Life and Science of Clemens C. J. Roothaan∗

GENERAL ARTICLE Life and Science of Clemens C. J. Roothaan∗

Shridhar R. Gadre, Subhas J. Chakravorty

Clemens Roothaan, a Nazi concentration camp survivor and Professor Emeritus of the University of Chicago, passed away on 17 June 2019, 10 months after celebrating his 100th an- niversary. For his doctoral thesis, Roothaan developed the matrix version of the Hartree–Fockequations. These Hartree– Fock–Roothaanequations form the cornerstoneof atomic and Shridhar Gadre served for 30 molecular structure theory. In addition, Roothaan devised years as a Professor of Chemistry at the Savitribai computing methods for quantum chemistry, physics, and other Phule Pune University scientific fields from the early years. After 1988, he actively (SSPU), and later 6 years at helped Hewlett Packard develop the Intel Itanium processor. IIT Kanpur. In 2016, he This article presents the highlights of the life and science of joined as a Distinguished Professor at SPPU. His Roothaan. research interests are Clemens Roothaan, Louis Block Professor Emeritus of Physics theoretical and computational chemistry and Chemistry at the University of Chicago, died 17 June 2019 in Chicago. He was born on 29 August 1918 and celebrated his 100th birthday, about ten months before his death. Roothaan developed a foundational model for computing electronic orbitals in Subhas Chakarvoty did his atoms and molecules, particularly known as the ‘Hartree–Fock– PhD under the supervision of Roothaan equations’ [1, 2]. These equations and several compu- Shridhar Gadre. His postdoctoral training was tational methodologies devised by him with his associates are the with Enrico Clementi at foundation of scientific exploration of the electronic structure of IBM-Kingston (1987–1990), atoms and molecules. He led the University of Chicago Compu- and Ernest Davidson at tation Centre in its early years and devised new digital comput- Indiana University (1990–1996). Thereafter, he ing methods for quantum chemistry, physics and other scientific has been involved in research fields [3, 4]. In his later retired years, he actively helped Hewlett in the field of Packard develop the Intel Itanium processor and served as the cheminformatics and drug company’s liaison with the Large Hadron Collider. discovery and is currently employed at a major Clemens Roothaan was a native of Nijmegen, Netherlands. He pharmaceutical company. had enrolled in the University of Delft in 1935 to study electrical

∗Vol.26, No.6, DOI: https://doi.org/10.1007/s12045-021-1178-0

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Figure 1. Professor Roothaan, at his home in Hyde Park, Chicago, receiving congratulations on his 100th birthday on 27 August 2018, from the Pres- ident, Society of Catholic Scientists. Photo courtesy, Society of Catholic Scientists.

Keywords engineering. However, he became dissatisﬁed with the course in LCAO-MO theory, Roothaan 1938–39 and spent the spring semester auditing courses on elec- equations, MCSCF theory, H 2 tronics and acoustics at the Technical University of Karlsruhe in molecule, ﬂoating-point arithmetic unit, vector transcendental math Germany. In the fall of 1940, Clemens returned to Delft as a library. graduate student of physics in the laboratory directed by H. van Leeuwen. She is well known for her work in establishing the quantum mechanical nature of magnetism. In late 1942, the allies were gaining strength, and there were major German defeats in North Africa and Stalingrad. As a consequence, the Germans declared martial law in the Netherlands, which was under their occupation since 1940. Roothaan [3] has described how young men were captured and forced to work for German factories. In order to escape from this, Clemens and his As a member of the brothers, Victor and John, were back at their parental home in Ni- Computation Chamber, jmegen. Victor escaped arrest, but SS captured Clemens and John he carried out a despite their non-involvement in the resistance movement. In Au- Kronig-Penney calculation on a gust 1943, they were moved to the Vught concentration camp. In one-dimensional crystal this camp, not far from Eindhoven, the Philips company had the and later calculated the manufacturing operations of electric razors and short wave ra- elastic constants in a dios. The academically inclined prisoners of war (PoW), includ- classical crystal. ing Clemens, were assigned to the ‘Computation Chamber’. As a member of the Chamber, he carried out a Kronig-Penney

