Tag: science

Blog Entry (c) Friday, April 11, 2025, by James Pate Williams, Jr. Multiplication and Division of Finite Power Series

x = 0.25
N = 5
Series Cosine(x)  = 0.968912421711
C++ cos(x)        = 0.968912421711
Series sine(x)    = 0.247403959255
C++ sin(x)        = 0.247403959255
Series Tangent(x) = 0.255341921221
C++ tan(x)        = 0.255341921221
Series sin(2x)    = 0.479425538604
C++ sin(2x)       = 0.479425538604
C++ 2sin(x)cos(x) = 0.479425538604
End app ? y = yes = n
x = 0.5
N = 5
Series Cosine(x)  = 0.877582561890
C++ cos(x)        = 0.877582561890
Series sine(x)    = 0.479425538604
C++ sin(x)        = 0.479425538604
Series Tangent(x) = 0.546302489844
C++ tan(x)        = 0.546302489844
Series sin(2x)    = 0.841470984807
C++ sin(2x)       = 0.841470984808
C++ 2sin(x)cos(x) = 0.841470984808
End app ? y = yes = y
x = 0.75
N = 5
Series Cosine(x)  = 0.731688868808
C++ cos(x)        = 0.731688868874
Series sine(x)    = 0.681638760020
C++ sin(x)        = 0.681638760023
Series Tangent(x) = 0.931596460023
C++ tan(x)        = 0.931596459944
Series sin(2x)    = 0.997494986509
C++ sin(2x)       = 0.997494986604
C++ 2sin(x)cos(x) = 0.997494986604
End app ? y = yes = n
 
x = 1.00
N = 5
Series Cosine(x)  = 0.540302303792
C++ cos(x)        = 0.540302305868
Series sine(x)    = 0.841470984648
C++ sin(x)        = 0.841470984808
Series Tangent(x) = 1.557407730344
C++ tan(x)        = 1.557407724655
Series sin(2x)    = 0.909297423159
C++ sin(2x)       = 0.909297426826
C++ 2sin(x)cos(x) = 0.909297426826
End app ? y = yes = n
x = 1.25
N = 5
Series Cosine(x)  = 0.315322332275
C++ cos(x)        = 0.315322362395
Series sine(x)    = 0.948984616456
C++ sin(x)        = 0.948984619356
Series Tangent(x) = 3.009569952151
C++ tan(x)        = 3.009569673863
Series sin(2x)    = 0.598472085108
C++ sin(2x)       = 0.598472144104
C++ 2sin(x)cos(x) = 0.598472144104
End app ? y = yes = n
x = 1.50
N = 5
Series Cosine(x)  = 0.070736934117
C++ cos(x)        = 0.070737201668
Series sine(x)    = 0.997494955682
C++ sin(x)        = 0.997494986604
Series Tangent(x) = 14.101472846329
C++ tan(x)        = 14.101419947172
Series sin(2x)    = 0.141119469924
C++ sin(2x)       = 0.141120008060
C++ 2sin(x)cos(x) = 0.141120008060
End app ? y = yes = y


// DivMulPowerSeries.cpp (c) Tuesday, April 8, 2025
// by James Pate Williams, Jr.
// https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/10%3A_Power_Series/10.02%3A_Properties_of_Power_Series
// https://en.wikipedia.org/wiki/Formal_power_series

#include <iostream>
#include <iomanip>
#include <string>
#include <vector>

std::vector<double> Multiplication(
	std::vector<double> c,
	std::vector<double> d,
	int N) {
	std::vector<double> e(N + 1);

	for (int n = 0; n <= N; n++) {
		double sum = 0.0;

		for (int k = 0; k <= n; k++) {
			sum += c[k] * d[n - k];
		}

		e[n] = sum;
	}

	return e;
}

std::vector<double> Division(
	std::vector<double> a,
	std::vector<double> b,
	int N) {
	double a0 = a[0];
	std::vector<double> c(N + 1);

	for (int n = 0; n <= N; n++) {
		double sum = 0.0;

		for (int k = 1; k <= n; k++) {
			sum += a[k] * c[n - k];
		}

		c[n] = (b[n] - sum) / a0;
	}

	return c;
}

double Factorial(int n) {
	double nf = 1.0;

	for (int i = 2; i <= n; i++) {
		nf *= i;
	}

	return nf;
}

std::vector<double> Cosine(double x, int N, double& fx) {
	std::vector<double> series(N + 1);

	fx = 0.0;

	for (int n = 0; n <= N; n++) {
		int argument = 2 * n;
		double coeff = pow(-1, n) / Factorial(argument);
		series[n] = coeff;
		fx += coeff * pow(x, argument);
	}

