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Ab initio energy band structure of polycytosine
Volume 61A, number 7
PHYSICS LETTERS
27 June 1977
AB INITIO ENERGY BAND STRUCTURE OF POLYCYTOSINE S. SUHAL Lehrstuhl fr~rTheoretische Chemie, Universitiit G&ringen, G~ttingen,BRD
Ch. MERKEL Lehrstuhl fuir Theoretische Chernie, Technische Universit2it Miinchen, Miinchen, BRD
and J. LADIK Lehrstuhl fur Theoretische Chemie, Universitit Erlangen-Iviirnberg, Erlangen, BRD Received 8 November 1976 The results of an ab initio LCAO Hartree-Fock crystal orbital study are reported for the homopolynucleotide polycytosine. The obtained energy bands show significant deviations from the results of earlier semiempirical calculations.
A number of different semiempirical quantum chemical methods were used in the past to study the electronic structure of systematically built more and more complicated periodic models of DNA [1]. As the next step in this series of investigations we applied the LCAO crystal orbital method in its ab initio form [2] to calculate the energy band structure of the homo-polynucleotide polycytosine. The main motivation behind this study was, besides to get more reliable information about the polymer in question itself, to facilitate critical of semi-calempirical results the on the basisreexamination of the first principle culations. The model of polycytosine investigated was built from cytosine molecules forming a Watson-Crick-type helix according to the structural data of DNA B [3]. The distance between the planar stacked cytosine molecules is 3.36 A in this model and they are rotated by 36°around the helix axis. The wavefunction of the polymer was taken as an antisymmetrized product of one-electron Bloch orbitals of the form ‘I’ (k r~ n
‘
~‘
M
=M112
in
~ ~ii~ exp(ikfa)C
1 ~(k)~1(r—R1 ~).
j1 1=1
‘
‘
where n is the band index, k stands for the wavenumber, M and m give the number of elementary cells and that of the atomic orbitals (AO’s) in one cell, respec-
tively. ~1(r
R1,/) denotes the lth AO centered on the atom with position vector R1 in the .ith elementary cell. The linear combination coefficients C1,~(k) are determined by solving the Hartree-Fock equations for all inequivalent k-values within the first Brilouin zone [2]. A minimal atomic basis set was used in this calculation, i.e. five orbitals on each first row atom and one orbital on the hydrogens. The atomic orbitals themselves were taken as appropriate linear combinations3/4 of gaussian lobe functions of the form ~(r) = (2~’ir)orX exp (—r?r2). The exponents of the uncontracted bitals as well as the contraction coefficients were taken from [4] for the heavy atoms (4s2p basis) and from [5] for the hydrogens (3GTO basis applying the usual scaling factor of 1.25). Since our system has a combined symmetry operation (translation along the helix axis and a rotation around it) we had to rotate the local atomic coordinate systems in different elementary cells with the molecules to preserve the cyclic property of the hamiltonian hypermatrix of the crystal (this is necessary for its block-diagonalization [2]). Since the basis set applied is rather limited in its —
,~
size we performed with it a number of comparative calculations for different molecules among others for the cytosine itself to test its ability to predict different physical properties. According to our experiences [6] though this basis set is less rich in “hard” gaussians 487
Volume 61A, number 7
PHYSICS LETTERS
(with large r~values) and consequently the molecular total energies are considerably higher than the values obtained previously by a 7s3p [7] and a 3GTO [8] basis, respectively, the overall description of the Valence shell one-electron properties (the decisive valence shell contributions to the dipole moments, ionization potentials, etc.) is equivalent or even better with this basis set, In building the Fock-matrix according to [2] we applied the “nearest-neighbour interaction” approximation in the strict sense, ~e. we calculated only those one- and two-electron integrals in which no such two orbitals occur which are centered on atoms belonging to non-neighbouring cells. Furthermore, we applied the correct Hartree-Fock exchange term without any approximation to it. In this way all the integrals in absolute value larger than the threshold value of 10—8 at.u. were