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calculation on a one-dimensional crystal and later calculated the elastic constants in a classical crystal [2]. The ﬁnal report of this work was transmitted to the Philips management in Eind- hoven, a few days before the camp was evacuated on 5 Septem- ber 1944. The PoW’s, including the two Roothaan brothers, were transferred to the Sachsenhausen camp in Germany. Near the end of the war, the camp inmates, including Clemens, were sent on a death march. John, his younger brother, was sent in an earlier transport to the infamous camp of Bergen-Belsen, where he suc- cumbed to deprivation and typhoid in March 1945. The death march of tens of thousands of undernourished prisoners was con- ducted by SS troops, chased by the Russian army, which lasted 12 days, covering 160 km. Clemens later estimated that one-third of these prisoners perished. Nijmegen was razed to the ground, killing many residents and wounding almost all others. Clemens survived, and after the war, his work in the ‘Computation Cham- ber’, and his technical report was accepted by TU Delft as his master’s thesis. He was awarded a master’s degree in physics in

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1945. In January 1946, Clemens arrived at the University of Chicago with a postgraduate fellowship. During this period, Chicago was the most exciting place to be for physicists. The first self-sustaining nuclear chain reactor and a physics faculty, with heavyweights like Enrico Fermi, Edward Teller, Maria Goeppert-Mayer, and Robert Mulliken. Nuclear research was, of course, de rigueur! Yet, Roothaan arrived in Chicago and opted for research in molecular structure and spectra as a research student of Robert Mul- liken. He spent the period between January 1946 and June 1947 learning quantum mechanics, linear algebra and group theory, passing the PhD candidacy examination in physics. Shortly thereafter, Roothaan was offered a faculty position in physics at the Catholic University of America in Washington, DC. He spent the next two years teaching in Washington, while working on his the- Roothaan realized that sis, making frequent trips to Chicago to confer with Mulliken on although the linear the semiempirical molecular orbital (MO) calculations on substi- combination of atomic tuted benzenes as his thesis subject. After publishing a research orbitals (LCAO) approximation of MO’s article on UV spectra of benzene and borazol jointly with Mul- was not new, entering liken in the Journal of Chemical Physics in 1948, he became dis- this approximation satisfied with semiempirical MO theory. formally into the N-electron variation He realized that although the linear combination of atomic or- principle provided a new bitals (LCAO) approximation of MO’s was not new, entering this and much better approximation formally into the N-electron variation principle mathematical point of foundation for electronic provided a new and much better mathematical point of founda- structure calculations on tion for electronic structure calculations on molecules (See Box molecules. 1). When he communicated his insight to Mulliken, who fully understood its significance, he encouraged Roothaan to devote his PhD thesis to this topic [1]. This work was later published as a paper in the Reviews of Modern Physics [2] and is regarded as a seminal contribution of Roothaan to quantum chemistry. Due to space constraint, it is not possible to describe here the full technical details of this paper of Roothaan. Only a brief and simplified account of the important aspects is presented, using the notations in [5]. For this, it is necessary to be familiar with the basics of the Hartree–Fock theory, which are given below.

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The starting point in the non-relativistic treatment of molecules is the Born–Oppenheimer approximation. Since nuclei are much Since nuclei are much heavier than electrons, they move more slowly. Hence, the kinetic heavier than electrons, energy of the nuclei can be neglected and the repulsion between they move more slowly. Hence, the kinetic the nuclei can be considered to be constant for a fixed nuclear energy of the nuclei can configuration. The Schrödinger equation for the electronic mo- be neglected and the tion is hence separated out and can be written as repulsion between the nuclei can be considered to be constant for a fixed HêΨe({ri}; {RA}) = ǫeΨe({ri}; {RA}). (1) nuclear configuration.

