represent the most important cf’s of theground state of the C2 molecule. The strength of the C−−Cbond of C2 depends on the extent of the mixing of these twokinds of cf’s. At infinite separation, the contribution of Eq. (28)becomes zero since the ground state of the carbon atom is the3P(2s22p2) state. The contribution of the cf given by Eq. (28)increases as the two carbons come closer. In the above, weomitted the cf’s representing the contributions of the ionicterms, but now, it will be easily written down. When one studiesonly near the equilibrium geometry, the local bond forms areuseful for both σ and π bonds. The detailed study will bereported in a separate paper.

The above examples were given only for some of thevalence atomic states. There are many other states particularlyfor the states including carbon. The Rydberg atomic states lieat rather high energy levels, and their orbital exponents arediffuse in comparison with the valence states. Therefore, theircontributions to the lower electronic states are limited.

Next is the fourth stage. We describe our CFT wave func-tion by a linear combination of the molecular cf’s, as givenby Eq. (19). Note that the CFT is not a theory for the groundstate alone, but it describes also the excited states of the targetmolecule. Usually, as the energy becomes higher, the accuracydecreases because we limit the energy range of the atomic cf’sto 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 variationaldetermination of the ground and excited states. The groundstate is the most tightly bound state in which the stabilizationenergy relative to a sum of the atomic separate states is usedto strengthen the chemical bonds within the molecule. Thehigher excited states consist of the chemical bonds whose sumof the bond energies is less than that of the ground state: somebonds are weaken in some way, which might be interested inthe reactions of excited states.

The CFT is a general electronic structure theory for atomsand molecules that try to construct the electronic structures ofthe target system as naturally as possible like in the chemicalformulas. Atomic and molecular CFT describes both groundand excited states lying in the energy range of interest. Molec-ular CFT wave functions are produced by making chemicalbonds from the ground and excited states of the constituentatoms. For the reacting system, the states of interest alongthe reaction coordinate are calculated referring to the chemi-cal reaction formula. These bond formation steps are probablythe most important steps in CFT, and we use both VB andMO knowledge. The locality concept of VB is common toCFT and useful for reducing the number of the cf’s. For mostσ-framework, this would be valid. At the same time, if thedelocalized nature is the essence of the bond, we must use itsince by doing so the convergence to the exact solution of theSE will be accelerated. These CFT wave functions are used tostudy molecular structural properties and the potential energysurfaces along the reaction. The central concept of CFT isthe locality concept of the electronic structure, its transferabil-ity, and from-atoms-to-molecule concept. The central principleof CFT is the variational principle so that the resultant wavefunctions satisfy the expressions, as given by Eq. (10). As inthe chemical formula, the locality concept for bonds has stillsome ambiguity, which might be considered as an aspect ofthe diversity which is also an essential nature of chemistry andbiology.

A motivation of CFT was to construct the wave functiontheory that mimics the chemical formula. Then, by applyingthe FC theory to the initial functions obtained with the CFT,we will be able to get the exact solution of the SE easier thanstarting from other initial functions. In the following section,we apply the FC theory to the initial functions obtained bythe CFT. This level of theory is referred to as FC-CFT (free-complement chemical-formula theory).

IV. FREE COMPLEMENT THEORY APPLIEDTO THE CFT WAVE FUCTION: FC-CFT

For the importance of the chemical formula in chemistryand biochemistry, the CFT would be a useful initial theory forsolving the SE. For this purpose, we use the CFT as

ψFC-CFTexact = FC(ψCFT

0 ), (31)

where we use the FC theory as if it is an operator trans-forming the CFT function to the exact wave function. Inthis section, we describe in some detail about this operation.One important note is given initially. The CFT wave func-tion is obtained by applying the variational principle to thecf’s of CFT, as explained with Eq. (12) for atoms and with

Eq. (19) for molecules. We use them as the initial functionof the FC theory. However, since the FC theory is linear onthe cf’s, the operation is done to the elementary cf’s {φA

0,m}of Eq. (12) for atoms and {φMol

0,I } of Eq. (19) for molecules.In other words, the operation of the FC theory is done tothese elementary cf’s, not to the optimized CFT wave func-tion, because their coefficients are reoptimized again by the FCtheory.

