Two-body Interaction Energies Research Articles

REAlgo: Rapid and efficient algorithm for estimating MP2/CCSD energy gradients for large molecular clusters.

This work reports the development of an algorithm for rapid and efficient evaluation of energy gradients for large molecular clusters employing correlated methods viz. second-order Møller-Plesset perturbation theory (MP2) theory and couple cluster singles and doubles (CCSD). The procedure segregates the estimation of Hartree-Fock (HF) and correlation components. The HF energy and gradients are obtained by performing a full calculation. The correlation energy is approximated as the corresponding two-body interaction energy. Correlation gradients for each monomer are approximated from the respective monomer-centric fragments comprising its immediate neighbours. The programmed algorithm is explored for the geometry optimization of large molecular clusters using the BERNY optimizer as implemented in the Gaussian suite of software. The accuracy and efficacy of the method are critically probed for a variety of large molecular clusters containing up to 3000 basis functions, in particular large water clusters. The CCSD level geometry optimization of molecular clusters containing ∼800 basis functions employing a modest hardware is also reported.

Benchmarking two-body contributions to crystal lattice energies and a range-dependent assessment of approximate methods.

Using the many-body expansion to predict crystal lattice energies (CLEs), a pleasantly parallel process, allows for flexibility in the choice of theoretical methods. Benchmark-level two-body contributions to CLEs of 23 molecular crystals have been computed using interaction energies of dimers with minimum inter-monomer separations (i.e., closest contact distances) up to 30 Å. In a search for ways to reduce the computational expense of calculating accurate CLEs, we have computed these two-body contributions with 15 different quantum chemical levels of theory and compared these energies to those computed with coupled-cluster in the complete basis set (CBS) limit. Interaction energies of the more distant dimers are easier to compute accurately and several of the methods tested are suitable as replacements for coupled-cluster through perturbative triples for all but the closest dimers. For our dataset, sub-kJ mol-1 accuracy can be obtained when calculating two-body interaction energies of dimers with separations shorter than 4 Å with coupled-cluster with single, double, and perturbative triple excitations/CBS and dimers with separations longer than 4 Å with MP2.5/aug-cc-pVDZ, among other schemes, reducing the number of dimers to be computed with coupled-cluster by as much as 98%.

Accurate three-body noncovalent interactions: the insights from energy decomposition.

An impressive collection of accurate two-body interaction energies for small complexes has been assembled into benchmark databases and used to improve the performance of multiple density functional, semiempirical, and machine learning methods. Similar benchmark data on nonadditive three-body energies in molecular trimers are comparatively scarce, and the existing ones are practically limited to homotrimers. In this work, we present a benchmark dataset of 20 equilibrium noncovalent interaction energies for a small but diverse selection of 10 heteromolecular trimers. The new 3BHET dataset presents complexes that combine different interactions including π-π, anion-π, cation-π, and various motifs of hydrogen and halogen bonding in each trimer. A detailed symmetry-adapted perturbation theory (SAPT)-based energy decomposition of the two- and three-body interaction energies shows that 3BHET consists of electrostatics- and dispersion-dominated complexes. The nonadditive three-body contribution is dominated by induction, but its influence on the overall bonding type in the complex (as exemplified by its position on the ternary diagram) is quite small. We also tested the extended SAPT (XSAPT) approach which is capable of including some nonadditive interactions in clusters of any size. The resulting three-body dispersion term (obtained from the many-body dispersion formalism) is mostly in good agreement with the supermolecular CCSD(T)-MP2 values and the nonadditive induction term is similar to the three-body SAPT(DFT) data, but the overall three-body XSAPT energies are not very accurate as they are missing the first-order exchange terms.