Here the electronic wave function Ψe depends explicitly on the electronic co-ordinates, {ri}, but has an implicit, parametric dependence on the position of the nuclei, viz. {RA}. In Eqn. (1),

Hˆ e is the electronic hamiltonian, with the eigenvalue ǫe. Eqn. (1) is written using the atomic units (viz. me = |e| = ~ = 1 and so on), describing the motion of N electrons in the ﬁeld of M nuclei, treated as point charges, viz.

N 1 N M Z N N 1 Hˆ = − ∇ˆ 2 − A + . (2) e 2 i r r Xi=1 Xi=1 XA=1 iA Xi=1 XJ=i i j In Eq. (2), the subscripts i, j are used for denoting the electron indices and A, B stand for the nuclear ones. The first term is the electronic kinetic energy operator, the second one denotes the nuclear-electron attraction and the third one, the electron-electron repulsion energy. The eigenfunction, Ψe in Eq. (1) for a fixed nuclear configuration, has a further dependence on the spins of all the electrons. The combination of the spatial co-ordinate (ri) and the spin (ωi) of the electron i is jointly represented as the space-spin variable, xi. The spin part is normally described as α(ω) or β(ω), i.e., spin up (↑) and spin down (↓) [5]. The spin orbital, χ(x) is the product of a spatial part Ψ(r) and a spin part, viz. α(ω) or β(ω). A traditional physical and chemical model and simplification is to approximate an orbital for each electron χ1(x1), χ2(x2),...χn(xn) and to represent Ψe({ri}; {RA}) as a product of the individual orbitals by Hartree. Then, the Hartree model reduces Eq. (1) to a very simple set of equations to solve, viz.

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ˆ χ ˆ ˆ χ ǫ χ , fi(1) i(1) = h(1) + Ji j(1) i(1) = i i(1) (3)  Xj,i      where the one-electron operator (the so-called ‘core Hamiltonian’) is given by

1 Z hˆ(1) = − ∇ˆ 2 − A . (4) 2 1 |r − R | XA 1 A

An additional requirement on the electronic wave function needs to be imposed, viz. the electrons need to be indistinguishable and

Ψe should be antisymmetric with respect to the interchange of the electronic coordinates x (both space and spin). This requirement led to an ingenious construction of an antisymmetric wave function Ψe(ri; RA) in terms of the spin orbitals χ1, χ2,...χN and given by Fock, Dirac and Slater, now commonly known as a Slater de- terminant:

− 1 Ψe(x1, x2, ··· , xN) = (N!) 2 detM, (5)

where M is the (N × N) matrix whose (i, j)-th entry is χi(x j).

Despite the complexity of the wave function Ψe(x1, x2, ··· , xN ) above, the corresponding simpliﬁcation of Eq. (1) is closely re- sembled in Eq. (3) with an additional term, the exchange operator

Kˆ b (1): the Hartree–Fock equations are

ˆ χ = ˆ + ˆ − ˆ χ = ǫ χ . f (1) a(1) h(1) Jb(1) Kb(1) a(1) a a(1) (6)  Xb,a Xb,a      The Coulomb (Jˆb) and exchange (Kˆ b) operators are deﬁned below.

∗ −1 Jˆb(1)χa(1) = [dx2χ (2)r χb(2)]χa(1), (7) Z b 12

and

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∗ −1 Kˆ b(1)χa(1) = [dx2χ (2)r χa(2)]χb(1). (8) Z b 12

For closed shells, the calculation of molecular orbitals becomes equivalent to the problem of solving the spatial integro-diﬀerential equation

fˆ(1)Ψa(1) = [hˆ(1) + {2Jˆb(1) − Kˆ b(1)}]Ψa(1) = ǫaΨa(1). (9) Xb,a

Eq. (9) is a pseudo-eigenvalue equation (although it looks like an eigenvalue equation) since the Fock operator has a functional dependence on the solutions Ψa through Eq. (7) and (8). Thus the Hartree–Fock equations are really nonlinear equations and need to be solved by an iterative procedure known as the self- consistent ﬁeld (SCF) [1,5] procedure. Hartree, in 1928, solved these integro-diﬀerential equations in Eq. (3) numerically for atomic systems, and the solution to Eq. (9) later became rou- tine for atoms. However, the numerical solutions of Eq. (9) were, and still are, formidable for molecules, especially for polyatomic molecular systems.