A. Atoms

For atoms, the cf’s of the FC theory are produced usingthe recursion formula given by Eq. (7) with the scaling func-tion given by Eq. (6). More simply, they can be obtained bymultiplying the elements of the g operator to φ0. Then, the cf’s{φA

l,m} of atoms are written as

{φAlm ,m} = φ

A0,m, riφ

A0,m, rijφ

A0,m, rjriφ

A0,m, rkrijφ

A0,m,

rklrijφA0,m, . . . , (32)

where i, j, . . . denote the electrons and ri is actually riA withA denoting the nucleus. We define Lm to be the number of thecf’s produced from the initial cf φA

0,m and lm to be the indexrunning from 0 (initial cf) to Lm. With these definitions, theFC wave function for the Ith state of the atom A is written as

ψA,FC-CFTI =

M∑m

Lm∑lm

cIlm ,mφ

Alm ,m (I , m = 0, . . . , M). (33)

The number of the cf’s included in this expression, NACF,

becomes

NACF =

M∑m=0

Lm. (34)

Thus, the SE of the atom A is solved for all the electronic states.In comparison with the CFT equation given by Eq. (12), theFC theory produced the cf’s denoted by the suffix lm, and withthese functions, each atomic state becomes exact, interactingwith other atomic states. Then, the results satisfy⟨

ψA,FC-CFTI

���ψA,FC-CFTJ

⟩= δIJ ,⟨

ψA,FC-CFTI

���H���ψ

A,FC-CFTJ

⟩= EA,FC-CFT

I δIJ ,(35)

which is essentially the same quality equation, as given byEq. (10).

The above formulation shows the motivation of the CFTfor the use as the initial functions of the FC theory. We com-pare Eqs. (12) and (13) for the CFT with Eqs. (33) and (35)for the FC-CFT. In both formulations, we use each atomicstate as a basis for improving the set of the atomic states(I,m = 0, . . ., M). This originates from our experience ofstudying the ground and excited states with the SAC-CI(symmetry-adapted-cluster configuration-interaction) formal-ism.49,50 Rather than using some arbitrary basis function set,it would be more efficient if we use both ground and excitedstates of the system as their interacting basis. Then, each statebecomes better with the interactions among them and finallybecomes exact, as the FC-CFT result. Looking at the chem-ical formula, we imagined the formulation to be as close aspossible as the nature does and formulated the CFT and FC-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 theintermediate stage, as will be explained later.

For obtaining the chemical accuracy, it is necessary toconsider to the second order, but we usually include r3

i andr3

ij, but neglect the second-order terms of rij when elec-trons i and j belong to different shells. The term rklrij isincluded only when all four electrons are different. However,depending on each atom, we modify this rule based on ourexperiences.

B. Molecules

The locality of the electronic structures of the CFT bringsmuch simplicity to the calculations of the exact solution ofthe SE of molecules. First, we show the simplicity producedto the g-operator of the FC theory. As shown by Eq. (5),the g-operator was introduced to avoid the collisions betweencharged particles in a molecule that causes the divergence dif-ficulty of the integrals in the variational solution of the SE. Itwas originally defined by Eq. (6). However, for the locality, theg-operator is much simplified. The electron belonging to atomA, iA, has almost no chance to collide with the other nucleiB. Likewise, iA and iB do not have a chance to collide witheach other. Taking these facts into account, the g-operator tobe applied to the molecular initial functions of CFT is proposedas

g =∑A,iA

(riAA + xiAA + yiAA + ziAA) +∑

A,iA>jA

riAjA

+A-B,bond∑

A,iA,B

riAB +A-B,bond∑

iA,jB

riAjB , (36)

where the first two terms are the ordinary terms and the last twoterms are sometimes added. Because molecules are asymmet-ric, we added xiAA, et cetera in addition to the spherical termriAA. They work to polarize the orbital. The last two terms areexceptional since they are two-center riAB and riAjB terms forthe A-B bond. They might be unnecessary because they areessentially proportional to the AB bond distance and thereforemight cause a redundancy. So, we consider them only when thehydrogen atom is involved in the bond since the bond distanceis usually small.