Crystallization to the Square Lattice for a Two-Body Potential

We consider two-dimensional zero-temperature systems of N particles to which we associate an energy of the form $$\begin{aligned} \mathcal {E}[V](X):=\sum _{1\leqq i<j\leqq N}V(|X(i)-X(j)|), \end{aligned}$$ where $$X(j)\in \mathbb R^2$$ represents the position of the particle j and $$V(r)\in \mathbb R$$ is the pairwise interaction energy potential of two particles placed at distance r. We show that under suitable assumptions on the single-well potential V, the ground state energy per particle converges to an explicit constant $$\overline{\mathcal E}_{\mathrm {sq}}[V]$$ , which is the same as the energy per particle in the square lattice infinite configuration. We thus have $$\begin{aligned} N{\overline{\mathcal E}_{\mathrm {sq}}[V]}\leqq \min _{X:\{1,\ldots ,N\}\rightarrow \mathbb R^2}\mathcal E[V](X)\leqq N{\overline{\mathcal E}_{\mathrm {sq}}[V]}+O(N^{\frac{1}{2}}). \end{aligned}$$ Moreover $$\overline{\mathcal E}_{\mathrm {sq}}[V]$$ is also re-expressed as the minimizer of a four point energy. In particular, this happens if the potential V is such that $$V(r)=+\infty $$ for $$r<1$$ , $$V(r)=-1$$ for $$r\in [1,\sqrt{2}]$$ , $$V(r)=0$$ if $$r>\sqrt{2}$$ , in which case $${\overline{\mathcal E}_{\mathrm {sq}}[V]}=-4$$ . To the best of our knowledge, this is the first proof of crystallization to the square lattice for a two-body interaction energy.

Linearization of moment tensor potentials for multicomponent systems with a preliminary assessment for short-range interaction energy in water dimer and trimer

Moment tensor potentials have been recently proposed as a promising novel method of polynomial expansion for the systematic approximation of molecular potential energy surfaces. However, its current formulation for multicomponent systems has not been fully linearized and requires nonlinear optimization techniques for parameter estimation. We propose an alternative relaxed formulation of the original potential energy function where parameter optimization is expressed as a linear sparse approximation problem. The main difficulty arising in sparse approximation is finding a suitable subset of predictors in highly multi-collinear variable space where the number of variables largely exceeds the size of the training set. To efficiently reduce the number of descriptors to an optimal size and prevent overfitting, we present a simple heuristic that is based on importance ranking of variables and Bayesian information criterion. For the empirical assessment of our approach, we employed published data on short-range components of water two-body and three-body interaction energies that have previously been used for the comparison of various potential energy representations. Numerical experiments suggest that our proposed methodology allows achieving accuracy that is comparable to other popular interpolation and machine learning techniques and requires significantly less time for model training than nonlinearly parameterized formulation.

A Fragment Quantum Mechanical Method for Metalloproteins.

An accurate energy calculation of metalloprotein is of crucial importance and also a theoretical challenge. In this work, a metal molecular fractionation with conjugate caps (metal-MFCC) approach is developed for efficient linear-scaling quantum calculation of potential energy and atomic forces of metalloprotein. In this approach, the potential energy of a given protein is calculated by a linear combination of potential energies of the neighboring residues, two-body interaction energy between non-neighboring residues that are spatially in close contact and the potential energy of the metal binding group. The calculation of each fragment is embedded in a field of point charges representing the remaining protein environment. Numerical studies were carried out to check the performance of this method, and the calculated potential energies and atomic forces all show excellent agreement with the full system calculations at the M06-2X/6-31G(d) level. By combining the energy calculation with molecular dynamic simulation, we performed an ab initio structural optimization for a zinc finger protein with high efficiency. The present metal-MFCC approach is linear-scaling with a low prefactor and trivially parallelizable. The individual fragment typically contains about 50 atoms, and it is thus possible to be calculated at higher levels of the quantum chemistry method. This fragment method can be routinely applied to perform structural optimization and ab initio molecular dynamic simulation for metalloproteins of any size.

Concentration for Coulomb gases on compact manifolds

We study the non-asymptotic behavior of a Coulomb gas on a compact Riemannian manifold. This gas is a symmetric n-particle Gibbs measure associated to the two-body interaction energy given by the Green function. We encode such a particle system by using an empirical measure. Our main result is a concentration inequality in Kantorovich-Wasserstein distance inspired from the work of Chafa\"i, Hardy and Ma\"ida on the Euclidean space. Their proof involves large deviation techniques together with an energy-distance comparison and a regularization procedure based on the superharmonicity of the Green function. This last ingredient is not available on a manifold. We solve this problem by using the heat kernel and its short-time asymptotic behavior.

Comparison of permutationally invariant polynomials, neural networks, and Gaussian approximation potentials in representing water interactions through many-body expansions.