Box 1. Mulliken on the Doctoral Work of Roothaan

“I tried to induce Roothaan to do his Ph.D. thesis on Hückel-type calculations on substituted benzenes. But after carrying out some very good calculations on these, he revolted against the Hückel method, threw his excellent calculations out the window, and for his thesis developed entirely independently, his now well- known all-electron LCAO SCF self-consistent-field method for the calculation of atomic and molecular wave functions...”

–Robert S. Mulliken, Nobel Lecture (1966).

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Box 2. The Breakthrough Moment

“I went with a group of friends, including my future wife, to a concert by the Chicago Symphony Orchestra at its open-air summer home in Ravinia Park, north of Chicago. Lying in the grass, listening to the music and looking at the stars, it suddenly ﬂashed through my head that standard MO theory....should start....with molecules, with the additional constraint that the MOs are constructed as linear combinations of atomic orbitals (LCAO).”

–C.C.J. Roothaan [3]

With this background of the Hartree–Fock theory, let us return to the work of Roothaan. During these developments, chemists were already using linear combination of atomic orbitals (LCAO) for describing the molecular orbitals (MO), known since the time of Hückel, for π electrons. Other semi-empirical theories were also using such a framework wherein all MO calculations started with a “one-electron Hamiltonian”. At that time, Roothaan could not find a satisfactory definition of the “one-electron Hamiltonian” used in these theories. This was the time Roothaan “revolted” against the Hückel theory, as described by his thesis supervisor, Professor Mulliken (see Box 1). Indeed, the time was ripe for a newer and better methodology founded on rigor. Roothaan fig- ured out that solution to the Hartree Fock equations for molecules could be reached. using linear combination of atomic orbitals (see Box 2). The contribution of Roothaan [5] was to show how by in- troducing a basis set of atomic orbitals into the differential equation in Eq. (6), the Hartree–Fock equations would be converted to a set of matrix equations and solved by standard matrix tech-

niques. The unknown molecular orbitals Ψi are simply written as a linear expansion of the K known basis functions (or atomic orbitals), {ϕµ(r)|µ = l, 2,..., K}

K Ψi(r) = Cµiϕµ(r); i = 1, 2,..., K. (10) Xµ=1

A consequence of this approximation requires the deﬁnition of

744 RESONANCE | June 2021 GENERAL ARTICLE matrix elements of the operators involved, e.g.,

∗ Fµv = dr1ϕµ(1) fˆ(1)ϕv(1). (11) Z

core µ σλ 1 µλ σ . Fµv = Hµv + Pλσ[( v| ) − ( | v)] (12) Xλσ 2

core where the core-Hamiltonian matrix elements Hµv are given by

core ∗ Hµ = dr1ϕµ(1)hˆ(1)ϕv(1). (13) v Z

The elements of the core-Hamiltonian matrix are integrals involv- The elements of the ing the one-electron operator hˆ (1) describing the kinetic energy core-Hamiltonian matrix and nuclear attraction of an electron, as given in Eq. (4). are integrals involving the one-electron operator Eq. (14) gives a short-hand notation for the two-electron repul- hˆ (1) describing the sion integrals, (µv|λσ). kinetic energy and nuclear attraction of an electron. ∗ −1 ∗ (µv|λσ) = dr1dr2ϕµ(1)ϕv(1)r ϕλ(2)ϕσ(2), (14) Z 12 and the Fock matrix is obtained as [1]

core µ σλ 1 µλ σ . Fµv = Hµv + Pλσ[( v| ) − ( | v)] (15) Xλσ 2

Here, the term Pλσ is an element of the Mulliken charge density- bond order matrix 2 i ciλciσ. And the ﬁnal Hartree–Fock equations within Roothaan’sP treatment emerge as the matrix equation

FC = SCǫ, (16) where S is the (K × K) overlap matrix of the basis functions and ǫ is a diagonal matrix containing the eigenvalues of the Fock matrix as the diagonal elements. Roothaan noted the close resemblance to Eq. (9) albeit now in a matrix equation. Thus, Roothaan’s treatment was able to replace the integro-diﬀerential form of the

