dna and rna nucleobases contain the genetic 
 information of all 
 living cells and play a vital role in the development and functioning 
 of all known living organisms . because of their importance in maintaining 
 the genome integrity during cell replication , increasing attention 
 continues to be devoted to understanding the specific mechanisms ( particularly 
 ionization and electron - impact radiation damage ) that can dramatically 
 alter their electronic and structural states .
an important first step 
 in understanding the reactivity and damage mechanisms in dna and rna 
 is the ability to both efficiently and accurately predict electronic properties such as ionization energies and electron 
 affinities .
while numerous theoretical studies using high - level wave - function - based 
 techniques have been used to accurately predict the electronic properties 
 of nucleobases , ( see ref ( 1 ) and references within ) , their immense computational costs 
 prevent their routine use for complex biological environments or for 
 large geometries ( i.e. , a fully periodic dna / rna strand is currently 
 not possible with wave - function - based methods ) .
because of its favorable 
 balance between accuracy and efficiency , density functional theory 
 ( dft ) has become the most widely used quantum mechanical method for 
 obtaining electronic structure information of molecules and solids . 

however , selecting the best exchange - correlation functional continues 
 to be a difficult task due to the large number of functionals available .
herein , we assess the accuracy of a nonempirically tuned range - separated 
 ( long - range corrected ) density functional theory ( lc - dft ) method for predicting both the quasiparticle properties ( homo  lumo 
 fundamental gaps ) and the excitation energies in dna and rna nucleobases . 
 the fundamental gap is rigorously defined as the difference in energy 
 between the first ionization potential ( ip ) and the first electron 
 affinity ( ea ) , whereas the excitation / optical gap is the difference 
 in energy between the lowest dipole - allowed excited state and the 
 ground state . in general , the true optical excitation energy is smaller 
 than the fundamental gap due to excitonic effects that arise from 
 the coulombic attraction between the excited electron and hole within 
 the molecule ( see figure 7 in ref ( 5 ) for a clear pictorial example of the fundamental 
 gap and exciton binding energy ) . as discussed extensively by izmaylov 
 and scuseria ,
semilocal functionals are 
 incapable of accurately predicting excitonic effects whereas exchange - correlation 
 kernels that include a portion of nonlocal long - range exact exchange 
 ( such as the lc - dft kernels used in this work ) give rise to a nonzero 
 exciton binding energy , resulting in an accurate prediction of the 
 optical gap . in this study
, we investigate the quasiparticle properties 
 for guanine , adenine , cytosine , thymine , and uracil ( molecular structures 
 shown in figure 1 ) , and we evaluate the performance 
 of the nonempirically tuned lc - dft approach against both wave - function - based 
 and conventional dft methods . finally ,
using the same nonempirical 
 tuning procedure , we assess the accuracy of this approach and discuss 
 the implications for predicting the excited - state energies and properties 
 in these systems .
over the past few years , the 
 use of range - separated functionals for both dft and time - dependent dft ( tddft ) applications has significantly 
 grown in popularity . in particular , we and other researchers have 
 shown that these functionals show a dramatic improvement for strong 
 charge - transfer systems and 
 also surprisingly show improved accuracy even for relatively simple 
 valence excitations .
in contrast to conventional hybrid functionals which incorporate 
 a constant fraction of hartree  fock exchange , the range - separated 
 formalism mixes exchange densities nonuniformly by partitioning the 
 electron repulsion operator into short - range ( first term ) and long - range 
 ( second term ) contributions as1the  erf  term denotes the standard 
 error function , r12 is the interelectronic 
 distance between electrons 1 and 2 , and  is the range - separation 
 parameter in units of bohr . for a pure density 
 functional ( i.e. , blyp or pbe ) which does not already include a fraction 
 of nonlocal hartree 
fock exchange , the exchange - correlation 
 energy according to the lc formalism is2where ec , dft is the dft correlation functional , ex , dftsr is the short - range dft exchange functional , and ex , hflr is the hartree 
fock contribution to exchange computed 
 with the long - range part of the coulomb operator .
the modified ( nonlocal ) ex , hflr term can be analytically evaluated with gaussian 
 basis functions , and the short - range ex , dftsr contribution is computed with a modified exchange kernel specific 
 for each generalized gradient approximation .
recently , baer , kronik , and others and srebro and autschbach have demonstrated that the range - separation parameter in these exchange - correlation 
 kernels is highly system dependent but can be nonempirically tuned 
 for a given system .
this tuning process ensures that the negative 
 of the homo energy is equal to the ionization potential ( ip ) of the n electron system , which is a fundamental condition within 
 the kohn
this condition would naturally be satisfied 
 if the exact exchange - correlation functional were known ; as a result , 
 tuning  in a self - consistent manner to satisfy this fundamental 
 constraint is both intuitive and theoretically justified .
more rigorously , 
 the ip of a given system can be determined from the difference between 
 the ground - state energy of the n electron and the n  1 electron systems .
this energy difference corresponds 
 to the energy required to remove an electron from the system and , 
 according to janak s theorem , should be equal to the negative 
 of the homo energy .
as mentioned , this condition can be satisfied 
 ( or approximately satisfied ) with an optimal range - separation parameter 
 that can be obtained by minimizing the following function:3the second term in this function takes into 
 consideration the n + 1 system to indirectly tune the lumo of the n electron system . 