The accurate representation of multidimensional potential energy surfaces is a necessary requirement for realistic computer simulations of molecular systems. The continued increase in computer power accompanied by advances in correlated electronic structure methods nowadays enables routine calculations of accurate interaction energies for small systems, which can then be used as references for the development of analytical potential energy functions (PEFs) rigorously derived from many-body (MB) expansions. Building on the accuracy of the MB-pol many-body PEF, we investigate here the performance of permutationally invariant polynomials (PIPs), neural networks, and Gaussian approximation potentials (GAPs) in representing water two-body and three-body interaction energies, denoting the resulting potentials PIP-MB-pol, Behler-Parrinello neural network-MB-pol, and GAP-MB-pol, respectively. Our analysis shows that all three analytical representations exhibit similar levels of accuracy in reproducing both two-body and three-body reference data as well as interaction energies of small water clusters obtained from calculations carried out at the coupled cluster level of theory, the current gold standard for chemical accuracy. These results demonstrate the synergy between interatomic potentials formulated in terms of a many-body expansion, such as MB-pol, that are physically sound and transferable, and machine-learning techniques that provide a flexible framework to approximate the short-range interaction energy terms.

Two-component, ab initio potential energy surface for CO2-H2O, extension to the hydrate clathrate, CO2@(H2O)20, and VSCF/VCI vibrational analyses of both.

We report an ab initio, full-dimensional, potential energy surface (PES) for CO2-H2O, in which two-body interaction energies are fit using a basis of permutationally invariant polynomials and combined with accurate potentials for the non-interacting monomers. This approach which we have termed "plug and play" is extended here to improve the precision of the 2-body fit in the long range. This is done by combining two separate fits. One is a fit to 47 593 2-body energies in the region of strong interaction and approaching the long range, and the second one is a fit to 6244 2-body energies in the long range. The two fits have a region of overlap which permits a smooth switch from one to the other. All energies are obtained at the CCSD(T)-F12b/aug-cc-pVTZ level of theory. Properties of the full PES, i.e., stationary points, harmonic frequencies of the global minimum, etc., are shown to be in excellent agreement with direct CCSD(T)-F12b/aug-cc-pVTZ results. Diffusion Monte Carlo calculations of the dimer zero-point energy (ZPE) are performed, and a dissociation energy, D0, of 787 cm-1 is obtained using that ZPE, De, and the rigorous ZPEs of the monomers. Using a benchmark De, D0 is 758 cm-1. Vibrational self-consistent field (VSCF)/virtual state configuration interaction (VCI) MULTIMODE calculations of intramolecular fundamentals are reported and are in good agreement with available experimental results. Finally, the full dimer PES is combined with an existing ab initio water potential to develop a potential for the CO2 hydrate clathrate CO2(H2O)20(512 water cage). A full normal-mode analysis of this hydrate clathrate is reported as are local-monomer VSCF/VCI calculations of the fundamentals of CO2.

Full QM Calculation of RNA Energy Using Electrostatically Embedded Generalized Molecular Fractionation with Conjugate Caps Method.

In this study, the electrostatically embedded generalized molecular fractionation with conjugate caps (concaps) method (EE-GMFCC) was employed for efficient linear-scaling quantum mechanical (QM) calculation of total energies of RNAs. In the EE-GMFCC approach, the total energy of RNA is calculated by taking a proper combination of the QM energy of each nucleotide-centric fragment with large caps or small caps (termed EE-GMFCC-LC and EE-GMFCC-SC, respectively) deducted by the energies of concaps. The two-body QM interaction energy between non-neighboring ribonucleotides which are spatially in close contact are also taken into account for the energy calculation. Numerical studies were carried out to calculate the total energies of a number of RNAs using the EE-GMFCC-LC and EE-GMFCC-SC methods at levels of the Hartree-Fock (HF) method, density functional theory (DFT), and second-order many-body perturbation theory (MP2), respectively. The results show that the efficiency of the EE-GMFCC-SC method is about 3 times faster than the EE-GMFCC-LC method with minimal accuracy sacrifice. The EE-GMFCC-SC method is also applied for relative energy calculations of 20 different conformers of two RNA systems using HF and DFT, respectively. Both single-point and relative energy calculations demonstrate that the EE-GMFCC method has deviations from the full system results of only a few kcal/mol.

Electrophilicity of oxalic acid monomer is enhanced in the dimer by intermolecular proton transfer.