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Hartree–Fock equations by their matrix counterpart and he pro- posed that these equations be adopted to calculate wave functions for molecules, using the familiar iterative procedure, henceforth to be called the “LCAO-SCF method”. There was a parallel development in Cambridge, England, by Hall [6] between 1950 and 1951. Roothaan pointed out that these two treatments are very similar, yet there is an important diﬀerence. In Hall’s work M, the number of atomic orbitals, equals N, the number of occupied molecular orbitals. Dealing with the general case, with M > N, Roothaan was able to show the “equivalence” to the Hartree–Fock equations spanning the space of the M atomic orbitals. In his paper [1], Roothaan also discussed some inequal- ities on the Coulomb and exchange integrals, as well as the use of symmetry in molecular calculations. The so-called ‘Roothaan A quick check with equations’ received great acclaim in the following years. A quick Scopus reveals 4000 check with Scopus reveals 4000 citations of this work (and many citations of Roothaan’s of the older citations may still be missing). This long-lasting work work (and many of the > older citations may still is cited by 250 research articles during the last 4 years (2016– be missing). This 2019) even after 68 years of its publication! long-lasting work is cited by > 250 research Let us take a closer look into the events between 1949 and 1960 articles during the last 4 and into Roothaan’s research activities. In June 1949, he returned years (2016–2019) even to Chicago, as a research associate of Mulliken. At that time, after 68 years of its he was in the company of potential giants of molecular struc- publication! ture theory, Jake Bigeleisen, Mike Kasha, Christopher Longuett– Higgins, Al Matsen, Harden McConnell, Bob Parr, and Bill Price. And, during that time, he completed his thesis. In his thesis work, he addressed phenomena such as ionization and excitation processes. The essential chemical concepts of shells, molecular symmetry and group theory were infused and blended naturally into the discretized Hartree–Fock equations. A lofty standard set for him by the dissertation committee, and can be surmised from their playful insistence, “never mind all those beautiful formal- ities – what can you really calculate?” Roothaan complied their challenge with an LCAO-SCF calculation (although admittedly rather crude) on the benzene molecule in Chapter IV of his PhD thesis. This calculation was, however, dropped out from the pub-

746 RESONANCE | June 2021 GENERAL ARTICLE lished version of his thesis in [1]. After graduation, the University of Chicago invited him to join the faculty of the Physics Department, an extraordinary step, con- sidering the usual practice of renowned academic institutions. Roothaan now embarked under the auspices of the Laboratory for Molecular Structure and Spectra (LMSS) to deliver the methodology, computer programs, and the electronic structure calculations of atoms and molecules. Although the goal was to cover polyatomic molecules, the work began with diatomic molecules. His first major task was to systematize the calculation of one- and two-electron integrals over the basis functions from which the MOs are constructed. Slater-type functions rather than atomic orbitals were adopted as more suitable building blocks for the construction of MOs. This work got a significant boost from Klaus Rüdenberg, who had come to Chicago from Zurich. He tack- led the most demanding class of integrals, namely the two-center two-electron exchange integrals [3]. In 1950, a group of chemists and physicists fell into a discussion of the inadequacies of valence theory and the several difficulties in valence calculations. In 1951, Mulliken organized the Shelter conference on quantum chemistry, sponsored by the National Academy of Sciences, to bring together the researchers involved in the study of the electronic structure of molecules. Parr and Crawford’s account in the Proceedings of the Shelter Conference summarized it with an en- couraging note “that a frontal hopeful rather than desperate attack was, at last, being made on these equations.” Much of the credit to that then hopeful and now amply recognized as a most success- ful attack can be attributed to Roothaan, his many associates and the members of LMSS.

