Energy studies, beryllium atom [ 20 — 28 ] lower excited states fine, and hyperfine structure play an important role in multielectron atoms excited states theory development, and better correlation effects understanding between electrons. In this work, we present a theoretical characterization of a confined beryllium atom in a spherical box with impenetrable walls. The intent is to determine the effects due to confinement such physicochemical properties and electronic configuration.
It was confirmed that electronic configuration of the ground state of confined beryllium atom, is different from that of the free beryllium atom depending on the confinement region. We proposed the direct variational method to do this calculation, which is computationally much simpler in comparison with FDT or Hartree—Fock, due to it requires fewer operations and it does not need a specialized software, pointing that we use a four-element basis; giving energy values inside the experimental precision.
Using different effective atomic numbers for each hydrogen function, it is improved considerably the energy values precision due to electron screening effect consideration. This section is dedicated to studying confined beryllium atom inside a box with spherical symmetry and impenetrable walls. Hydrogenic functions will be used instead free-system functions, as well as introducing a cutoff function to ensure that the trial function is zero at the boundaries [ 5 — 13 , 16 — 18 , 30 — 32 ]. Hamiltonian operator for a four-electron atom in atomic units, using Born—Oppenheimer approximation and dismissing spin-orbit interaction, is given by:.
A trial wave function will be used, in Slater's determinant form, using spin—orbital hydrogen functions for 1 s and 2 s orbitals. From previously performed variational approaches [ 14 ], we know that a better energy values approximation is obtained using different effective atomic numbers for different each orbital so that electron screening effect is taken into account. Therefore, we will use different variational parameters for each hydrogenic function. Using spatial part and spin functions orthonormality conditions, it is possible to obtain the energy functional:.
After integrals calculation and plugged them into equation 11 , it follows a numerical minimization process for each variational parameter provided nuclear charge Z value and confinement radius r 0 , namely:. In order to improve confined atom energy approximate calculation, a slight modification was considered for the 2 s functions, adding a different variational parameter to each of them, to give them more flexibility, being as follows:. Once hydrogenic functions have been modified, 2 s functions nodes are properly adjusted to reduce energy value.
This change only affects integrals values, the energy functional form remains unchanged. It is worth to remember that the variational method can be used to estimate excited states energy value, as long as it is ensured that the trial wave function is normalized and orthogonal to lowest states [ 33 ] wave function.
Since 2 p orbital has three projections 2 p Z , 2 p Y , 2 p X ; in this work, the projections in -direction and -direction will be the one considered, the hydrogenic functions for that state are:. In our case the trial wave function is orthogonal to the ground state function; this is due to hydrogenic functions angular part being orthogonal. The energy functional form remains unchanged, due to the change residing solely in the 2 p orbital function, only modifying the integrals values. There is no angular dependence in the wave function ground state because orbitals only depend on the radial coordinate.
Because of this, the Laplacian operator, which acts on the function, depends exclusively on the radial coordinate. Once 2 p Z orbital with radial and angular dependence has been obtained, it is important to be very careful when calculating terms for kinetic energy so as not to make mistakes. The so-obtained variational energy and the trial wave function, make it possible to calculate some confined beryllium atom properties. Average pressure exerted by system boundaries is given by the expression [ 16 , 18 ].
An important physical quantity to calculate is polarizability, Kirkwood's [ 18 ] approximation was used:. This section presents results associated with the variational method, as well as few confined beryllium atom physical properties, to describe pressure effect in the system electronic structure. Using four and six variational parameters, energy values and their respective variational parameters for beryllium atom's electronic configuration 1 s 2 2 s 2 are shown in tables 1 and 2 , using the direct variational method and an antisymmetric wave function, where confinement radius r 0 is measured in Bohr and energy E H in Hartrees.
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Table 1. Direct variational calculation for confined beryllium atom's electronic configuration 1 s 2 2 s 2 using four variational parameters. Table 2. Direct variational calculation for confined beryllium atom's electronic configuration 1 s 2 2 s 2 using six parameters.
The 1 s orbital electrons experience higher nuclear charge than those in the 2 s orbital. This is reflected in variational parameters values. As confinement radius decreases, system energy increases as expected, and the difference between values in variational parameters becomes smaller.
A significant correction is observed in the energy using six variational parameters with respect to those obtained with four parameters. This is due to new parameters included in 2 s hydrogenic functions, which allow for greater flexibility when energy value minimization is looked for. Figure 1. Four and six variational parameters were used in this work. Using six variational parameters, they fluctuate between 0. Bohr for both cases. This difference has to do with the use of SCF approximation to the Hartree—Fock method, where the use of a sufficient number of basis functions is needed in order to calculate the analytical wave function precisely and, at the same time, to optimize orbital exponents, making calculations more complex and increasing computing time and effort.