calculated and nine different points of k in the first Brillouin zone were taken into account, The energy band structure obtained consists of forty-five bands of which twenty-nine are doubly filled. The correspondence between the individual molecular energy levels (eM0) and between the calculated bands is always unambiguous as it can be seen from table 1 where the lower and upper limits (e~~ and ~ respectively) as well as the widths of nine bands around the Fermi level are shown. Inspection of the wavefunctions shows that though the symmetry of the original MO’s is broken in polycytosine because of the stacked arrangement of the units there remains still a possibility to define quasi-it-type bands as was the case previously for all-valence electron
27 June 1977
wavefunctions [1]. Though the eight such quasi-iltype bands are located mainly around the Fermi level, the condition of a c-ir separation is not fulfilled in cytosine. It is interesting to note that the physically most important valence- and conduction-bands resulting from the present ab initio calculation are much broader (— 0.5 and 1 .2 eV, respectively) than those obtained with different semiempirical crystal orbital methods (0.1—0.3 eV [1]). If this trend will be valid also for other periodic DNA models and also for calculations with larger basis sets then it will be inevitable to reconsider the transport properties of DNA according to the new band structures. Concerning the band positions it should be mentioned that though the description of the filled bands seems to be satisfactory (the theoretical ionization potential of 9.11 eV obtained by Koopmans’ theorem can be quite well compared with the experimental value of 8.90 eV [9]) the forbidden gap between the highest filled and lowest unfilled band is clearly too large in this calculation (~Eg— 10.65 eV). This very large gap is in our opinion a direct consequence of the failure of the Hartree-Fock method in describing virtual levels with the aid of a VN potential instead of the appropriate VN1 potential [10]. Calculations are in progress in our laboratory to determine the correct theoretical value of the gap by using an excitation hamiltonian [11] in which the Fock operator used in this study will be complemented by an operator of the form OAO where the operator 0 projects onto the space of the virtual states and
Table 1 The physically most important energy bands around the Fermi level of the periodic polycytosine model. The original molecular energy levels (~MO),the band minima and maxima (e~~ and ~ respectively) with their location (ka) and the corresponding bandwidths are given in eV ~MO (type)
e~(kmjna)
e~(kmaxa)
8.8569 4.5849 1.9292 —9.7657 —11.4881 —11.5615 —11.9996 —14.3642 —16.2444
7.9045 4.8 134 1.5346 —9.6650 —11.1153 —11.7738 —11.8146 —14.1220 —15.8444
8.8351 5.129 1 2.7754 —9.1126 —10.8976 —11.2513 -—1L3275 —13.8009 —15.8362
a Highest filled level.
488
(a) (it) (it) (it)
(a) (it)
(a) (it)
(a)
a
(it)
(a) (a) (it)
(a) (a) (a) (it)
(ir/2)
(n/2)
(it)
0.9306 0.3157 1.2408 0.5524 0.2177
(it)
0.5225
(it) (it)
(a)
(11)
(a) (a)
0.4871 0.3221 0.0082
Volume 61A, number 7
PHYSICS LETTERS
A is chosen so that the virtual levels see the correct V~”~ potential. We should like to express our gratitude to Professor G.L. Hofacker for his interest and support of this project. One of us (S.S.) would like to express his special thanks to Professor W.A. Bingel for his continuous interest and support in this work,
References [11 J. Ladik,Adv. Quantum Chem. 7(1973) 397; S. Suhai and J. Ladik, mt. J. Quant. Chem. 7 (1973) 547 [21 G. Del Re, 3. Ladik and G. Bicz6, Phys. Rev. 155 (1967) 997.
27 June 1977
[31 S.
Arnott, S.D. Dover and A.J. Wonacott, Acta Cryst. B25 (1969) 2192. [4] B. Mely and A. Pullman, Theor. Chim. Ada (Berl.) 13 (1969) 278. [5] S. Huzinaga, J. Chem. Phys. 42 (1965) 1293. [6] S. Suhai, Ch. Merkel and J. Ladik, 3. Chem. Phys., to be published.
[71 E.
Clementi, J.-M. André, M.-CI. André, D. Klint and D. Hahn, Acta Phys. Hung. 27 (1969) 493. [8] J.D. Goddard, P.G. Mezey and 1G. Csizmadia, Theor. Chim. Acta (Ben.) 39 (1975) 1. [91C. Lifschitz, E.D. Bergmann and B. Pullman, Tetrahedron Lett. 46 (1967) 4583. [101 W.J. Hunt and W.A. Goddard III, Chem. Phys. Letters 3(1969) 414; H.P. Kelly, Adv. Chem. Phys. 14 (1969) 129. 1111 T.C. Collins and A.B. Kunz, mt. 3. Quant. Chem. 8 (1974) 437.