We have analyzed the effect of excess electron attachment on the network of hydrogen bonds in the oxalic acid dimer (OA)2. The most stable anionic structures may be viewed as complexes of a neutral hydrogenated moiety HOA˙ coordinated to an anionic deprotonated moiety (OA-H)-. HOA˙ acts as a double proton donor and (OA-H)- as a double proton acceptor. Thus the excess electron attachment drives intermolecular proton transfer. We have identified several cyclic hydrogen bonded structures of (OA)2-. Their stability has been analyzed in terms of the stability of the involved conformers, the energetic penalty for deformation of these conformers to the geometry of the dimer, and the two-body interaction energy between the deformed HOA˙ and (OA-H)-. There are at least seven isomers of (OA)2- with stabilization energies in the range of 1.26-1.39 eV. These energies are dominated by attractive two-body interaction energies. The anions are vertically bound electronically by 3.0-3.4 eV and adiabatically bound by at least 1.6 eV. The computational predictions are consistent with the anion photoelectron spectrum of (OA)2-. The spectrum consists of a broad feature, with an onset of 2.5 eV and spanning to 4.3 eV. The electron vertical detachment energy (VDE) is assigned to be 3.3 eV.

Wannier functions using a discrete variable representation for optical lattices.

We propose a numerical method using the discrete variable representation (DVR) for constructing real-valued Wannier functions localized in a unit cell for both symmetric and asymmetric periodic potentials. We apply these results to finding Wannier functions for ultracold atoms trapped in laser-generated optical lattices. Following S. Kivelson [Phys. Rev. B 26, 4269 (1982)], for a symmetric lattice with inversion symmetry, we construct Wannier functions as eigenstates of the position operators x̂, ŷ, and ẑ restricted to single-particle Bloch functions belonging to one or more bands. To ensure that the Wannier functions are real-valued, we numerically obtain the band structure and real-valued eigenstates using a uniform Fourier grid DVR. We then show, by a comparison of tunneling energies, that the Wannier functions are accurate for both inversion-symmetric and asymmetric potentials to better than 10 significant digits when using double-precision arithmetic. The calculations are performed for an optical lattice with double-wells per unit cell with tunable asymmetry along the x axis and a single sinusoidal potential along the perpendicular directions. Localized functions at the two potential minima within each unit cell are similarly constructed, but using a superposition of single-particle solutions from the two lowest bands. We finally use these localized basis functions to determine the two-body interaction energies in the Bose-Hubbard model and show the dependence of these energies on lattice asymmetry.

Large effective three-body interaction in a double-well optical lattice

We study ultracold atoms in an optical lattice with two local minima per unit cell and show that the low-energy states of a multiband Bose-Hubbard (BH) Hamiltonian with only pairwise interactions are equivalent to an effective single-band Hamiltonian with strong three-body interactions. We focus on a double-well optical lattice with a symmetric double well along the $x$ axis and single-well structure along the perpendicular directions. Tunneling and two-body interaction energies are obtained from an exact band-structure calculation and numerically constructed Wannier functions in order to construct a BH Hamiltonian spanning the lowest two bands. Our effective Hamiltonian is constructed from the ground state of the $N$-atom Hamiltonian for each unit cell obtained within the subspace spanned by the Wannier functions of the two lowest bands. The model includes hopping between ground states of neighboring unit cells. We show that such an effective Hamiltonian has strong three-body interactions that can be easily tuned by changing the lattice parameters. Finally, relying on numerical mean-field simulations, we show that the effective Hamiltonian is an excellent approximation of the two-band BH Hamiltonian over a wide range of lattice parameters, in both the superfluid and Mott insulator regions.

Dependence of the Many-Body Interaction Strength in Water Clusters (H&lt;sub&gt;2&lt;/sub&gt;O)&lt;sub&gt;&lt;i&gt;n&lt;/i&gt;&lt;/sub&gt; on the Water-Water Distance

使用高精度从头算方法（含基组重叠误差校正）计算了水团簇（H2O）n（n=8,10,16,20,22,24）中的所有二体、三体和四体作用能,分析了水团簇中的多体效应.研究表明,二体作用对体系总作用能的贡献高达70%以上,三体作用对总作用能的贡献可高达25%,四体作用在总作用能中所占比例不超过3%,五体及以上多体作用能在总作用能中所占比例更小,不超过0.5%.本文研究还表明,两个水分子间距小于0.68 nm的二体作用、三个和四个水分子中最近的两个水分子间距小于0.31 nm的三体和四体作用对体系总作用能的贡献高达99.4%.因此,以生物体系为对象的分子模拟方法应该具备准确地模拟两个水分子间距小于0.68 nm的二体作用、三个和四个水分子中最近的两个分子间距小于0.31 nm的三体和四体作用的能力。

Intermolecular interactions between molecules in various conformational states: the dimer of oxalic acid.