However, when we use MO theory, the above simplifica-tions are not realized: since all electrons are delocalized over allatoms in a molecule, we cannot assign an electron i to an atomA so that the number of the cf’s increases. In Sec. III, we haveconsidered the cases where the delocalized MO descriptionmight be better or easier than the local VB-type description.However, such cases occur usually for the system which hasa large π-electron framework. Most σ-electron framework isstill described by the local two electron bonds. We shouldchoose the description as local as possible when we preparethe initial functions of the FC theory.

For molecules, the method of constructing the molecularcf’s {φMol

0,I } of the CFT was explained in Sec. III B, and theexamples were given for some diatomic molecules. Here weuse another example of the acetylene molecule, which is alinear closed shell molecule

H − C ≡ C − H.

The principal part of the initial function for the ground statemay be written as

φAcetylene0,G = A

[(1sC1 )2(1sC2 )2(σ(h1C1))2(σ(C1C2))2

× (π1(C1C2))2(π2(C1C2))2(σ(C2h2))2] . (37)

The first two are inner 1s cores, and the followings are theσ andπ local bonds. Each of the local σ and π bonds in Eq. (37) mayinclude covalent, ionic, and local MO forms, giving indepen-dent cf’s. Next, we apply the g function given by Eq. (36). Forchemical accuracy, we produce the molecular cf’s mostly toorder two, though some third-order terms are added when theyare important. In addition to this standard general rule, we usethe following rules to reduce the computational labors. Mostof them are the results of the compromise between accuracyand costs.

1. When we study molecules in their equilibrium structure,the local bonds in the initial functions like the one givenby Eq. (37) may be written in the local MO form: thereason is that when we use the sampling method of thelocal SE method,51,52 the local MO form is twice fasterthan the covalent form. Then, each local bond is replaced,one by one order, with the covalent form and also withthe ionic terms, if necessary.

2. When the functions riA, xiA, and rij are applied to theelectrons belonging to the local MO bond, the two-centerlocal MO is changed to the covalent VB form to make itmatch with the local treatment.

3. For the 2s orbital, we add exp(−α2Sr) at the first orderwhen we use a nodeless 2s orbital in the initial function.

4. For the second order function riAjA rkAlA , we include onlywhen all four suffixes are different. The function riAjA rjAkA

may also be included, when it is thought not to benegligible.

5. For the two center terms for the bond between atoms Aand B, like riAB and riAjB , originating from the last twoterms of the g function given by Eq. (36), only the firstorder term is considered when A is the hydrogen atom.

6. Similarly, only up to the first order is considered for theinter-pair terms: between 1s and 2s pair, between 1s andbond pair, etc.

7. The diagonal terms are considered up to the third order,like riAA, r2

iAA, r3iAA and riAjA , r2

iAjA, r3

iAjA, when they are

thought to be important.

When we have several initial cf’s, they are grouped in order,according to the order of their “parent” initial function. Forthe first-order cf’s, the g function is applied only to the firstorder. All cf’s generated with these rules are again localfunctions.

The cf’s thus produced by the FC theory for the molecule,Mol, may be written in a general form as

φMollI ,I (1, 2, . . . , N) = A[{rij

mlI ,Iij } · u1

lI ,I (1)u2lI ,I (2) · · · uN

lI ,I (N)]

(38)

including the initial functions, where the ith electron functionulI ,I

i (i) is obtained from the initial functions by multiplying thefunctions like riA, xiA, and/or rij. In Eq. (38), the suffix I of lI

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-ferent for each initial function I. Note that when φlI ,I includesthe two-electron terms rij, the functions ulI ,I

i (i) and ulI ,Ij (j) are

connected by rij, where usually the electrons i and j belong tothe same atom because only in such a case the electrons i andj can collide. mlI I

ij 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 K th state of

the molecule is written as

ψMol,FC-CFTK =

M∑I

LI∑lI

cKlI ,Iφ

MollI ,I (K , I = 0, . . . , M). (39)

Thus, the SE of the molecule is solved for the groundand excited states. The number of the cf’s included in thisexpression, NMol

cf , is given by

NMolcf =

M∑I=0

LI . (40)

In comparison with the CFT equation given by Eq. (19), theFC theory produced the functions denoted by the suffix lI , andwith these additional cf’s, the ground and lower excited statesbecome exact, interacting with the other states at the sametime. Then, the results satisfy⟨

ψMol,FC-CFTK

���ψMol,FC-CFTL

⟩= δIJ ,⟨

ψMol,FC-CFTK

���H���ψ

Mol,FC-CFTL

⟩= EMol,FC-CFT

K δKL,(41)

which is important showing the quality of the ground andexcited states calculated.