489
PHYSICS LETTERS
27 June 1977
AB INITIO ENERGY BAND STRUCTURE OF POLYCYTOSINE S. SUHAL Lehrstuhl fr~rTheoretische Chemie, Universitiit G&ringen, G~ttingen,BRD
Ch. MERKEL Lehrstuhl fuir Theoretische Chernie, Technische Universit2it Miinchen, Miinchen, BRD
and J. LADIK Lehrstuhl fur Theoretische Chemie, Universitit Erlangen-Iviirnberg, Erlangen, BRD Received 8 November 1976 The results of an ab initio LCAO Hartree-Fock crystal orbital study are reported for the homopolynucleotide polycytosine. The obtained energy bands show significant deviations from the results of earlier semiempirical calculations.
A number of different semiempirical quantum chemical methods were used in the past to study the electronic structure of systematically built more and more complicated periodic models of DNA [1]. As the next step in this series of investigations we applied the LCAO crystal orbital method in its ab initio form [2] to calculate the energy band structure of the homo-polynucleotide polycytosine. The main motivation behind this study was, besides to get more reliable information about the polymer in question itself, to facilitate critical of semi-calempirical results the on the basisreexamination of the first principle culations. The model of polycytosine investigated was built from cytosine molecules forming a Watson-Crick-type helix according to the structural data of DNA B [3]. The distance between the planar stacked cytosine molecules is 3.36 A in this model and they are rotated by 36°around the helix axis. The wavefunction of the polymer was taken as an antisymmetrized product of one-electron Bloch orbitals of the form ‘I’ (k r~ n
‘
~‘
M
=M112
in
~ ~ii~ exp(ikfa)C
1 ~(k)~1(r—R1 ~).
j1 1=1
‘
‘
where n is the band index, k stands for the wavenumber, M and m give the number of elementary cells and that of the atomic orbitals (AO’s) in one cell, respec-
tively. ~1(r
R1,/) denotes the lth AO centered on the atom with position vector R1 in the .ith elementary cell. The linear combination coefficients C1,~(k) are determined by solving the Hartree-Fock equations for all inequivalent k-values within the first Brilouin zone [2]. A minimal atomic basis set was used in this calculation, i.e. five orbitals on each first row atom and one orbital on the hydrogens. The atomic orbitals themselves were taken as appropriate linear combinations3/4 of gaussian lobe functions of the form ~(r) = (2~’ir)orX exp (—r?r2). The exponents of the uncontracted bitals as well as the contraction coefficients were taken from [4] for the heavy atoms (4s2p basis) and from [5] for the hydrogens (3GTO basis applying the usual scaling factor of 1.25). Since our system has a combined symmetry operation (translation along the helix axis and a rotation around it) we had to rotate the local atomic coordinate systems in different elementary cells with the molecules to preserve the cyclic property of the hamiltonian hypermatrix of the crystal (this is necessary for its block-diagonalization [2]). Since the basis set applied is rather limited in its —
,~
size we performed with it a number of comparative calculations for different molecules among others for the cytosine itself to test its ability to predict different physical properties. According to our experiences [6] though this basis set is less rich in “hard” gaussians 487
Volume 61A, number 7
PHYSICS LETTERS
(with large r~values) and consequently the molecular total energies are considerably higher than the values obtained previously by a 7s3p [7] and a 3GTO [8] basis, respectively, the overall description of the Valence shell one-electron properties (the decisive valence shell contributions to the dipole moments, ionization potentials, etc.) is equivalent or even better with this basis set, In building the Fock-matrix according to [2] we applied the “nearest-neighbour interaction” approximation in the strict sense, ~e. we calculated only those one- and two-electron integrals in which no such two orbitals occur which are centered on atoms belonging to non-neighbouring cells. Furthermore, we applied the correct Hartree-Fock exchange term without any approximation to it. In this way all the integrals in absolute value larger than the threshold value of 10—8 at.u. were calculated and nine different points of k in the first Brillouin