the lumo can not be directly incorporated in eq 3 as there is no formal equivalent of janak s theorem for the 
 lumo ; that is , janak s theorem does not explicitly relate the 
 electron affinity to the negative of the lumo .
as we will see , the 
 tuning of  , besides being theoretically rigorous , significantly 
 improves the ability to predict homo and lumo levels , fundamental 
 gaps , and even excitation energies . in order to maintain a consistent 
 comparison across previously 
 published benchmark calculations ,
identical molecular geometries obtained 
 from ref ( 1 ) were used 
 for this work .
optimal  values 
 were determined for guanine , adenine , cytosine , thymine , and uracil 
 with the long - range corrected blyp functional ( lc - blyp ) using a polarized triple- basis set ( cc - pvtz ) . 

we also investigated the effect of including diffuse functions using 
 a larger aug - cc - pvtz basis but found that the use of larger or more 
 diffuse basis sets did not significantly change our overall findings 
 ( see results and discussion section for further 
 details ) .
the optimal range - separation values were determined by varying 
  from 0.05  0.9 in increments of 0.05 ( increments of 
 0.01 were used from 0.2  0.4 ) . in order to determine j , single - point calculations were carried out 
 for the n and n  1 states 
 for each  value .
figure 2 graphically 
 illustrates j as a function of  
 for the different nucleobases .
the minimum ( optimal  ) of each 
 curve was obtained by spline interpolation , and these optimal values 
 are reported in table 1 .
these optimal range - separation 
 parameters were used for all subsequent lc - blyp calculations .
all 
 calculations were carried out with the gaussian 09 package using default scf convergence criteria ( density 
 matrix converged to at least 10 ) and the default 
 dft integration grid ( 75 radial and 302 angular quadrature points ) .
j ( eq 3 ) as 
 a function of  for the different dna / rna nucleobases as determined 
 using the lc - blyp functional and cc - pvtz basis . the inset shows a 
 magnified view of j in the 0.2 <  
 < 0.4 range .
the ability of the nonempirically 
 tuned lc - blyp method to predict 
 fundamental gaps and molecular orbital energies is investigated by 
 comparing results to recently reported gw , caspt2 , and experimental 
 values . the gw method is based on a green s function formalism which self - consistently 
 corrects the kohn  sham eigenvalues to give significant improvements 
 in the orbital energies .
the caspt2 ( complete active - space second - order 
 perturbation theory ) method is a multiconfigurational 
 perturbation technique which gives an estimate to the full configuration 
 interaction ( ci ) energy .
the theoretically predicted homo1 , 
 homo , and lumo ( ev ) levels ( in ev ) and the computed fundamental gaps 
 ( lumo  homo ) for the five nucleobases considered are reported 
 in table 2 .
it should be noted that the  homo  
 and  lumo  quasiparticle energies are not well - defined 
 quantities in wave - function - based caspt2 calculations , and the reported 
 homo / lumo energies in table 2 are , in fact , 
 vertical ionization energies and electron affinities , respectively . 