We considered stability of the dimer of oxalic acid. The global minimum energy structure identified by us is stabilized by two inter- and four intramolecular hydrogen bonds, whereas the most stable structure identified in previous studies is supported by two inter- and three intramolecular hydrogen bonds. The latter structure proves to be less stable by 25 meV than the former. The global minimum stability results from a balancing act between a moderately attractive two-body interaction energy and small repulsive one-body terms. We have analyzed zero-point vibrational corrections to the stability of various conformers of oxalic acid and their dimers. We have found that minimum energy structures with the most stabilizing sets of hydrogen bonds have the largest zero-point vibrational energy, contrary to a naive anticipation based on red shifts of OH stretching modes involved in hydrogen bonds.

Fragment quantum mechanical calculation of proteins and its applications.

Conspectus The desire to study molecular systems that are much larger than what the current state-of-the-art ab initio or density functional theory methods could handle has naturally led to the development of novel approximate methods, including semiempirical approaches, reduced-scaling methods, and fragmentation methods. The major computational limitation of ab initio methods is the scaling problem, because the cost of ab initio calculation scales nth power or worse with system size. In the past decade, the fragmentation approach based on chemical locality has opened a new door for developing linear-scaling quantum mechanical (QM) methods for large systems and for applications to large molecular systems such as biomolecules. The fragmentation approach is highly attractive from a computational standpoint. First, the ab initio calculation of individual fragments can be conducted almost independently, which makes it suitable for massively parallel computations. Second, the electron properties, such as density and energy, are typically combined in a linear fashion to reproduce those for the entire molecular system, which makes the overall computation scale linearly with the size of the system. In this Account, two fragmentation methods and their applications to macromolecules are described. They are the electrostatically embedded generalized molecular fractionation with conjugate caps (EE-GMFCC) method and the automated fragmentation quantum mechanics/molecular mechanics (AF-QM/MM) approach. The EE-GMFCC method is developed from the MFCC approach, which was initially used to obtain accurate protein-ligand QM interaction energies. The main idea of the MFCC approach is that a pair of conjugate caps (concaps) is inserted at the location where the subsystem is divided by cutting the chemical bond. In addition, the pair of concaps is fused to form molecular species such that the overcounted effect from added concaps can be properly removed. By introducing the electrostatic embedding field in each fragment calculation and two-body interaction energy correction on top of the MFCC approach, the EE-GMFCC method is capable of accurately reproducing the QM molecular properties (such as the dipole moment, electron density, and electrostatic potential), the total energy, and the electrostatic solvation energy from full system calculations for proteins. On the other hand, the AF-QM/MM method was used for the efficient QM calculation of protein nuclear magnetic resonance (NMR) parameters, including the chemical shift, chemical shift anisotropy tensor, and spin-spin coupling constant. In the AF-QM/MM approach, each amino acid and all the residues in its vicinity are automatically assigned as the QM region through a distance cutoff for each residue-centric QM/MM calculation. Local chemical properties of the central residue can be obtained from individual QM/MM calculations. The AF-QM/MM approach precisely reproduces the NMR chemical shifts of proteins in the gas phase from full system QM calculations. Furthermore, via the incorporation of implicit and explicit solvent models, the protein NMR chemical shifts calculated by the AF-QM/MM method are in excellent agreement with experimental values. The applications of the AF-QM/MM method may also be extended to more general biological systems such as DNA/RNA and protein-ligand complexes.

Electrostatically Embedded Generalized Molecular Fractionation with Conjugate Caps Method for Full Quantum Mechanical Calculation of Protein Energy