zone were taken into account, The energy band structure obtained consists of forty-five bands of which twenty-nine are doubly filled. The correspondence between the individual molecular energy levels (eM0) and between the calculated bands is always unambiguous as it can be seen from table 1 where the lower and upper limits (e~~ and ~ respectively) as well as the widths of nine bands around the Fermi level are shown. Inspection of the wavefunctions shows that though the symmetry of the original MO’s is broken in polycytosine because of the stacked arrangement of the units there remains still a possibility to define quasi-it-type bands as was the case previously for all-valence electron
27 June 1977
wavefunctions [1]. Though the eight such quasi-iltype bands are located mainly around the Fermi level, the condition of a c-ir separation is not fulfilled in cytosine. It is interesting to note that the physically most important valence- and conduction-bands resulting from the present ab initio calculation are much broader (— 0.5 and 1 .2 eV, respectively) than those obtained with different semiempirical crystal orbital methods (0.1—0.3 eV [1]). If this trend will be valid also for other periodic DNA models and also for calculations with larger basis sets then it will be inevitable to reconsider the transport properties of DNA according to the new band structures. Concerning the band positions it should be mentioned that though the description of the filled bands seems to be satisfactory (the theoretical ionization potential of 9.11 eV obtained by Koopmans’ theorem can be quite well compared with the experimental value of 8.90 eV [9]) the forbidden gap between the highest filled and lowest unfilled band is clearly too large in this calculation (~Eg— 10.65 eV). This very large gap is in our opinion a direct consequence of the failure of the Hartree-Fock method in describing virtual levels with the aid of a VN potential instead of the appropriate VN1 potential [10]. Calculations are in progress in our laboratory to determine the correct theoretical value of the gap by using an excitation hamiltonian [11] in which the Fock operator used in this study will be complemented by an operator of the form OAO where the operator 0 projects onto the space of the virtual states and
Table 1 The physically most important energy bands around the Fermi level of the periodic polycytosine model. The original molecular energy levels (~MO),the band minima and maxima (e~~ and ~ respectively) with their location (ka) and the corresponding bandwidths are given in eV ~MO (type)
e~(kmjna)
e~(kmaxa)
8.8569 4.5849 1.9292 —9.7657 —11.4881 —11.5615 —11.9996 —14.3642 —16.2444
7.9045 4.8 134 1.5346 —9.6650 —11.1153 —11.7738 —11.8146 —14.1220 —15.8444
8.8351 5.129 1 2.7754 —9.1126 —10.8976 —11.2513 -—1L3275 —13.8009 —15.8362
a Highest filled level.
488
(a) (it) (it) (it)
(a) (it)
(a) (it)
(a)
a
(it)
(a) (a) (it)
(a) (a) (a) (it)
(ir/2)
(n/2)
(it)
0.9306 0.3157 1.2408 0.5524 0.2177
(it)
0.5225
(it) (it)
(a)
(11)
(a) (a)
0.4871 0.3221 0.0082
Volume 61A, number 7
PHYSICS LETTERS
A is chosen so that the virtual levels see the correct V~”~ potential. We should like to express our gratitude to Professor G.L. Hofacker for his interest and support of this project. One of us (S.S.) would like to express his special thanks to Professor W.A. Bingel for his continuous interest and support in this work,
References [11 J. Ladik,Adv. Quantum Chem. 7(1973) 397; S. Suhai and J. Ladik, mt. J. Quant. Chem. 7 (1973) 547 [21 G. Del Re, 3. Ladik and G. Bicz6, Phys. Rev. 155 (1967) 997.
27 June 1977
[31 S.
Arnott, S.D. Dover and A.J. Wonacott, Acta Cryst. B25 (1969) 2192. [4] B. Mely and A. Pullman, Theor. Chim. Ada (Berl.) 13 (1969) 278. [5] S. Huzinaga, J. Chem. Phys. 42 (1965) 1293. [6] S. Suhai, Ch. Merkel and J. Ladik, 3. Chem. Phys., to be published.
[71 E.
Clementi, J.-M. André, M.-CI. André, D. Klint and D. Hahn, Acta Phys. Hung. 27 (1969) 493. [8] J.D. Goddard, P.G. Mezey and 1G. Csizmadia, Theor. Chim. Acta (Ben.) 39 (1975) 1. [91C. Lifschitz, E.D. Bergmann and B. Pullman, Tetrahedron Lett. 46 (1967) 4583. [101 W.J. Hunt and W.A. Goddard III, Chem. Phys. Letters 3(1969) 414; H.P. Kelly, Adv. Chem. Phys. 14 (1969) 129. 1111 T.C. Collins and A.B. Kunz, mt. 3. Quant. Chem. 8 (1974) 437.
489