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Box 3. Galaxy of Researchers at Chicago Group

Prominent research associates and doctoral students at LMSS, most of whom later became well-known quantum chemists. Enrico Clementi, Megumu Yoshimine, Paul Bagus, Doug McLean, Bowen Liu, Rein- hart Ahlrichs, Paul Cade, Serafin Fraga, Juergen Hinze, Sigeru Huzinaga, Wlodzimierz Kolos, Sigrid Pey- erimhoff, Bernie Ransil, Robert Sack, Lutek Wolniewicz, Howard Cohen, John Detrich, Winnifred Huo, Yong-ki Kim, Gulzari Malli, Charles Scherr, Chris Wahl, Andy Weiss and Bill Worley. Several of the members of the Roothaan group later led by Enrico Clementi formed a missive at IBM, San Jose in the 1960s and continued this close collaboration to develop further the art of scientific computing and performed several landmark computations of atoms and molecules.

Another laudable Another laudable undertaking of Roothaan’s was to develop math- undertaking of ematical algorithms and libraries for the inchoate computer hard- Roothaan’s was to ware in 1960, which lacked floating point abilities. These old develop mathematical algorithms and libraries computer systems only consisted of binary input and output rou- for the inchoate tines. Assemblers, compilers, and floating point hardware were computer hardware in unavailable at that time. In Roothaan’s first LCAO-SCF model, 1960, which lacked he dealt with closed-shell systems. And, at that time, the numer- floating point abilities. ical Hartree–Fock methods side-stepped the open-shell dilemma and worked directly with angular momentum coupling coefficients and spherical harmonics as basis sets. Roothaan’s solution [2] to this puzzle was another instance of clarity and sheer brilliance, where he formulated the method to ask the right question and modified the variation principle so that all degenerate N-electron wave functions, which jointly represent the open shell, partici- pated equally in the variational process. While this extension of LCAO-SCF theory did not lead to a simple scheme for all con- ceivable cases, it paved the way for a general solution to open- shell problems, for both atoms and molecules subject to the suc- cessful calculation of vector coupling coefficients. Vector coupling coefficients in the context of molecular orbitals arise when degenerate orbitals are partially occupied, needing a modification to the expectation value of the energy.

The total open shell wave function Ψe is, in general, a sum of several antisymmetrized products, each of which contains a closed-

shell core ϕc, and a partially occupied open shell chosen from a

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set ϕo containing N0 open shell orbitals. The different antisymmetrized products are constructed containing different subsets of ϕo, The combined set of orbitals {ϕ}, is defined by {ϕc,ϕo} and is orthonormal. In referring to the individual orbitals, the indices i, j indicate the closed-shell orbitals and m, n, the open-shell ones. The expectation value of the energy is given by

N/2 N/2 N/2 N0 E0 = 2 i (i|h|i) + i j 2[(ii| j j) − (i j| ji)] + f { m 2(m|h|m) P PN0 NP0 P + f m n [2a(mm|nn) − b(mn|nm)] / P NP2 N0 +2 i m 2[(ii|mm) − (im|mi)]} (17) P P where a, b, and f are numerical constants depending on the spe- cific open shell case. The first two sums in Eq. (17) represent the closed-shell energy, the next two sums the open-shell energy, and the last sum the interaction energy of the closed and open shell. The number f is, in general, the fractional occupation of the open shell, that is, it is equal to the number of occupied open- shell spin orbitals divided by the number of available open-shell spin orbitals; obviously 0 < f < 1. The numbers a and b differ for different states of the same configuration. Although f, a and b were treated and indicated as constants, one knows today for the general open shell energy case, these are actually orbital- and representation-dependent. Curiously though, Roothaan left that possibility open by leaving a and b inside the sum expressions. The algebraic simplification of these vector coupling coefficients for atoms using the spherical harmonic basis and their connection to the numerical Hartree–Fock f and g coupling coefficients was worked out by Roothaan’s student Gulzari Malli and later fully exploited by Paul Bagus in the LMSS atom SCF package and its several avatars particularly Clementi’s ATOMSCF. The reader may refer to the original paper and recent works to follow the expression of unknown molecular orbitals Ψi in Eq (17) when linear expanded into K known basis functions {ϕµ(r)|µ = l, 2,..., K} to yield coupled equations:

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Figure 3. Mulliken, Ran- sil, and Roothaan meeting in the LMSS discussion area. Thanks are due to Professor Paul Bagus for making this picture available to the authors. Photo courtesy: M. Yoshimine and Ms. Hidemi Suzuki.