An electrostatically embedded generalized molecular fractionation with conjugate caps (EE-GMFCC) method is developed for efficient linear-scaling quantum mechanical (QM) calculation of protein energy. This approach is based on our previously proposed GMFCC/MM method (He; et al. J. Chem. Phys. 2006, 124, 184703), In this EE-GMFCC scheme, the total energy of protein is calculated by taking a linear combination of the QM energy of the neighboring residues and the two-body QM interaction energy between non-neighboring residues that are spatially in close contact. All the fragment calculations are embedded in a field of point charges representing the remaining protein environment, which is the major improvement over our previous GMFCC/MM approach. Numerical studies are carried out to calculate the total energies of 18 real three-dimensional proteins of up to 1142 atoms using the EE-GMFCC approach at the HF/6-31G* level. The overall mean unsigned error of EE-GMFCC for the 18 proteins is 2.39 kcal/mol with reference to the full system HF/6-31G* energies. The EE-GMFCC approach is also applied for proteins at the levels of the density functional theory (DFT) and second-order many-body perturbation theory (MP2), also showing only a few kcal/mol deviation from the corresponding full system result. The EE-GMFCC method is linear-scaling with a low prefactor, trivially parallel, and can be readily applied to routinely perform structural optimization of proteins and molecular dynamics simulation with high level ab initio electronic structure theories.

Coupled-cluster, Möller Plesset (MP2), density fitted local MP2, and density functional theory examination of the energetic and structural features of hydrophobic solvation: Water and pentane

The interaction potentials between immiscible polar and non-polar solvents are a major driving force behind the formation of liquid:liquid interfaces. In this work, the interaction energy of water-pentane dimer has been determined using coupled-cluster theory with single double (triple) excitations [CCSD(T)], 2nd order Möller Plesset perturbation theory (MP2), density fitted local MP2 (DF-LMP2), as well as density functional theory using a wide variety of density functionals and several different basis sets. The M05-2X exchange correlation functionals exhibit excellent agreement with CCSD(T) and DF-LMP2 after taking into account basis set superposition error. The gas phase water-pentane interaction energy is found to be quite sensitive to the specific pentane isomer (2,2-dimethylpropane vs. n-pentane) and relative orientation of the monomeric constituents. Subsequent solution phase cluster calculations of 2,2-dimethylpropane and n-pentane solvated by water indicate a positive free energy of solvation that is in good agreement with available experimental data. Structural parameters are quite sensitive to the density functional employed and reflect differences in the two-body interaction energy calculated by each method. In contrast, cluster calculations of pentane solvation of H(2)O solute are found to be inadequate for describing the organic solvent, likely due to limitations associated with the functionals employed (B3LYP, BHandH, and M05-2X).

Examination of the hydrogen-bonding networks in small water clusters (n = 2–5, 13, 17) using absolutely localized molecular orbital energy decomposition analysis

Using the ωB97X-D and B3LYP density functionals, the absolutely localized molecular orbital energy decomposition method (ALMO-EDA) is applied to the water dimer through pentamer, 13-mer and 17-mer clusters. Two-body, three-body, and total interaction energies are decomposed into their component energy terms: frozen density interaction energy, polarization energy, and charge transfer energy. Charge transfer, polarization, and frozen orbital interaction energies are all found to be significant contributors to the two-body and total interaction energies; the three-body interaction energies are dominated by polarization. Each component energy term for the two-body interactions is highly dependent on the associated hydrogen bond distance. The favorability of the three-body terms associated with the 13- and 17-mer structures depends on the hydrogen-donor or hydrogen-acceptor roles played by each of the three component waters. Only small errors arise from neglect of three-body interactions without two adjacent water molecules, or beyond three-body interactions. Interesting linear correlations are identified between the contributions of charge-transfer and polarization terms to the two and three-body interactions, which permits elimination of explicit calculation of charge transfer to a good approximation.

Effects of Cu Addition on Behavior of Nanoclusters during Multi-Step Aging in Al-Mg-Si Alloys

Two types of nanoclusters, i.e. Cluster (1) and Cluster (2), play a strongly important role in the bake-hardening (BH) response in Al-Mg-Si alloys. Different formation behaviors of two types of nanoclusters were studied by means of hardness, differential scanning calorimetry (DSC), electrical resistivity measurement, transmission electron microscopy (TEM) and high resolution TEM observation in the both Cu-free and Cu-added Al-Mg-Si alloys. As the results, Cluster (1) formed during natural aging at room temperature causes a deleterious effect, whereas Cluster (2) formed by the pre-aging at 100°C is effective for the suppression of the negative effect on the two-step aging behavior of the both Cu-free and Cu-added alloys. On the other hand, the microalloying of Cu strongly affects the nanocluster formation due to the strong interaction with Mg, Si atoms and vacancies. The first-principle calculation for the two-body interaction energies provides quite useful information to understand the early stage of the phase decomposition. The effects of nanoclusters and the Cu addition on the age-hardening behaviors of Al-Mg-Si alloys are discussed based on the multi-step age-hardening phenomena.