In contrast, in this work, we obtained sufficient energies to study a confined atom behavior with only six parameters. The use of such small base, composed of only four hydrogen-like functions, dramatically reduces calculation difficulty and execution time when minimizing energy values. Compared to Rodriguez-Bautista's work [ 41 ], which used Roothaan's approach to solve the Hartree—Fock equations, there was a 0. They used a new basis set for Hartree—Fock calculations related to many-electron atoms confined by soft walls, and reported that orbital energies present one behavior totally different to that observed for confinements imposed by hard walls.
Inner orbital energies do not necessarily go up when the confinement is applied, contrary to the increments observed when the atom is confined by walls of infinite potential. This is because for atoms with large polarizability, like beryllium and Potassium, external orbitals are delocalized when confinement is imposed. Consequently, internal oribatls behave as if they were in ionized atom. As expected, as confinement radius r 0 decreases, kinetic energy system increases due to the system pressure effect.
This can be seen in table 3. Table 3. Pressure and kinetic energy using a four variational parameters and b six variational parameters. First excited state experimental value for beryllium atom is unknown, therefore data obtained in this work will be compared to approximate results.
The energy results obtained for different radius of confinement for the confined beryllium atom's electronic configuration 1 s 2 2 p Z 2 s , as well as values obtained in different papers, are shown in table 4. It is evident from variational parameters that for the electron in 2 p orbital, the core is more shielded compared to other electrons. Its energy also rises when confinement radius decreases, same as ground state case. It is the largest, as well. Table 4. Comparison between energy values for confined beryllium atom's electronic configuration 1 s 2 2 p Z 2 s and values obtained by Hibbert [ 20 ], Weiss [ 21 ], Chao Chen [ 35 ].
Lower energy values obtained in this work have a 0. Hibbert and Weiss reported a set of large-scale configuration interaction CI calculations for the states, which can give an accurate approximation for each state, but it may tend to obscure the global picture of the spectrum which is so transparent in the other approach.
On the other hand, energies and wave functions for the beryllium atom are calculated with the full-core plus correlation wave functions by Chao Chen [ 35 ], obtaining a better approximation because of the use of many relevant angular and spin couplings which greatly contribute to the final energy values. Besides, Hibbert and Weiss did not include any intra-shell correlation in the shell, because their calculations were those of transitions in outer subshells.
The purposes of these works were to obtain the energy values in a precise way though in our case we tried to find acceptable energy values to calculate atomic properties which were energy-dependent, plus, we consider the case of the non-free confined atom as Chao Chen [ 35 ], Hibbert [ 20 ] Weiss [ 21 ] did. All of which adds an additional potential due to confinement, which in turn influences on the difference among the values with respect to those ones already mentioned.
These methods are more expensive in terms of computation compared to the direct variational method because the CI basis sets expansion grows factorially and hundreds sometimes thousands of terms are needed in order to obtain the precision desired. Montgomery [ 39 ], Dolmatov [ 44 ] and Saul Goldman [ 40 ] report that for a strong confinement regime the behavior of the orbitals is different from that in which the confinement is weak.
For small confinement radii in the hydrogen atom, the energies of orbitals are lower than those of orbitals; energies different to the ones in the free atom. The crossing intersection of orbital energy for confined atoms was also reported by Garza et al [ 42 ], in particular for the Kr atom. It is well known that confinement overestimates the energies of the systems. Aquino et al [ 43 ] reported that a more physical way to simulate spherical compression would be accomplished by using soft, penetrable walls. Table 5 shows the energy results obtained for different radius of confinement for the beryllium atom with configuration In figure 2 , we show that the change in the beryllium atom's configuration confined to the ground state takes place in the region Bohr radius, now being the configuration for the ground state.
Consequently, for Bohr radius the energy values in tables 1 and 2 refer to the second excited state. The change in the electronic configuration of the beryllium atom, when decreases below is one of the principal effects of spatial confinement [ 42 ], and can produce important changes in the physical properties such as electronegativity, softness, and hardness.
Table 5. Direct variational calculation for confined beryllium atom's electronic configuration 1 s 2 2 p 2.
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When an atom is confined, the energy of its ground state rises, as was showed above in figure 2. The same is true for the first and the second excited state, but the rise is much smaller. As a result, there is always a crossing point for cavities smaller than a critical size: the ground state of the atom lies higher in energy than the first confined states of the atom. Evidently, the ground state of the atom is no longer stable when it lies higher in energy than the other states. When this occurs, the ground states transform into autoionizing states in the confined atom.