Very often, the calculations are limited only to the groundstate of a molecule. In such a case, the calculation can besimplified. The number of the cf’s given by Eq. (40) can bereduced. There are two choices. One is to limit the calculationsto the ground state from the beginning. Then, we calculate onlythe ground state, starting from the most probable initial func-tion. Usually, we know from the experience that what atomicstate is most important for molecular formation. For example,for carbon, we use the sp3 configuration rather than the s2p2

configuration, though the former is the excited state, whenthe symmetry of the molecule permits this choice. Then, weconstruct the initial wave functions using the idea of the CFTand apply the FC theory to this initial function. Even in thiscase, we can get the exact wave function and the energy withthe FC-CFT because the FC theory leads to the exact wavefunction from any initial function, if it has an overlap with theexact wave function, though we may need a larger number ofcf’s than the other cases we explain below. This is a simpleway of performing the FC-CFT: the locality concept and otherimplications of the chemical formula are included in the ini-tial wave function. Actually, for oxygen, for example, only thes2p4 configuration lies below 73 000 cm−1 so that this choiceand the other choices may be the same.

The second choice is, for the carbon case, to choose theatomic states arising from both ground s2p2 and excited sp3

configurations for making bonds with other atoms. Then, weconstruct all cf’s at the CFT level and apply the FC theory tothem to proceed the FC-CFT. The diagonalization at the CFTlevel is optional, if only the FC-CFT is the purpose, though werecommend doing so, because we can get some understanding

and information at the CFT level. If we need only the groundstate, we may apply the FC theory only to those cf’s that areimportant in the ground-state wave function at the CFT levelto reduce the labor of the calculations.

Like the initial functions, the higher-order cf’s of the FC-CFT also have local and transferable structures. Therefore,the final FC wave functions as the solutions of the SE havethe local and transferable structures. This is an important fea-ture of the present theory to obtain not only accurate but alsounderstandable results. Then, the following quantities

ψexactFC-CFT − ψ

CFT0 , (42)

∆ΓexactFC-CFT = Γ

exactFC-CFT(1, 2, · · · , N) − ΓCFT(1, 2, · · · , N), (43)

and∆ρexact

FC-CFT = ρexactFC-CFT(1) − ρCFT (1) (44)

are of special interest for chemists, where Γ and ρ are theN-electron density matrix and electron density, respectively.These difference quantities are the modifications that the SEhas given over the CFT quantities. The FC-CF theory is basedon the non-orthogonal set of functions so that the formulationsof Lowdin53 would be useful for analyzing the quantities givenby Eqs. (43) and (44). Since the SE is the governing principleof chemistry and biology, these quantities might give what wewanted to know. We want to learn from these quantities whatare necessary to make quantum chemical calculations useful inthe research of chemistry. The exact concept obtained with theFC-CFT would be useful since it is reliable. In the chemicalformula, the descriptions of bonds and interactions are rathervague in comparison with the atomic concepts. Therefore, wewant to learn from the results of the present theory about theirexact nature and concepts for the system of interest or underthe course of chemical reaction.

V. POSSIBILITY OF THE INTERMEDIATE THEORY:FC-CFT-V

The cf’s {φI } produced by the FC-CF theory includeboth integratable and non-integratable functions. We groupthem into those integratable, which is called the i-set, andthose non-integratable, which is called the n-set. The i-set plusn-set forms the complete set. The n-set cf’s are characterizedby the existence of the odd-powers of rij terms that makes themnon-integratable. However, these cf’s characterized by the rij

terms may not be so important for chemical researches, thoughthey work to lower the energy. For example, both MO and VBtheories do not include rij terms, but they are very importantfor studying chemistry.