FCCC = SCCǫC ; F0C0 = SC0ǫ0. (18)

The ROHF ATOMSCF package and the tabulation of atomic wave functions for ground and excited states has been and continues to be a pervasive starting point of many research endeavours. While this refinement and development of SCF were ongoing, Kołos and Roothaan Kołos and Roothaan [7] developed a unique computer program, developed a unique for solving the electronic Schrödinger equation for the H2 molecule computer program, for to spectroscopic accuracy. They employed the explicitly corre- solving the electronic Schrödinger equation for lated basis introduced earlier by James and Coolidge. Later, this the H2 molecule to basis was improved and generalized by Kołos and Wolniewicz, spectroscopic accuracy. resulting in a seminal work that theoretically predicted the vibra- They employed the tional spectrum of the H molecule with unprecedented accuracy. explicitly correlated 2 basis introduced earlier In this famous development, Gerhard Herzberg had to revise his by James and Coolidge. experimental value of the dissociation energy of the H2 molecule, in order to explain a small deviation of 3.8 cm−1 from the theoretical value published by Kołos and Wolniewicz. This was the first demonstration that quantum chemists could solve the Schrödinger equation for a molecule with an accuracy that surpassed the best spectroscopic measurements. From 1962 to 1968, Roothaan worked as the director of the Uni- versity of Chicago Computation Center. He spent considerable time on the design and development of computer hardware and software for large-scale scientific calculations. Prominent research

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Figure 4. Clemens Roothaan in the University of Chicago Computation Center. Photo by Virginia Weissman. Reproduced with permission, Special Collections Research Cen- ter, University of Chicago.

associates and graduate students, who later became stalwarts (See Box 3) in atomic and molecular structure theory, contributed to LMSS and Roothaan’s broad-based development of computational quantum chemistry.

Roothaan’s official retirement was in 1988. Of course, such bril- Roothaan’s official liant, painstaking and meticulous scientists never retire! His ex- retirement was in 1988. pertise in scientific computing was harnessed in the development Of course, such brilliant, painstaking and of the Itanium-based platform while he continued his work on the meticulous scientists development and finessing of the Roothaan SCF theory. Roothaan never retire! His and his family lived in Hyde Park, Chicago, a home (a former fra- expertise in scientific ternity house) that he purchased for $30,000 in 1967. He himself computing was harnessed in the did a meticulous remodeling of it, with the help of a full-time development of the carpenter for two years. Now, this historic Hyde Park property is Itanium-based platform estimated to be worth $1.6 million. while he continued his work on the development Despite his several stints with the development of technology and and finessing of the strategic computing intiatives and projects with the US govern- Roothaan SCF theory. ment labs (the man-machine project), Roothaan had a procilivity towards physics and quantum chemistry. During the mid-1970’s he collaborated intensively with his former student John Detrich, on relativistic effects in atoms and molecules, and from about

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1980, again with John Detrich, on the MCSCF (multi-conﬁguration self-consistent ﬁeld) model of electronic structure. John Detrich, was himself building the massive research computers called as Loosely Coupled Array of Processors at Clementi’s IBM facil- ity in Kingston, New York between 1985 and 1991 to be used

for the Hylleraas CI calculations on H3. Their general multi- configuration self-consistent field (MCSCF) model permits the determination of wave functions for a collection of different physical states, achieves all possible group-theoretical reductions, and exhibits fast convergence. This general algebraic formalism which implements the MCSCF model for a wide variety of electronic systems published in 1983 can be considered the result and cul- mination of a development in MCSCF theories over a fifteen-year period by the Roothaan group.