	return series;
}

std::vector<double> Sine(double x, int N, double& fx) {
	std::vector<double> series(N + 1);
	
	fx = 0.0;

	for (int n = 0; n <= N; n++) {
		int argument = 2 * n + 1;
		double coeff = pow(-1, n) / Factorial(argument);
		series[n] = coeff;
		fx += coeff * pow(x, argument);
	}

	return series;
}

std::vector<double> Tangent(double x, int N, double& fx) {
	double fc = 0.0, fs = 0.0;
	std::vector<double> seriesC = Cosine(x, N, fc);
	std::vector<double> seriesS = Sine(x, N, fs);
	std::vector<double> seriesT = Division(seriesS, seriesC, N);
	fx = fs / fc;

	return seriesT;
}

std::vector<double> Sine2x(double x, int N, double& fx) {
	double fc = 0.0, fs = 0.0;
	std::vector<double> seriesC = Cosine(x, N, fc);
	std::vector<double> seriesS = Sine(x, N, fs);
	std::vector<double> series2 = Multiplication(seriesS, seriesC, N);
	fx = 2.0 * fs * fc;

	return series2;
}

int main()
{
	while (true) {
		char line[128] = { };
		std::cout << "x = ";
		std::cin.getline(line, 127);
		std::string str1(line);
		double x = std::stod(str1);
		std::cout << "N = ";
		std::cin.getline(line, 127);
		std::string str2(line);
		int N = std::stoi(str2);
		double cx = 0.0, sx = 0.0, tx = 0.0, xx = 0.0;
		std::vector<double> cSeries = Cosine(x, N, cx);
		std::vector<double> sSeries = Sine(x, N, sx);
		std::vector<double> tSeries = Tangent(x, N, tx);
		std::vector<double> xSeries = Sine2x(x, N, xx);
		std::cout << std::fixed << std::setprecision(12);
		std::cout << "Series Cosine(x)  = " << cx << std::endl;
		std::cout << "C++ cos(x)        = " << cos(x) << std::endl;
		std::cout << "Series sine(x)    = " << sx << std::endl;
		std::cout << "C++ sin(x)        = " << sin(x) << std::endl;
		std::cout << "Series Tangent(x) = " << tx << std::endl;
		std::cout << "C++ tan(x)        = " << tan(x) << std::endl;
		std::cout << "Series sin(2x)    = " << xx << std::endl;
		std::cout << "C++ sin(2x)       = " << sin(x + x) << std::endl;
		std::cout << "C++ 2sin(x)cos(x) = " << 2.0 * sin(x) * cos(x);
		std::cout << std::endl;
		std::cout << "End app ? y = yes = ";
		std::cin.getline(line, 127);

		if (line[0] == 'Y' || line[0] == 'y') {
			break;
		}
	}

	return 0;
}

Blog Entry (c) Friday, October 18, 2024, by James Pate Williams, Jr. Ab Initio Quantum Chemical Calculation

On Wednesday, October 16, 2024, I bought an Amazon Kindle book named “Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory” by Attila Szabo and Neil S. Ostlund. It cost me $10.69 which is a real bargain. In Appendix B there is a listing for a FORTRAN program to perform an ab initio Hartree-Fock Self Consistent Field calculation for a two-electron heteronuclear molecule namely the helium-hydrogen cation. I successfully translated the program from FORTRAN to C++. I had to remember that FORTRAN stores matrices in column major order and C/C++ stores matrices in row major order. I took the transposes of two FORTRAN COMMON matrices to get the correct C++ storage. The authors of the book did an extensive treatment of the test calculation. The application is only 823 lines of monolithic C++ source code. I used FORTRAN like array indexing starting at 1 instead of the C initial beginning index of 0. I think I will try to get in touch with the authors to get permission to post the source code and results on my blog. 

P. S. I got permission from Dover Books to publish my source code and results. I think I will reconsider posting the C++ source code. The actual ground state energy of the cation is -2.97867. My calculation and the book’s computation are in percentage errors of about 4%. The book’s value is a little closer to the exact value than my result. The book calculation was done in FORTRAN double precision on a Digital Equipment Corporation PDP-10 minicomputer. My recreation of the book’s endeavor was executed on an Intel Itanium Core 7 and Windows 10 Professional machine using Win32 C++. The C++ compiler was from Microsoft Visual Studio 2019 Community Version.

Note I added a calculation for a homonuclear molecule, namely, the hydrogen diatomic molecule.

Ab Initio Quantum Chemical ResultsDownload