This suggests a possibility of the intermediate theory. Weconsider only the i-set cf’s at some order of the FC theory andapply the variational principle to them. We refer to this theoryas FC-CFT variational (FC-CFT-V), where “V” indicates vari-ational. The accuracy of the FC-CFT-V can be improved byraising the order of the FC theory and by expanding the realmof the integral method. Because of its variational nature, wecan study extensively the chemistry of the system of interest.Furthermore, when we start from the CFT, we generate the cf’snot only for the ground states but also for the excited states.These ground and excited states satisfy the orthogonality andHamiltonian orthogonality. This relation is important for the

114105-11 Nakatsuji, Nakashima, and Kurokawa J. Chem. Phys. 149, 114105 (2018)

study of the ground and excited states. The quantities definedby Eqs. (42)–(44) are useful to investigate the chemistry understudy.

The FC-CFT-V is an advanced theory of the CFT. Sincethe FC theory always improves the theory, FC-CFT-V isalways more accurate than the CFT itself. When integrationsare possible for all cf’s, the FC-CFT-V is just FC-CFT andhighly accurate results are obtained. Such cases were alreadyreported for small systems like the He atom54–56 and H2

molecule.57 For H2+, highly accurate BO and non-BO results

were reported.58,59 In particular, highly accurate analyticalpotential curves of H2

+ were derived from the non-BO wavefunction.59

Then, we propose the two-step approach. After studyingthe basic chemistry with the FC-CFT-V theory, we study onlythe most important subject by considering the effects of then-set cf’s by performing FC-CFT. Then, we can extend theaccuracy to the exact level and we can predict and performchemistry with confidence. For this purpose, we may use thelocal Schrodinger equation (LSE) method.51,52 It is interestingto formulate the LSE method to be useful at this stage of theFC theory.

VI. POSSIBILITIES OF THE CFT AND FC-CFTFOR LARGE AND GIANT MOLECULES

When we deal with large and giant molecules, the localnature of the electronic structure of the CFT and the FC-CFTleads to much simplifications. This arises from the nature of theantisymmetry requirement for the electron pairs in molecules.The inter-exchange (iExg) theory40 tells us that as the distanceof the electron pair increases, the antisymmetry rule betweenthem fades away. As the system becomes larger, the numberof the distant electron pairs in the molecule increases rapidlyso that much simplifications result since antisymmetrizationsare usually very time-consuming processes. Here, we discussthis important effect in large and giant molecules.

A. iExg theory for antisymmetrization

Electrons are fermions, and therefore the electronic wavefunctions must satisfy the antisymmetry requirement, asexpressed by Eq. (2). We have proposed an efficient anti-symmetry theory, called Nk theory,60 which is useful forcorrelated theory including rij explicitly. This algorithm isdeterminant-based, and its order is roughly N3-4. Therefore, forlarge molecules, the antisymmetrization of electrons becomesa time-consuming step, and with the Nk theory alone, it is diffi-cult to apply theories to very large molecules. To overcome thissituation, we have proposed the inter-exchange (iExg) theoryof antisymmetrization40 based on the locality of the molecularelectronic structure. We briefly explain just the essence of theiExg theory for applying it to large and giant molecules.

For explaining the iExg theory, let us consider a diatomicmolecule AB. The antisymmetrizer AAB for the molecule ABcan be rewritten as

AAB = AAAB(1 + E(1)AB + E(2)

AB + · · · + E(k)AB + · · · + E(K)

AB ), (45)

where E(k)AB is the k-electron exchange operator between A and

B, K is the maximum number of electrons belonging to either Aor B, and AA and AB are the antisymmetrizers for the electronsbelonging to the atoms A and B, respectively. Note that eachelectron belongs to the orbital of an exponential form thatbelongs to either A or B. When we apply the one-electronexchange operator E(1)

AB to the molecular wave function ψAB,

the magnitude of E(1)AB ψAB becomes exponentially smaller than

the magnitude ofψAB itself since by the exchange, the electron-nucleus distances riA and rjB become longer by about the bonddistance R. Then, as the distance R increases, the result of theexchange becomes smaller exponentially and finally becomesnegligibly small, when the following Eq. (46) is satisfied. Sucha situation can be measured by using the overlap Tab betweenthe squares of orbitals χa and χb to which the electron pairbelongs,