Prediction of Physicochemical Parameters by Atomic Contributions
To calculate polarizability, Kirkwood's approximation [ 18 ] is used equations 13 and Values obtained for free beryllium atom using different confinement radii and other reported results given by Komasa [ 36 ], Sahoo and Das [ 37 ] and Porsev and Derevianko [ 38 ], are shown in table 6. Polarizability is measured in units of and as polarizability exact value is not reported for beryllium atom, we had to compare it to approximate data.
The SVM focuses upon the small subset of examples SVs that are critical and informative to the classification and throws out the remaining examples. Removal of SVs changes the location of the separating hyperplane and alters the efficiency of the classification. This observation is easy to understand by comparing the size of sequence space and structure folding space. Even very different protein sequences can fold to a similar three-dimensional structure. On the other hand, relatively similar sequences can have different folds in proteins.
Protein structures can be robust for point mutations Taverna and Goldstein, Still numerous diseases arise due to single mutations. Residues within a protein take part in complex and non-linear interaction networks, which makes single sequence-based predictions difficult, especially when accounting for residues located far away in the sequence but close together in the folded structure.
Two sequence-based approaches, using single or multiple sequences, can be utilized for analyzing amino acid mutations. Amino acid mutations are constrained evolutionarily by two factors: structural and functional constraints. The score matrices reflect both of these constraints. Many of the functional residues locate on the surfaces of proteins, which may have little effect on the protein structure and stability.
For the classification of stabilizing and destabilizing mutations, it may not be necessary to account for the functional constraint.
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Therefore, the matrices which include both structural and functional information may in fact decrease the discriminative power of the classifier. In Fig. It is clear that no simple tendencies or relations could be found. Therefore, a non-linear model is necessary to describe and predict the relationships.
Our method uses single sequence information. Previously, this kind of classification has been based upon alphabet attributes and SVM machine learning Capriotti et al. The highest accuracy is for the method of Cheng et al. Of note is the high variation, especially in the MCC values. The developers of the previous methods have used fold cross validation, where the test set is smaller compared to training set. The present model improves the prediction in two aspects. First, we use the cleaned data set for the training and testing of our model.
We analyzed the classification result of the previous work by Capriotti et al. This site lists the data set used for training and testing their SVM models. The majority of the misclassifications cases, i. The analysis of the previous prediction results by Capriotti et al.
Capriotti et al. Only the misclassified cases are shown, most of which For the other improvement, we use the numeric physicochemical properties of amino acids instead of the alphabet. The properties of amino acids as attributes better explain the characteristics of residues. Protein folding and stability are mainly determined by the properties of amino acids in the primary sequence Anfinsen, Furthermore, the similarity of amino acids has to be learnt from the training set. According to the conventional rule of thumb for pattern recognition, the number of training samples should be 5—10 times the number of model parameters Kanal and Chandrasekaran, Optimization revealed the window size of 13 as the best classifier.
With fewer parameters, our model is more robust against over-fitting and has better generality. The problem of protein structure availability in the whole genome era has been redefined as the determination of representative structures of conserved protein families Redfern et al. But even with the success of this program, the demand for computational methods for discerning structural information from primary sequences will be higher than ever, since the protein sequences are widely diverse.
Site-directed mutagenesis is a commonly used technique. However, the relationship between a mutation and protein stability is often still an unresolved and difficult problem. Many studies are based on limited data and empirical rules. As the amount of data is continuously increasing, our task is to improve the prediction accuracy and refine the model.
Our study indicates a promising approach to predicting mutation effects based solely on single sequence information. Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide. Sign In or Create an Account. Sign In. Advanced Search.
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Article Navigation. Close mobile search navigation Article Navigation. Volume Article Contents. Materials and methods. Results and discussion. Physicochemical feature-based classification of amino acid mutations Bairong Shen. Oxford Academic. Google Scholar. Jinwei Bai. Mauno Vihinen. Cite Citation. Permissions Icon Permissions. Abstract A huge quantity of gene and protein sequences have become available during the post-genomic era, and information about genetic variations, including amino acid substitutions and SNPs, is accumulating rapidly. Open in new tab Download slide. Table I. Open in new tab.
Table II. Point mutations total in the 68 proteins used in the classification. The quality of the prediction is described by four parameters: accuracy, recall, precision and MCC. Table III. Accuracy values for predictions with different input parameters a. Table IV. Search ADS. Google Preview. Edited by Jane Clarke. Published by Oxford University Press.
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