in addition to the mentioned methods , results using the popular b3lyp and standard lda functionals are also reported .
the fundamental gap is determined by the difference 
 between the homo and lumo orbital energies .
the mae ( mean absolute 
 error ) values are with respect to the caspt2 values ; the values in 
 parentheses are with respect to the gw values .
the reported caspt2 homo / lumo energies 
 are actually vertical ionization energies and electron affinities , 
 respectively ( see text )
. using the caspt2 method as our benchmark , we find 
 that the nonempirically 
 tuned lc - blyp method produces the overall best results , even outperforming 
 the gw method .
the gw method predicts the homo energy levels slightly 
 better ; however , the nonempirically tuned lc - blyp method is more accurate 
 for the homo1 energy levels ( 0.42 and 0.02 ev mae , respectively ) 
 and the fundamental gaps ( 0.12 and 0.06 ev mae , respectively ) .
the 
 lc - blyp method even slightly outperforms for predicting lumo energy 
 levels .
as mentioned in the theory and methodology section , we also investigated the effect of including diffuse functions 
 and found that our lc - blyp / aug - cc - pvtz results actually show a better 
 agreement with experimental results while deviating slightly more 
 from the gw and caspt2 benchmarks ( see tables s1 and s2 in the supporting information ) .
we attribute the deviations 
 between the lc - blyp / aug - cc - pvtz and gw / caspt2 calculations to the 
 nondiffuse basis functions used in the computationally demanding wave - function - based 
 methods .
for this reason , we only show the cc - pvtz calculations in 
 table 2 since this comparison allows a fair 
 and consistent evaluation since all of these theoretical methods use 
 similarly sized basis sets .
a quick analysis of table 2 shows that both the lda and b3lyp methods perform extremely 
 poorly for predicting orbital energy levels and fundamental gaps compared 
 to the gw and lc - blyp methods .
strikingly , in all cases , both the 
 lda and b3lyp calculations are qualitatively incorrect , predicting the wrong sign of the lumo .
the mean 
 absolute errors ( mae ) for the molecular orbitals and fundamental gaps 
 are reported in table 2 .
the lc - blyp method 
 produces results very comparable to those of the more rigorous gw 
 method , and the mae between the two methods is only 0.06 ev for predicting 
 the fundamental gaps .
furthermore , the nonempirically tuned lc - blyp 
 functional significantly outperforms lda and b3lyp , and qualitative 
 improvements are even achieved .
it is important to mention at 
 this point two additional aspects 
 that make the nonempirical tuning procedure more efficient than the 
 computationally demanding wave - function - based methods .
first , although 
 we calculated several single - point energies to determine the optimal 
 range - separation parameter ( i.e. , 37 different  values were 
 used for each nucleobase ) , we found that substantially fewer calculations 
 are actually required , and a coarse grid of  values ( in increments 
 of 0.05 or even larger ) fit to a smooth spline gives nearly 
 identical results .
the large number of calculations used to generate 
 figure 2 were carried out for completeness . 

second , it is important to note that even if one chooses to use a 
 fine grid of  values , each of the 37 different calculations 
 is completely independent of each other and can actually be calculated 
 separately on different cpus .
in contrast , the prohibitive computational 
 scaling of wave - function - based methods ( the eom - ccsd formalism used 
 later in this work scales as n ) prevents 
 their routine use even if several processors are used in parallel . 

for example , a single lc - blyp / cc - pvtz calculation for guanine takes 
 less than an hour on 8  2.93 ghz intel nehalem cpus , whereas 
 a calculation at the eom - ccsd / cc - pvtz level of theory takes over five 
 days using the same computational resources .
this combination of efficiency 
 with the accuracy demonstrated previously demonstrates a clear benefit 
 of the nonempirically tuned range - separated formalism .
the true 
 test of any computational method is the ability to accurately 
 reproduce experimental results . in figure 3 ,
the ionization energies determined by the various computational 
 methods are graphically compared to the experimental ranges ( numerical 
 values are reported in table 2 ) .
it is visually 
 apparent that the caspts , gw , and lc - blyp methods all produce results 
 in good agreement with the experimental values ; however , the lda and 
 the popular b3lyp methods show a dramatic underestimation of energies 
 ( errors > 2.0 ev ) . again
, this example demonstrates that traditional / popular 
 density functionals can fail significantly for predicting ionization 
 energies ( homo ) ; however , significant improvements 
 can result from the use of nonempirically tuned lc - dft functionals .
ionization 
 energies ( ev ) of the five nucleobases considered determined 
 by the negative of the homo orbital at different levels of theory 
 compared to the experimental range . finally , in addition to our detailed study of ( ground - state ) 

quasiparticle 
 properties , we also analyzed the accuracy of nonempirically tuned 
 range - separated methods for predicting optical excitation energies 
 using time - dependent density functional theory ( tddft ) . as benchmarks 
 for assessing the quality of the various tddft methods , we calculated 
 equation - of - motion couple - cluster ( eom - ccsd ) 
 excitation energies with the same geometries and cc - pvtz basis set 
 used previously in our tddft calculations .
it is important to mention 
 that the lowest excited states in these nucleobases are largely dominated 
 by    * valence excitations and have been shown 
 by the krylov group to be well - described with the cc - pvtz basis set .
for both the tddft and eom - ccsd calculations , we also compute the 
 oscillator strength since this provides another stringent benchmark 
 test for assessing excited - state properties . 