Tab =⟨χ2

a��� χ

2b

⟩≤ λ. (46)

We use Tab instead of Sab2 to avoid the case of Sab = 0 by sym-

metry. For the pair of valence electrons of carbon atoms, λ isabout 10−6 and the distance Rλ corresponds to the four bonds ofhydrocarbons. For the k electron exchange E(k)

AB, this decrease

is k times faster than that of E(1)AB. In large molecules, there are

many atomic pairs for which E(n)ABψ is negligibly small: then

all terms of k ≥ n are negligible. Furthermore, the operationcounts for AAAB also reduce from NA! × NB! to NA! + NB!and further to NA

3 + NB3 with the use of the Nk theory. Thus,

the labor of antisymmetrizations can be reduced much with acombinatorial use of the iExg and Nk theories.

Now let us consider what the iExg theory implies whenwe apply the FC theory to large molecules and even to giantmolecular systems.

B. Simplification in large molecules

As molecules become larger, the number of electronsincreases, but at the same time, the number of distant atomicpairs in the molecule also increases. With the use of the expres-sion of the cf given by Eq. (38), the matrix elements betweendifferent cf’s are written as

〈φI |H | φJ〉 =

N∑i,j=1

⟨ui

I (1)��� −12∆1 −

∑A

ZA

rA1

���ujJ (1)

⟩D(Ii | Jj)

+N∑

i<jk<l

[⟨ui

I (1)ujI (2)

�����1

r12

�����uk

J (1)ulJ (2)

⟩

−⟨ui

I (1)ujI (2)

�����1

r12

�����ul

J (1)ukJ (2)

⟩]D(IiIj |JkJl ),

(47)

where the minors D(Ii | Jj) and D(IiIj | JkJl) are defined asusual.53 Here and below, the upper suffix l, I of ui

l,I is denotedby I alone for simplicity. In the above formula, we did notconsider the effect of rij included in some cf’s, but such cf’sdo not affect the following discussions because rij is actuallyriAjA , as explained before: rij connects electrons only withinthe same atom.

114105-12 Nakatsuji, Nakashima, and Kurokawa J. Chem. Phys. 149, 114105 (2018)

With this equation, let us consider the behaviors of theintegrals when the orbitals involved are so distant that theexchange of the electrons between these orbitals is negligi-ble as the iExg theory implies. For the one-electron part, whenthe orbitals uI

i and uJj are far apart, the integral is negligibly

small since each orbital is local and centered on a nucleusand decays exponentially from the center of the orbital, whilethe kinetic and nuclear attraction operators have little effecton these behaviors of the integrals. For the two-electron part,when all four orbitals, uI

i , uIj , uJ

k , and uJl , are far apart to

each other, the integral is negligible. When three of them arefar apart, again the integral is negligible. When two of them,(i, k) and (j, l), in parentheses are close, but (i, l) and (j, k)are far distant, the first term remains but the second term isnegligible. This case occurs when uI

i and uJk belong to one

atom and uIj and uJ

l belong to other different atom and the twoatoms are far apart in the molecule. Similarly, when one is farapart though three are close, again both integrals are negli-gible. Both integrals survive only when all orbitals are closeto each other. Similar behaviors are also seen for the overlapmatrix elements involved in the co-factor matrices D(Ii

��Jj) andD(IiIj |JkJl ). The above observations clearly imply that muchsimplification occurs when the molecule is very large: we canexpect that roughly order-N behavior of the computationallabor would result.

We apply the above argument to the interaction of thetwo atoms A and D that lie far apart from each other in alarge molecule. We assume that both A and D have only oneelectron, and the distance between A and D is so far apartthat the electron exchange between A and D does not occur inchemical accuracy. Then, the direct interaction term betweenA and D is written as

〈ψ |H |ψ〉A,D =ZAZD

RAD−

∫ZD

r1ADρA(r1A )dr1A

−

∫ZA

r1DAρD(r1D )dr1D

+∫ρA(r1A )

1r1A2D

ρD(r2D )dr1A dr2D , (48)

where r1A represents the coordinate of an electron belong-ing to atom A and ρA(r1A ) is the electron density distributionassociated with atom A,

ρA(r1A ) =∫ψ(r1A r2B · · · rNG )ψ(r1A r2B · · · rNG ) dr2B · · · drNG .