The Roothaan–Detrich The Roothaan–Detrich variational energy is expressed in terms of variational energy is orbital and configuration vectors that are not redundant. And the expressed in terms of variational method optimizes these vectors to the desired mini- orbital and configuration vectors that are not mum energy and state. The orbital and configuration expansion redundant. And the are optimized simultaneously (see [8] for a review). The Taylor variational method series expansion of the energy and limiting it to the second degree optimizes these vectors and neglecting the higher-order terms is thus linearized, yielding to the desired minimum energy and state. in the following form:

Hx = −g (19)

This truncation of the energy expression to second-order term requires one to iterate just akin to standard SCF methods for a subsequent quadratic convergence of the orbital vectors and the conﬁguration vector. Roothaan spent considerable time alternating between physics, quantum chemistry, and computer development till the end. It is important that Roothaan’s contributions to the development of computers are illustrated. After he retired from the University of Chicago in 1988, he worked for two decades for Hewlett Packard Laboratories in Palo Alto, California, contributing to the development of the mathematical coprocessor routines for the Itanium

752 RESONANCE | June 2021 GENERAL ARTICLE chip. Most time-consuming calculations of today are iterative procedures. One may appreciate one of Roothaan’s invention better by knowing e.g. that modern floating point arithmetic units use the modified Newton—Raphson technique for division and square root operations which are also iterative procedures and these are well known to be much more efficient than the funda- mental restoring and non-restoring division and square root algorithms. Yet, iterative procedures are notorious for yielding in- accurate results due to an accumulation of round-off error, especially when using floating point operations. Note that single- and double-precision representation of floating point numbers corre- spond to 7 and 16 significant digits, respectively. The MCSCF and accurate CI calculations require higher floating point precision, specifically quad precision! Although quad precision hardware can be built, it would slow down the computers, and also increase the complexity of the circuits. Therefore, quad precision and higher-order precision are implemented using machine- level software programming on the existing hardware for spe- cific computations. For the predominant portions of the rest of the computer program, double and single precision is adequate. Roothaan and his HP coworkers’ invention enabled the floating- Roothaan and his HP point arithmetic unit to flexibly perform quad precision arithmetic coworkers’ invention on hardware designed for double-extended precision. Probably, enabled the floating-point arithmetic Roothaan’s real purpose in this important digression may have unit to flexibly perform been to deliver the most crucial elements of the MCSCF method. quad precision These include the coefficients of the orbitals, the configurations, arithmetic on hardware the matrix elements and coupling coefficients between configura- designed for double-extended tion state functions, requiring the extended precision calculation precision. of 3n-j symbols. Clemens Roothaan, a giant pioneer of modern quantum chemistry, also did significant work in computing technology. He was recognized by the computing giant companies HP, Intel and SGI with a Gelato Federation Lifetime Achievement Award. Among the groundbreaking works in the development of modern computers are his vector transcendental math library for the Itanium architecture, contribution to the man-machine project with Los

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Figure 5. Professor Roothaan sitting outside Hewlett Packard headquar- ters in Palo Alto, California in 2000. Photo courtesy of Robin Mitchell, UChicago News.

Alamos National Laboratory, and ﬁnally the work that resulted in the design and delivery of a ﬂoating point arithmetic unit, which contained the logic for quad precision arithmetic. Nowadays, quad precision is standard fare in modern compilers and essential for accurate atomic and molecular structure calculations. Roothaan’s legacy is a combination of tenacious work, mathematical and computing genius in pursuit of the solution of the daunt- ing quantum chemical equations to understand electronic structure and spectra. As a person, he was very hard-working, kind- hearted and intensely focused. A long time ago, Ernest Ruther- ford, known as the father of nuclear physics (Encyclopaedia Bri- tannica considers him to be the greatest experimentalist since Michael Faraday) had remarked “All science is either physics or stamp collecting”. We feel that Roothaan, a philatelist himself, supplanted the Rutherford pronouncement to: Science is physics and stamp collecting!

Acknowledgement

The authors thank Professor Paul S. Bagus, Department of Chem- istry, University of North Texas, a former student and close col- laborator of Roothaan for sharing many photographs, articles and anecdotes of the Chicago LMSS group.

754 RESONANCE | June 2021 GENERAL ARTICLE