table 3 compares both the lowest excitation 
 energy and oscillator strengths between b3lyp , lc - blyp , and eom - ccsd 
 for all five nucleobases .
overall , we find that the nonempirically 
 tuned lc - blyp results are in significantly better agreement with the 
 eom - ccsd benchmark calculations in all cases .
specifically , table 3 shows that the mae is reduced by more than 
 half for both the lc - blyp excitation energies and oscillator 
 strengths . in particular , we draw specific attention to the excitation 
 energies of adenine , thymine , and uracil , which are all significantly 
 underestimated ( by almost 0.5 ev ) using b3lyp . in order to understand 
 the possible cause of these dramatic errors , we calculated tozer s 
 lambda (  ) diagnostic at the b3lyp / cc - pvtz 
 level of theory for all five nucleobases .
this test numerically quantifies 
 the spatial overlap between the occupied and virtual orbitals involved 
 in an excitation . by construction ,
the diagnostic metric  is 
 bounded between 0 and 1 , with small values signifying a long - range 
 excitation and large values indicating a localized , short - range transition . 
 on the basis of their extensive benchmarks ,
if  is less than 
 0.3 , indicating little overlap and significant long - range charge transfer 
 character , hybrid functionals are predicted to yield inaccurate results . 
 as shown in the last column of table 3 , none 
 of the  values are less than 0.3 ; however , the results for 
 adenine , thymine , and uracil are nearly borderline cases with smaller 
  diagnostic values ( 0.4 ) . as a result , b3lyp exhibits 
 larger errors for these particular systems when compared to our benchmark 
 eom - ccsd results . in contrast
, it is remarkable to note that the nonempirically 
 tuned lc - blyp method demonstrates significant accuracy for these excited 
 states even though they have only been tuned to satisfy kohn 
sham ground - state constraints ( eq 3 ) without relying on any a priori knowledge of excited - state properties .
the lambda (  ) diagnostic 
 values were determined at the b3lyp / cc - pvtz level of theory .
in conclusion , we have shown that nonempirically 
 tuned range - separated 
 dft methods represent a significant improvement over traditional functionals 
 for predicting both the quasiparticle properties ( homo  lumo 
 fundamental gaps ) and excitation energies in dna and rna nucleobases . 

we have demonstrated that the nonempirically tuned lc - blyp method 
 accurately reproduces experimental ionization potentials for the various 
 nucleobases considered ; in addition , our results demonstrate an excellent 
 agreement with the more computationally intensive gw and caspt2 methods . 
 furthermore , even though this nonempirical tuning procedure has been 
 used to satisfy kohn 
sham ground - state constraints , 
 we have shown that this methodology also leads to significantly improved excited - state energies and properties in nucleobases , compared 
 to conventional dft methods .
we believe that these types of functionals 
 are at the forefront of density functional theory and will continue 
 to grow in popularity because of their improved accuracy , computational 
 efficiency , and theoretical rigorousness .
 using a nonempirically tuned range - separated dft approach , 
 we study 
 both the quasiparticle properties ( homo  lumo fundamental gaps ) 
 and excitation energies of dna and rna nucleobases ( adenine , thymine , 
 cytosine , guanine , and uracil ) . 
 our calculations demonstrate that 
 a physically motivated , first - principles tuned dft approach accurately 
 reproduces results from both experimental benchmarks and more computationally 
 intensive techniques such as many - body gw theory . 
 furthermore , in 
 the same set of nucleobases , we show that the nonempirical range - separated 
 procedure also leads to significantly improved results for excitation 
 energies compared to conventional dft methods . 
 the present results 
 emphasize the importance of a nonempirically tuned range - separation 
 approach for accurately predicting both fundamental and excitation 
 gaps in dna and rna nucleobases . 
Introduction
Theory and Methodology
Results and Discussion
Conclusion