(49)

The first term of Eq. (48) represents the nuclear repulsion,the second and third terms represent the attractions betweenthe electron density of one atom with the nucleus of anotheratom, and the last term represents the repulsion between theelectron density of the atoms A and D. Since the electronexchange between A and D is negligible, only the classicalCoulomb interactions remain, and they sum up to the abovesimple formula.

We note that r1A in Eq. (48) denotes the coordinate of theelectron that belongs to the atom A, but due to the overlapand antisymmetry principle, the electron delocalizes out ofthe atom A, but the delocalization limit is within the limiting

distance Rλ related to the parameter λ given by Eq. (46). Toeliminate the overlapping effect between A and D, the ADdistance necessary for Eq. (46) is roughly 2Rλ. For a carbonpair, it is about 8 bond separation, ∼24 a.u. (12.7 Å).

C. Further simplification in giant molecules

When we move to still larger systems, like giant molecularsystems, further simplification occurs. There, the distance ofthe AD pair can be much larger than the above case. When theAD distance is so large that the widths of the orbitals a andd are negligible, then the width of the orbital d looks like apoint when it is seen from the orbital a. In such a situation, theelectron density distribution ρA looks like an electronic chargeqA on atom A, where

qA =

∫ρA(r1A )dr1A (50)

and Eq. (48) is transformed to

〈ψ |H |ψ〉A,D =δAδD

RAD, (51)

where δA is the gross charge of atom A, δA = ZA − qA.Such simple electrostatic forms have been introduced in

many molecular dynamics (MD) simulation programs,44,45

though their gross charges are empirical and/or approximate.They were also used in the quantum mechanical models ofgiant molecular systems like the elongation method for poly-mers,61 the divide and conquer method,62,63 and the fragmentmolecular orbital method.64 It is remarkable that this expres-sion is valid even in the exact theory for solving the SE, whichis the present FC theory, when it is combined with the iExgtheory. Thus, the exact FC theory transforms its theoreticalstructure automatically to a lower-order N type theory for largemolecules and even to the theory of giant molecular systemsas the system size increases further. This transformation ofthe theoretical structure of the FC theory is ideal as a chem-ical theory. Intuitively, this is a natural transformation of thechemical theory that reflects the chemical formulas central tochemistry. The basis is the locality and the transferability ofchemistry combined with the iExg theory.

We note that the quantum-mechanical theories for giantmolecular systems, like the elongation method, divide-and-conquer method, and fragment MO method, cited above61–64

are not the theories one obtains when the MO theory isapplied to giant molecular systems, but these theories are theoriginal products of the authors invented based on the MOtheory. Furthermore, the correlation theories over Hartree-Fock are difficult to apply to giant molecular systems becauseintegral transformations necessary for the MO-based corre-lation theories are too time-consuming for giant molecularsystems.

The CFT for large and giant molecular systems mustdescribe the above simplifications well. Though this is thesubject of our separate paper, the FC-CFT based on such con-cepts must be designed to work to order N as the system sizeincreases to giant systems. Fine coding will be necessary tofully utilize the simplifications in large and giant molecularsystems.

114105-13 Nakatsuji, Nakashima, and Kurokawa J. Chem. Phys. 149, 114105 (2018)

VII. CONCLUDING REMARKS

We aimed in this paper to combine two major principles ofchemical science, Schrodinger equation and chemical formu-las, to produce useful predictive theory in chemistry. Chemicalformulas are used daily by chemists and biochemists as work-ing concepts and the sources of their ideas. Therefore, if thewave function of the exact theory can have the structure ofthe chemical formula, it must be very useful for chemists. TheFC theory has a merit that it produces the exact wave functionthat has the structure assumed to the initial function used atthe beginning. Therefore, in this paper, we first proposed theCFT that gives the wave function whose structure reflects thatof the chemical formula. Then, starting from CFT, we can pro-duce, by FC-CFT, the exact wave function that has the structuresimilar to the chemical formula by using the FC theory. The dif-ference of the results of the CFT and the FC-CFT is due to theexact SE itself. So, by analyzing the difference, the chemistsmay get deep understanding that assists their researches. Thus,the CFT is designed from the beginning to be used by the FCtheory as its initial functions so that it affects the descriptionsof the exact wave function.

In CFT, our basic idea is that the molecular electronicstates are produced from atomic electronic structures. Thedetailed knowledge of the atomic electronic structures is sum-marized in Moore’s book of Atomic Energy Levels,43 whichprovides us useful basis. The basic ideas on the natures of theinteractions between atoms are included in chemical formulas.It is not decisive, but suggestive and must be described well bythe methods of quantum chemistry. We use all real electronicstates, ground and excited states, and ionic and anion statesof atoms as if they are the basis functions for constructingthe molecular ground and excited states. Atoms react with thenearby ones using free valences with forming bonds for stabi-lization. The governing principle is the variational principle.We use both VB and MO concepts, but the basic and dom-inant local structures in chemical formulas are close to theVB concept. The delocalized π-framework may be describedwell with the MO concept. The resultant electronic states pro-duced by CFT satisfy the orthogonality and the Hamiltonianorthogonality between the different states.

This result of CFT is elevated to the exact level by theFC theory, which is referred to as FC-CFT. When we studyboth ground and excited states of the molecule with CFT, allof them are made exact by the FC theory. The CFT is designedto be used by the FC theory as its initial function so that thisconnection is smooth and natural: the important relation, theorthogonality and the Hamiltonian orthogonality expressed byEq. (10), is satisfied at all levels of CFT, FC-CFT, and FC-CFT-V, and only the first indices lm and lI , respectively, for atomsand molecules are used to represent the cf’s produces by theFC theory.

The cf’s of the FC-CFT consist of the integratable andnon-integratable functions, the latter characterized by theinclusion of the rij terms, which may not be important forchemical studies, as seen from the MO and VB theories whichalso do not include rij terms. So, even if we use only the inte-gratable functions, the chemistry of the system under studywould be described reasonably well. This is the FC-CFT-V

theory and is an advanced variational theory of the CFT. Forthe efficiency of the variational method, we can study chem-istry extensively. Afterwards, if it is necessary to go to the exactSE level, we add the non-integratable cf’s, perform FC-CFTfor both ground and excited states, if necessary, and obtain theexact-level of the results. The chemical formula concepts arevalid in all levels of the theory.

The examinations of the locality-based concepts for thepossibility toward very large and even giant molecular systemsgave some light for the future. Because of the locality-basedstructure of the present theory, the inter-exchange (iExg) the-ory of antisymmetrization introduced previously could pro-duce large simplifications in the treatment of large systems.There, the exchanges between distant electron pairs fade awayand Coulombic interactions become dominant. For still largergiant systems, even the electrostatic descriptions become pos-sible. Then, we have to develop the CFT for such systems con-sidering the simplifications investigated in this paper. The codeof the FC-CFT for exactly solving the SE must be designedto work to essentially order N for large and giant molecularsystems.

In applications of the present theory, the automated intel-ligence (AI) technology would be useful to simplify the inputs,to connect different interfaces of the FC theory, and to clar-ify the implications of the calculated results, given in chemicalformula language, obtained by the various levels of the presenttheories. Furthermore, the feedbacks from the experimental-ists about the merits and the faults of our theory will be usefulfor further development of our methodology. Such collabora-tions may be a key for the future progress of the theory andalgorithms.

ACKNOWLEDGMENTS

This paper is a tribute to the late Professor R. G. Parrwho has encouraged much one of the authors from thebeginning of the studies. The authors thank Research Centerfor Computational Science, Institute for Molecular Science,Okazaki, for their encouragement to our research project fromcomputational sides. We also thank Mr. Nobuo Kawakamifor his continuous support to the activities of QCRI. Thiswork was supported by the JSPS KAKENHI, Grant Nos.16H02257, 17H06233, 16H00943, 17H04867, 17KT0103,and 16K17864.

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